.

Advances in Applied Mechanics

Vol~ime 14

Editorial Board

T. BROOKE BENJAMIN Y. C. FIJNG

PAUL GERMAIN

L. HOWARTH

WILLIAM PKAGER

T. Y. WU

HANS ZIFGIIR

Contributors to Volume 14

BERNARD BUDIANSKY S C. COWIN

JOHN W. HUTCHINSON D A N I ~ L D. JCWPII

K STFWARTSON

A D V A N C E S I N

APPLIED MECHANICS Edited by Chia-Shun Yih

I ) F P A K I M E % T 01 APPLIED MEC'HAYICS AN11 FNGINTFRl%h;(; SCIFNCF

TtIF UNlVPRSlTY O F MICHIGAN

A N N ARBOR, MICHIGAN

V O L U M E 14

1974

AC A DE M I C PR E SS New York San Francisco London

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Contents

Theory of Buckling and Post-Buckling Behavior of Elastic Structures

Bernard Budian.skj~

1. Introduction 11. Simple Models

111. Functional Notation, Variational Calculus, and Frechet Dcrivativcs IV. Energy Approach

VI. Mode Interaction V. Stress, Strain, Displacement: Virtual-Work Approach

References

2 2

I I 16 41 sx 63

1. Introduction 67 11. Simple Models 70

V. Numerical Examples 132

111. Bifurcation Criterion 86 IV. Initial Post-Bifurcation Behavior for Donnell-Mushtari-Vlasov Theory 105

References 141

Multistructured Boundary Layers on Flat Plates and Related Bodies

K . Stewurtson

1. Introduction 11. The Basic Flow

111. The Triple Deck

I46 156 IS8

V

LIST OF CONTRIBUTORS

PRFFACE

Plastic Buckling

John W Hutchinson

Contents

vii ix

IV. The t'undamcntal Equation of thc Triplc Dcck

VI. Free Interactions in Supersonic Flow V. Transonic Free Interaction

VII. Expansive Free Interactions VIII. Compressive Free Interactions

IX. The Plateau

XI. Convex Corners in Supersonic Flow X. Comparison with Experinicnt

XI1. Injection into the Supersonic Boundary Laycr XII I . The Trailing Edge of a Symmetrically Disposed Flat Platc XIV. Trailing-Edge Flows for Bodies with Finite Thickness

X V I . Catastrophic Separation XV. Viscous Correction to Lift

XVII. Addendum Referenccs

Response Curves for Plane Poiseuille Flow

Daiiiel D. Joseph

1. Introduction 11. The Solution of the Basic Equations for Laminar Poiseuillc Flow

111. Global Stability of Laminar Poiseuille Flow I V . The Fluctuation Motion and the Mcan Motion

Vl. Laniiuar and Turbulent Comparison Theorems V. Steady Causes and Stationary Effects

VII. Turbulcnt Plane Poiseuille Flow- An Upper Bound for the Response Cui-vc VIII. The Response Function near the Point of Bifurcation

IX. Some Properties of the Bifurcating Solution X. Inferences and Coii.jectures

Appcndix: A Formal Bifurcation Theory for Nearly Pal-allel Flows Refci-ences

The Theory of Polar Fluids

S. C. Cowin

1. Introduction 11. Foundations of the Theory

111. Related Theories References Note Added in Proof

AUTHOR INDEX S U B J ~ C T INDFX

167 i69 171 I 74 176 184 188 194 200 213 222 225 229 23 I 234

24 1 245 246 249 250 25 I 253 256 26 1 263 268 276

279 290 335 344 347

349 354

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

BERNARD BUDIANSKY, Division of Engineering and Applied Physics,

S. C. COWIN, Department of Mechanical Engineering, Tulane University,

JOHN W. HUTCHINSON, Division of Engineering and Applied Physics,

DANIEL D. JOSEPH, Department of Aerospace Engineering and Mechanics,

K. STEWARTSON, Department of Mathematics, University College London,

Harvard University, Cambridge, Massachusetts ( 1 )

New Orleans, Louisiana (279)

Harvard University, Cambridge, Massachusetts (67)

University of Minnesota, Minneapolis, Minnesota (241)

London, England (145)

v i i

This Page Intentionally Left Blank

The first two articles, by Professors B. Budiansky and J. W. Hutchinson, treat elastic and plastic buckling of structures and are companion papers. The sense of unity given by two closely related and mutually supplementary articles presented in sequence is always desirable, but only rarely achieved: the appropriate authors have to be persuaded to write, and they must finish their articles in nearly the same time. In the present case we have achieved this by sheer determination and with good luck.

In the area of fluid mechanics the three articles by Professors K. Stewartson, D. D. Joseph, and S. C. Cowin give a different sense of unity-a unity by contrast, as it were. The Navier-Stokes equations for Newtonian fluids are solved by Professor Stewartson by matched asymp- topic expansions; boundary layers within boundary layers are found by him, and if the mathematics seems complicated, one must remember that the problem of laminar separation is still unsolved, and efforts such as those of Professor Stewartson are the main ones toward its solution. Professor Joseph, treating hydrodynamic stability and stable turbulence, also deals with the Navier-Stokes equations. But the approach is entirely different. For instance, he uses the energy method and the calculus of variations to determine the lower bounds of the Reynolds number at neutral stability. While the lower bounds can, with additional labor, be improved, they have the great merit of having been obtained by exact considerations which take, for instance, nonlinearity into consideration. The energy method provides much-needed information on hydrodynamic stability not obtainable from linear theories, and yet it does not rely on the solution of the Navier-Stokes eqliations per se. As to Professor Cowin’s article, it serves to show how far one can go in formulating the dynamics of polar fluids and in solving the resulting equations of motion. It also points out the need for experimental work to determine the additional boundary conditions needed for the differential equations of motion, which are of a higher order than the Navier-Stokes equations. Perhaps it is time for specifiers of

ix

X Preface

constitutive equations to consolidate their positions by some experimental work designed to discover the true boundary conditions before they continue their studies with ever increasing intensity.

CHIA-SHUN YIH

Theory of Buckling and Post-Buckling Behavior of Elastic Structures

B E R N A R D B U D I A N S K Y

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I 1 . Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2

A . Bifurcation and Post-Buckling Behavior . . . . . . . . . . . . . . 3 B . Initial Imperfections; Snap Buckling . . . . . . . . . . . . . . . 4 C . Imperfection Sensitivity . . . . . . . . . . . . . . . . . . . . . 7 D . Load-Shortening Relation . . . . . . . . . . . . . . . . . . . . 8 E . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

I11 . Functional Notation, Variational Calculus. and Frechet Derivatives . . . 1 1 A . State Variables . Functions. and Functionals . . . . . . . . . . . . 1 1 B . Norms. Linear and Multilinear Operators. Inner Products . . . . . . 12 C . Variations; Gateaux and Frechet Derivatives . . . . . . . . . . . . 13 D . Calculation of Frechet Derivatives . . . . . . . . . . . . . . . . 15

IV . Energy Approach . . . . . . . . . . . . . . . . . . . . . . . . . 16 A . Principle of Stationary Potential Energy . . . . . . . . . . . . . . 16

C . Post-Buckling Analysis . . . . . . . . . . . . . . . . . . . . . 19 D . Initial Imperfections . . . . . . . . . . . . . . . . . . . . . . 22 E . Load-" Shortening" Relation . . . . . . . . . . . . . . . . . . . 26 F . Other Stationary Functionals . . . . . . . . . . . . . . . . . . 30

G . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 V . Stress. Strain. Displacenient; Virtual-Work Approach . . . . . . . . . 41

A . Stress- Strain Relation; Virtual-Work Equation . . . . . . . . . . . 41 B . Bifurcation and Post-Buckling Analysis . . . . . . . . . . . . . . 43

D . Shallow Shells; Donnell-Mushtari Vlasov Shells . . . . . . . . . . 52 E . Initial Imperfections . . . . . . . . . . . . . . . . . . . . . . 55

V I . Mode Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 58 A . Simultaneous Buckling Modes; Post-Buckling Analysis . . . . . . . 58 B . Initial Imperfections . . . . . . . . . . . . . . . . . . . . . . 60 C . Nearly Simultaneous Modes . . . . . . . . . . . . . . . . . . . 61 Refcrcnces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B . Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . 17

C . Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . 46

1

2 Ber-nurd Budiansky

I. Introduction

The general theory of buckling and post-buckling behavior of elastic structures enunciated by Koiter (1945) has spawned a considerable amount of research in this field, especially during the last ten years. A recent comprehensive survey by Hutchinson and Koiter (1970) provides a very useful bibliography, together with an overview of the achievements, status, and goals of post-buckling theory. Their survey obviates the necessity of providing a similarly comprehensive guide to the literature in the present paper. Rather, the primary aim of this article is to provide a unified, general presentation of the basic theory in a form suitable for application to a wide variety of special problems. This will be done with the help of the succinct notation of functional analysis, which turns out to be remarkably appropriate for the purpose.

In addition to Koiter’s original work, very many papers on the general theory have emerged from the British school of post-buckling theorists (e.g., Sewell, 1968; Thompson, 1969), almost exclusively in the language of finite- dimensional systems. In the United States, variations of the Koiter approach have usually been based on continuum concepts, with a bias toward virtual work (Budiansky and Hutchinson, 1964; Budiansky, 1965, 1969; Budiansky and Amazigo, 1968; Fitch, 1968; Cohen, I968; Masur, 1973) rather than energy formulations (Seide, 1972). The derivations and results of the present exposition are equally applicable to continua and finite-dimensional systems; the virtual work and energy approaches are given separate treat- ments, but their equivalence is made explicit. Throughout the present paper, basic concepts of stability are relegated to a secondary role, in contrast to the central position they held in Koiter’s work. This, however, is largely a matter of taste, and will not affect essential conclusions concerning initial post-buckling behavior and imperfection sensitivity.

11. Simple Models

Very simple conceptual models can illustrate with remarkable verisimili- tude many of the essential characteristics of the buckling and post-buckling behavior of more complicated structural systems. Before undertaking a general analysis of arbitrary elastic structures, we will exploit such models in order to expose basic concepts of bifurcation buckling, snap buckling, imperfection-sensitivity, load-shortening relations, and stability.

Buckling Behavior of Elastic Structures 3

A. BIFURCATION A N D POST-BUCKLING BEHAVIOR

Consider the primitive modelt shown in Fig. 1, consisting of a vertical rigid rod of length L, fixed with respect to translation at its base, and elastically constrained against rotation Sy by a spring that supplies a restoring momentf'(<). In the presence of a vertical load 2 applied as shown, the rod is in a state of static equilibrium only if

FIG. 1. Simple model.

With ,f(t) given by

, f ( i ; ) = K l ( + K2Sy2 + K 3 t 3 + ... (Kl > 0) (2.2) thefunda/nentcil solution 4 = 0 of (2. I ) is available for all 2, but in addition the buckled states represented by

A = , f ( ( ) / L sin i; (2.3) exist for nonzero values of <. These states lie on an equilibrium path in the P-< plane (Fig. 2a) that is connected to the fundamental path < = 0 at the critic~il load 1, given by

3., = K,/L. (2.4)

t This is precisely the model used by Masur (1973) in his recent exposition of shell buckling, and by Koiter (1972) in his recent textbook.

In the vicinity of < = 0 (indeed, for I iJ I < TC) these buckled states lie on the path

1” = 2, + % ] { + A*{* + ’. . , (2.5)

A1/2, = ( K J K , ) , &/A, = ( K J K , + i), . . . . (2.6)

where

Figure 2a illustrates a case for which K , < 0.

x

( 0 ) ( b ) ( C )

FIG. 2. Equilibrium paths. (a) i, < 0; (b) i, = 0, i, > 0; (c) i, = 0, i2 < 0

The imagined transition at 2 = AC of the state of the structure from the fundamental equilibrium path to the buckled path (in either direction) is called hifurcution buckling, because the fundamental path can naively be viewed as having split into several branches. For ,il # 0, the bifurcation is usj~mrnetric; synimetric bifurcations, illustrated in Fig. 2b,c, occur for 2 , = 0, A, # 0. The phrase po.st-hiickli/ig hehaoior (or, more precisely, iiiitiul post- buckling behavior) refers to the character of the buckled equilibrium paths in the vicinity of the critical load i, .

The discovery of the equilibrium states in Fig. 2 does not, by itself, answer the question of what is the actual behavior to be expected of the structure modeled in Fig. I when it is subjected to a load that is increased slowly from zero. At least part of the answer is provided by the introduction of initial imperfections.

B. I N I T I A L IMPERFECTIONS; SNAP BUCKLING

Assume that in its unloaded state the rod must inevitably suffer some from a perfectly vertical position (Fig. 3); then, if { is the initial deviation

additional rotation produced by the load, equilibrium states must satisfy

2 =.f’(iJ)/L sin({ + z). (2.7)

Buckling Bekavior of’ Elastic Structures 5

FIG. 3. Initially imperfcct simple model.

These are shown schematically (for the case 2 , < 0) in Fig. 4a,b for positive and negative values of 5. The intersecting equilibrium paths of the perfect structures, shown dotted, have been deformed into disjoint branches by the presence o f t # 0, and with respect to loading that increases from A = 0 the branches that do not pass through the origin have evidently become irrelevant. For various values of z, the significant branches that do contain the origin make up the two families shown in Fig. 5a,b; as I 1 approaches zero, the members of each family approach different pairs of branches of the equilibrium paths associated with the perfect structure. For neither sign of 5 do paths from the origin ever approach the fundamental path above i = AC .

( 0 ) ( b )

Equilibrium paths for imperkct modcl. (a) 2 > 0 ; (b) < < 0. FIG. 4.

6 Bernard Budiansky

For a sufficiently small positive value of z, the associated equilibrium path (Fig. 5a) displays a local load maximum 4 < 1' at a rotation & that depends on 4. Obviously, if the actual loading is increased beyond I - , something nasty will happen; for 2 > I % , static equilibrium is impossible at values of 5 in the vicinity of (, and so a dynamic process involving initial loadings must ensue. If velocity-dependent damping is invoked in a conceptual attempt to bring the structure eventually to rest, the final static configuration (if it exists) cannot be arbitrarily close to & , no matter how slightly 4 has been exceeded. Accordingly, /2, is designated as the buckling load of the imperfect structure. This kind of buckling, associated with a local maximum in the static equilibrium path and the expected jump to a distant configuration when the maximum is exceeded, has variously been called stiiip buckling, snapping, limit-point buckliizg, and oil cannit?y.

E

yrc f

( 0 ) ( b )

FIG. 5. (b) 2 < 0.

Equilibrium paths for various values of initial imperfection r ( A l < 0). (a) > 0;

The curves of Fig. 5b, for < 0, do not suggest such a dramatic behavior. They do indicate that for loads in the vicinity of 2, significant increases occur in the rate of growth of rotation with load, but distinguished loads different from 1, that signal this increase are not usefully defined for various values of small j'. Thus, for < 0, A, still deserves to be called the buckling load of the structure, but this buckling is mild in comparison with the sharp, possibly catastrophic, snap buckling that occurs at 4 for z > 0.

These observations, for 2,/iC = K , / K , < 0, are readily extended to other combinations of parameters in f(0. If A, > 0, small negative values of 4 provoke snapping (Fig. 6a); in the case of a symmetric bifurcation, with %, = 0, snap buckling is induced by initial imperfections of either sign if 1, < 0 (Fig. 6b); 2, > 0 produces only mild buckling (Fig. 6c). Clearly, i t is


FIG. 6. Equilibrium paths for perfect and imperfect models. (a) i,, > 0; (b) 2 , = 0, i2 < 0: (c) i, = 0, i., > 0.

the initial post-buckling behavior of the idealized perfect structure-in particular, whether the load increases or decreases after bifurcation-that determines the kind of buckling to be expected in the imperfect one.

C. IMPERFECTION SENSITIVITY

I f snap buckling does occur, the magnitude of 4 can be significantly below A,. Maximizing /z with respect to 4 in (2.7) requires that

a L cos(4 + Z) = f '(t)

k/i, ZZ 1 - 2[-(A1/ic)Z]1'2

&/A, zz 1 - 3(-,1,/ ,1,) '~3(~~)2~3

(2.8)

(2.9)

(2.10)

and this, together with (2.7) provides the asymptotic result

for small (Z) and ( A l z) < 0. Similarly, for A , = 0, A, < 0,

for small z. In both cases, small imperfections can make the snap-buckling load 4 substantially lower than the critical load i, of the ideally perfect structure. Structures in which this can occur are said to be impeyjection sensitive. In the models studied, imperfection sensitivity is implied by A1 # 0, or by A1 = 0, A, < 0, and the seriousness of this sensitivity is then determined by the magnitudes of A1/Ac or i 2 / A c . [If A1 # 0 a d A, < 0, the approximation (2.9) may become inadequate for moderate values of 4; 4 must then be found from a more accurate solution for the maximum value of A implied by (2.7), which will be lower than the separate values given by (2.9) and (2.10).]

8 Bernard Burliansky

D. LOADSHORTENING RELATION

Instead of prescribing the load 2 on the imperfect structure, we could impose a vertical displacement, or “shortening,” A at the point of load application (Fig. 3), given by

A/L = COS(< + z) - cos < (2.11)

in terms of iJ and t. In the limiting case AIL % i t2 , and so, from (2.5)

= 0 of the perfect model,

A//$ z 1 + (i,/Ac)(2A/L)1’2 (( > 0) (2.12a)

z 1 - (Al/%c)(2A/L)1’2 (t < 0) (2.12b)

z 1 + 2(A2/&)(A/L) (21 = 0,22 # 0) (2.13)

for small A. The solid curves in Fig. 7 show typical load-shortening relations, together with associated A-( relations. Nole that asymmetric bifurcations

( C ) ( d )

FIG. 7 . Load-shortcniiig rclatioiis for simple model. (a) i., < 0. < 2 0 ; (b) i , < 0, < < 0 : ( c ) i , = 0, i z < 0, 2 2 0 ; (d) i , = 0. i z > 0. < 2 0.

Biickliny Behtrvior of Elastic Structures 9

give a vertical initial slope to the &A relation, whereas di/dA is finite for symmetrical bifurcations.

In the case of the imperfect structure, Eqs. (2.7) and (2.1 l) , for a given 2, supply the &A relation parametrically in terms of (, with typical results sketched as the dotted curves in Fig. 7. In those cases (Fig. 7a,c) that involve snap buckling when an increasing load 2 is imposed, the results indicate that prescription of a monotonically increasing shortening A would not provoke a violent response, but would simply induce a smooth peak in the magnitude of the associated load.

However, a slight modification of the model can render it susceptible to snapping under an imposed shortening. Replace the rigid bottom support by an elastic spring that supplies a vertical restoring force ce, where c is the vertical displacement of the bottom of the rod (Fig. Xa). Then A will, at any

x

1

A A

( C)

FIG. X. Load-shortening relations of (a) modified modcl. (b) i , < 0; ( c ) I., = 0, < 0.

load i, exceed that found for the original model by the amount l / c . For il < 0, and a sufficiently small 5 > 0, this leads to the possibility of a path in the &A plane (Fig. 8b) that displays a local maximum A,,, in shortening. Under increasing A, snapping must occur as soon as A exceeds A,, for then no neighboring static equilibrium state is to be found. The same thing can happen for 2 , # 0, %, < 0 (Fig. Xc) for small enough if, for 5 = 0, the initial post-buckling value of d%/dA = (l/c) + (2/2,/LlC) is positive-that is, if the initial post-buckling behavior of the perfect model shows a decrease in A as well as in 1.. In all cases, when snapping occurs under increasing A, the load 2 will already have decreased from its peak value I\.

10 Bernard Biidiansky

E. STABILITY

The analysis of bifurcation buckling, together with consideration of initial imperfections, has apparently provided an understanding of the behavior of the simple model, but from a practical viewpoint the study is really not complete without some attention to stability in the dynamic sense. For example, the presumption that the static equilibrium paths of Fig. 5a will be followed up to 2 = 1% involves the tacit assumption that if the prescribed loading /z increases slowly enough-or is applied in small enough increments-then time-dependent deviations A ( ( [ ) from the static equilibrium path induced by the loading will be damped out whenever further loading ceases and /z is held fixed. Also, with the load fixed, small enough accidental disturbances A< in orientation (or A[ in angular velocity) produced by an external agency should similarly be damped out if the static equilibrium path is to have any useful physical validity. These requirements are just those of stability (more precisely, asymptotic stability) in the Lya- punov sense. The same question of stability must be raised with respect to the curves for < 0 in Fig. 5b, in which 2 eventually exceeds the critical load & of the perfect model, and yet we presumed no practical difficulty in following the equilibrium paths by means of slow loading.

Fortunately, the stability of finite-dimensional elastic systems subjected to conservative loads (i.e., loads associated with an energy potential) can be assessed on the basis of the classical energy criterion. In static equilibrium, with the load fixed, the total potential energy 4 of the system must be stutionary with respect to its configurational arguments; but the equilibrium is stable if, and only if, this stationary point is actually a local nzinirnum. The argument for sufficiency is simply that a proper minimum in the potential energy provides an energy barrier between the initial configuration of static equilibrium and any alternative ones; by keeping initial disturbances small enough the energy increment needed to surmount the barrier can be denied to the system, and damping may then be relied upon to return the structure-asymptoticallyto its original configuration. Instahility is implied by the absence of a minimum because a configurational disturbance of arbitrarily small size, and sufficiently small kinetic energy, can always be found that would place the system out of static equilibrium into a state of lower total energy (potential plus kinetic); then damping can only decrease the total energy of the ensuing states still more. Even if the damping should ultimately lead the system toward a state of static equilibrium with no kinetic energy, this state cannot possibly coincide with the original state of higher potential energy. This is a rough, abbreviated version of Koiter’s (1965a) proof.?

The writer is informed by Prof. Koiter that this proof is actually due to Jouquet (1930).


In the case of our simple model under prescribed load 3,, the potential energy is given by

5 4 = I‘ , j-((‘) d5‘ + iL[cos(< + t) - cos t]

’ 0

and C?gi/d( =, f ( ( ) - AL sin(< + z) vanishes, by static equilibrium [Eq. (2.7)]. Stability is implied by a positive value for

P ( b / d < Z =, j - ’ (<) - AL cos(;. + Z), (2.14)

but along the equilibrium paths of Fig. 5a, i?24/i?<2 is positive at < = 0, and does not vanish until I+ reaches Is [see Eq. (2.8)]. This verifies stability for 0 I 3. < J, on these paths, and also on the paths for < 0 in Fig. 5b, along which (2.14) vanishes for no value of 2. Similar conclusions hold for the equilibrium paths for # 0 in Fig. 6; stability is the rule whenever d i / I d( I is positive. Furthermore, unstable equilibrium states correspond to those portions of the paths for which ~ l i / I d< I < 0, but this is not of much practical interest since these states could not be reached by means of a loading process in which A increases from zero. (The structure could, of course, be artificially set into one of these unstable configurations, and then the sligh- test disturbance would snap i t away.)

The stability characteristics of the perfect structure have some theoretical, if not practical, interest. It is easily verified that the fundamental path < = 0 is stable for < iC and unstable for /z > A c , and that on the bifurcated paths the rules relating the sign of dA/ I d( I to stability found for the imperfect case still apply. At Ib = Ac , a24/@ vanishes, and a decision on stability rests on the signs of higher derivatives; stability can thereby be verified at A = ;1, in case (c) of Fig. 6 and instability in cases (a) and (b). It is no accident that the boundary between stability and instability on the fundamental path occurs at the bifurcation load. Given any analytic function 4((, A), the vanishing of 84/dl on two intersecting paths of the I*-< plane implies that 8’4/?<’ (and also d2gi/d< dA) must vanish at the point of intersection.

111. Functional Notation, Variational Calculus, and Frechet Derivatives

A. STATE VARIABLES, FUNCTIONS, AND FUNCTIONALS

In conventional continuum mechanics, the state of an elastic structure may be described by the spatial distributions of stresses, strains, and displacements. In the approximate analysis of particular structures on the basis of special theories, generalized stress, strain, and displacement variables should always be identifiable. For example, in simple beam theory, the

12 Bern urd B ~ i d i ~ i n s k y

“stresses” are the bending moments, the generalized “strains” are the curvatures, and the lateral deflection along the beam is the pertinent displace- rnent variable. Similarly, in a pin-jointed truss the member loads and elongations may appropriately be identified as the stress- and strainlike quantities, and the values of displacement at the joints suffice to constitute the set of appropriate displacement-state variables. In the context of any such particular theory it will be useful to denote a distribution of generalized stress by the single symbol G, and to imagine that CT represents a “point” in a space that contains all stress distributions that are mathematically (if not physically) possible. Similarly, we will use I: and u to denote points in strain and displacement spaces.

These state variables will be related to each other in various ways by the field equations of the theory being used. Thus, the elastic constitutive equations permit the calculation of G in terms of c ; the procedure for so doing may be regarded as a transformation from strain space to stress space. We can then say that G is afuriction of F , and i t will be very convenient to use the single symbol CT to denote the name of this function of E , as well as the value of stress the function produces. Thus we will write CT = ~ [ c ] , and not insist on the more respectable form G = f [ c ] which carefully distinguishes the variable CT from the function.6 (The reader is warned, however, that our more casual notation is regarded with considerable distaste in certain circles.)

In structural mechanics, global scalar quantities arise, such as the potential energy 6, which can be expressed as a function of the generalized displacement u. Such a scalar-valued function is often called a,functionul and the value of 4 for a particular choice of u may be expressed as 4 = $[u] ; this represents a transformation from displacement space to the one-dimensional space of real numbers.

B. NORMS, LINEAR AND MULTILINEAR OPERATORS, INNER PRODUCTS

The generalization of length in the space of displacement is provided by the iiorni of u, written as lIul/, which is a measure of the “size” of u. A norm should have the properties that

ll4 > 0 for LI + 0.

1l4 = 0 for u = 0. and

11’4 = I @ I ll4 for all scalars c(. Also the triangle inequality

(3.la)

(3.lb)

Buckling Behuvior of‘ Ekistic Structures 13

should be satisfied. Subject to these restrictions, the choice of a norm, in a particular theory, is often a matter of convenience; some choices make it easier to prove things than others, but in this paper we will not deal seriously with matters that require precise discrimination among various possible norms.

A function (or functiona1)fis linear if f [ 4 = .f[uI (3.2a)

for all scalars CI, and

.f[Lll + u21 =. f [u , l +f”Li21 (3.2b)

for all u , , u 2 . For such a linear function, the notation

, f [ u ] = Au

may be introduced, and then A is called a linear operrrtor. This operator may in turn depend on another variable c, and to emphasize this we could write

, f [ u ; 111 = A[z:]u.

Similarly, a function g that is linear in each of the variables u , and u2 can be expressed in terms of a bilinear operator B by an expression of the form

9 [ 4 , Ll2l = Bu1u2 ‘

If Bu, u 2 = B u 2 u , for all u l , t i 2 , then B is a sjwmwtric bilinear operator. Finally, the expression Cu, u2 . . . u, may be used to denote a multilinear function of ul, u 2 , . . . , u, in terms of a multilinear operator C, and this operator is symmetric if the LI’S may be interchanged in any manner without affecting the result of the operation.

If Bu, u2 is a scalar and B is a symmetrical bilinear operator that satisfies

Bu,u , = Bu: > 0

for u 1 # 0, then Bu, 1i2 may be written in the iririer product form ( i l l , u 2 ) , and this may often be used to provide a convenient norm defined by

1 1 UII = (11, u )

C. VARIATIONS; GATEAUX AND FRECHET DERIVATIVES

In the classical calculus of variations, the variation of a function or functional is defined in terms of a variation 6u of its argument by

f[. + E 6111 - f [ l I ] 4 f [ u ] = lim (3 .3)

14 Brrrzurd Birdiunsky

where c is scalar. A suitable choice of norm in the space of,fis implicit in this definition; that is, a more precise statement specifying 2f is

f [ u + i: 6u] - , f [ c r ] lim csf'- ' 1 1 = 0. I - 0 l l 1; (3.4)

For sufficiently well-behaved functionsf; #'will be a linear function of 6u in the sense of Eqs. (3.2), and may be written as

;sf = f ' 6 1 1

in terms of a linear operator ,f', known as the Guteuiix derivative off: To emphasize the dependence o f f ' on u, we will often write 4 f = , f ' [ z r ] 6 u ; furthermore, it is not essential to retain the variational notation in the definition off ' , but simply write, for any fixed ul,

(3 .5 ) r - 0 1:

An alternative basic definition for the generalized derivative off; due to Frechet, requires that f ' be that linear operator satisfying

(3.6)

This may be somewhat more satisfying mathematically than the Gateaux definition in that there is no insistence that u , be fixed in shape as llul 1 1 + 0. But the two definitions are equivalent whenf' is continuous in if (Vainberg, 1964), and in this paper we will refer to , f ' as the Frechet derivative.

The relation (3.5) may be written conveniently as

f ' U 1 = (d/al:),f[U + F U l ] 11:=0 , (3.7) and this leads to a convenient definition for higher-order generalized derivatives o f f Thus

and, assuming sufficient continuity of the higher-order Frechet derivatives of , f ; this is the same as

, f"lflLfz = (c?Z/C?c, 21:,)f[u + C L I I 1 + c 2 1 r 2 ] I l : I = r z = O (3 .8)

wherein the order of differentiation is immaterial. It is evident, then, that , f " u , u2 = , f "uZ ul, so thatf'" is a symmetrical bilinear operator. Generalizing (3.8) we get the relation

Buckling Behavior of Elastic Stnictures 15

satisfied by the 12th-order Frechet derivative of ,f; which is a symmetrical multilinear operator. Note that identities like

f ” U 1 U 1 = ( i i ’ / ; E : ) , f [ l i + C1 l E l = 0

and

f ” ’ u 1 u 2 u 2 = (?2/&, ac$) . f [u + C ] 1 1 1 + c21i2] I F 1 - 1 . 2 = 0 follow from (3.9), and the convenient abbreviated notation f ” u : = ~ ‘ ’ I I ~ L I ~ ,

f.“’ul u: = f ’ ” u l u2 u2 will be used. This notation permits the succinct Taylor- series representation of an analytic function or functional in the familiar- looking form

f [ U ] = f [ U o ] +f’[UO](U - U 0 ) + ;,f”[Uo](Zl - 2 4 0 ) ~ + ” ’ . (3.10)

D. CALCIJLATION OF FRECHET DERIVATIVES

The formal definitions given for Frechet derivatives of any order are entirely equivalent to the familiar processes of “taking variations” in the calculus of variations. A few examples will illustrate such calculations.

Consider the functional . I

F = I [w3 + 3(c’)2 + X W ] dx.

l o 1-

‘ 0

The variable u is identified as the prrir of functions MI(X) , U ( Y ) . Recall that . I

6 F = 3w2 SW + 621’ f i r ? ’ + .X 6w] dx. Thus, letting 6w = wl, 6v = u l , we have

. I

F’u, = 1 [(3rv2 + x)wl + (6c’)c;l dx. ‘ 0

Next, taking another variation, with 6w = w 2 , 6c = c 2 , leads to 1

F”u1 142 = {, [ ~ w w , w2 + 6 4 ~ $ 1 dx.

Continuing, . 1

F“‘ul u2 ci3 = 6 w , \v2 u ’ ~ r1.x J 0

and then F @ ) = 0 for 11 2 4

16 Bernuril Budiaiisky

As another example, consider the nonlinear strain-displacement relation

c, = (('u/L?x) + $[((%/ix)' + (i~/i.)' + ( i i ~ / i x ) ~ ]

of elasticity theory. We get, with U as the designation of all three displacement components u, 0, w,

?u, C7u (711, (72. ?c, ?w ?HI, " U , = + +

i u i x ?u + ?x i'u ix i x

and

Note, finally, that the conventional higher-order variations h2F, h 3 F , of the calculus of variations are simply

(52F = fF"(61,)2, S 3 F = i F " ' ( d U ) 3 , h(")F = F ( " ) ( 6 u y , I1 !

so that the Taylor expansion (3.10) is the same as

f [ . + (su] =f"u] +f'"u]6u + +f"[U](6.)2 + " .

= ,f' + (sf' + 62f + ' . .. (3.1 1 )

Systematic, rigorous expositions of functional analysis may be found in Vainberg (1964) and Liusternik and Sobolev (1961); a very readable, elementary account has been given by Rall (1969).

IV. Energy Approach

A. PRINCIPLE. 01. STATIONARY POTENTIAL ENERGY

Koiter's (1945) general theory is based in large part on the principle of stationary potential energy, and the same approach is adopted in this section. With only a few exceptions, the results to be found are contained in Koiter's early work, and the major differences here are of style, emphasis, and notation.

The only state variable to be used in the energy approach is the generalized displacement u. The potential energy 6, of the structural system under contemplation is considered to be a functional of ti, but it will also be presumed to depend on a single scalar variable i. which determines the magnitude (and possibly the distribution) of prescribed external loads on the system. Informally, we will refer to A as the "load." It will be presumed that

Buckling Brhuz.ior qf Elustic Siructirrrs 17

certain geometric restrictions independent of 3, (typically, boundary or support conditions) are placed on the displacements. These restrictions, together with any more general requirements concerning continuity that may be appropriate, constitute arlrnissihilirj~ conrlitions that constrain 1 1 to lie in a well-defined subspace, from within which equilibrium states will be sought. This search will be based on the stationary energy principle which asserts that a particular u is an equilibrium state if, and only if, i t is admissible and

for all admissible variations 6u. The admissibility conditions on Csu are homogeneous, and follow simply from the requirement that, for any scalar CI,

14 + CI 611, as well as u, lie in the subspace of admissible displacements. Equa- tion (4. l ) is usefully regarded as a variational equation of equilibrium.

I t is to be emphasized that the equation 64 = 0, as the condition of equilibrium, is meant to reflect the use of a potential-energy functional appropriate to the special structural theory adopted for the analysis of the particular structure under consideration. Only those theories that enjoy such a principle of stationary potential energy are therefore susceptible to the present analysis. This is not, however, a serious restriction, since a good special theory should mimic general continuum mechanical theories in this respect.

B. BIFURCATION ANALYSIS

In order to discover conditions for bifurcation buckling, we assume first that there exists afirntlarnental soltrtiori uo that varies smoothly with i as the load increases from zero. The variational equation of equilibrium (4.1) requires that

(b"uo(3,); A]6u = 0 (4.2) for all admissible du. Now suppose that, for some range of A, there is another solution

11 = L&) + [.(A) (4.3) that intersects the fundamental one at ic, in the sense that

I t will be further assumed that u o ( i ) exists for i. greater than ic, so that a true bifurcation, rather than a limit point, is implied by (4.3) and (4.4).

18 Bernard Budiansky

The bifurcation buckling mode will be defined as

u t = 1im(v/llu[[), .I-&

where 1 1 1 1 represents a suitable norm; note that ( lu l l / = 1. Since the u given by (4.3) must satisfy equilibrium,

@[u,(A) + u ( A ) ; A] 8u = 0,

(4.5)

and, under the assumption that 4 is analytic in the vicinity of uO(Ac), a Taylor-series expansion gives

(4.6)

for sufficiently small 1 i - Ac 1 . But the first term vanishes by (4.2); dividing the rest of (4.6) by I / u / J , and letting 2 --f 2, then gives

$"uo; A] 6u + @"u,; 6u + $ V [ u , ; A]v2 6u + *. . = 0

@[uo(&); ic,lu, 6u = 0 (4.7)

as the variational equation governing the buckling mode u1 and the critical load LC . The notations

u0(A,) = u, and @'[u,(A,): A,] = 4: are convenient; then (4.7) becomes

4:u1 6u = 0. (4.8)

Equation (4.8) constitutes the variational statement of a homogeneous eigenvalue problem. Admissibility conditions on v, and hence on ulr coincide with the homogeneous conditions on 8u discussed earlier. This eigenvalue problem need not, of course, have a unique solution for A c , but Lc will be defined as the lowest positive eigenvalue. This lowest critical load, in turn, need not be associated with a unique buckling mode u l , but, for the present, uniqueness (except for sign) will be assumed.

Another derivation of (4.8) may be instructive. Since @[u; A] 6u = 0 all along any equilibrium path, differentiation of this equation with respect to time is permissible, and provides a variational equation for the displacement rate duldt. Thus, along the fundamental path,

duo d i , i3$"uo; 1.1 $'"uo; I"] - + - ( d t lit an -) 6u = 0

and on the bifurcated path

(4.9)

d i d4 ' [u , + v; A] - --6u = 0. (4.10)

d n

Buckling Behuoior of Elastic Structures 19

Letting 2 = Ac in each of these equations and subtracting gives

4;(dLl/dt), 6~ = 0 (4.11)

as the variational principle governing a nonzero displacement rate (dc /d t ) , that can be added to du,/dt on the fundamental path at 2, without violating equilibrium. This is simply another way to describe a bifurcation and (doldt), must be the same as the bifurcation mode u l , to within a multiplicative constant. Note that in the derivation of (4.1 1) there is no assumption that the loading rate di./dt associated with (dr /d t ) , must vanish. Casual derivations in the literature of buckling equations often refer to “ buckling under constant load,” meaning, presumably, the existence of a nonzero displacement du/dt consistent with dA/dt = 0. But formulation of the eigenvalue problem for bifurcation does not require an explicit assumption concerning dA/dr; and the eigenfunction u1 is proportional to the totul displacement rate du /d t = (du/dA)(dR/dt) + (dc /dr ) only if d)./dt is zero.

It may be noted that (4.11) also governs limit-point buckling; this follows from (4.9) for which a nontrivial solution du,/dt, with dA/dt = 0, becomes the limit-point-buckling mode.

C. POST-BUCKLING ANALYSIS

In order to proceed with a determination of c in (4.3) for i # ic , introduce

r‘ = (G U l ) , (4.12) where the bracket symbol represents any bilinear inner product; the only restriction on this inner product is that ( u l , u l ) # 0. Indeed, it is particularly convenient to choose the norm IluIl = ( u , u)”’ , for then ( u l , u l ) = 1 , and this choice will be assumed henceforth. Then it follows that

o = 5 U l + L‘, (4.13) where (C;, u l ) = 0. The parameter 4 is therefore a measure of the ‘‘ amount ” of buckling mode contained in the difference .(A) - uo( iL ) between the displacements on the bifurcated path and the fundamental path, at a given value of A. This amount depends, of course, on the particular choice made for the inner-product norm, but for any choice, 4 + O for A+&. Furthermore, since v/lloll = u1 + h, where h -+ 0 for A --+ I”,, it follows that / J v ( I = g/(l - iilhii’), and so u = { ( u , + h) / ( l - ~ ~ ~ 1 1 ~ ~ ’ ) ; hence, from (4.13), = ({(ul + h) / ( l - fllhll’) - u,) and therefore ij/{ -0 for A + 2,.

the scalar parameter ( defined by

20 Bernard Burliansky

Each term in the Taylor expansion (4.6) of the variational equation of equilibrium will now, in turn, be expanded about 3 , . We will use the notation

$;J $ 0 1 ) 1 $$) = p ) [ u O ( i ) ; 4, 0 A = & ,

4‘0) EE p J [ l l O ( n ) ; A], $!J = p 0 I i . = i .; , d

dx

d 2 (4.14)

and so on. Then, remembering that the first term of (4.6) vanishes, we have

&? = d i 2 d ( n ) [ 1 / o ( / 2 ) ; 4, 4:) = $ y ,

{$: + (1 - lC)& + +(n- icy$: + ”.)I’d, / + +I/( + ()- - j”,)& + ...),. 2 b l , + ‘I$” + ...; 1 ,3 du + . . . = 0. (4.15) 61 c

The aim of the analysis is now to discover how u = ~ ~ ( 2 ) + < u l + fi varies with 3. along the bifurcated path, and, with u0().) considered known, this would appear to dictate a search for (0.) and F ( i ) . A small change in viewpoint is, however, very effective; consider < to be the independent variable, and look for 2 and T> as functions of (. To this end, we will anticipate at least the asymptotic validity, for small <, of the perturbation expansions

1 = LC + /I,< + A 2 ( 2 + ‘..) i. = ( 2 ~ / 2 + c31r3 + . . . .

(4.16)5-

(4.17)

The full asymptotic expansion for the displacements is then

11 = “o(l”) + (lrl + s‘2,r2 + < 3 u 3 + ” ’ . (4.18)

Although we know that F/ ( + 0 for < --f 0, there is of course no guarantee that these asymptotic expansions must involve only integral powers of <; but it will turn out that this prescription, with rare exceptions, is self-consistent, in the sense that a systematic procedure can be developed for the calculation of the terms in (4.16) and (4.18).

Substituting I‘ = 11 - ~ ~ ( 3 . ) from (4.18) into (4.15), together with the expansion (4.16) for 2, and making use of the fact that &‘ ul 6 1 r = 0, gives

< 2 ( $ : ~ 2 + i, 4 : ~ ~ + f$:~:) csl/

+ (3(411(~1, + i,&:u2 + i24:111 + +jL:$:ul

+ (6:.1112 + +il ;b:u: + ie11:; f L 1 + . ’ . = 0. (4.19)

I t follows that the coefficients of t2 , C 3 , . . . must vanish separately, for all

+ 111 many carlicr papcrs, the notation i. = . , ( I + tr: + h:’ + ...) has been used insrcad.

Buckliiig Behauior qf Elastic Structures 21

admissible variations in 6i t . Set 6u = u l , and exploit the fact that

4:u2u1 = (i$u1u2 = 0 (4.20)

by (4.8), wherein 6u can be chosen as u, . Similarly, $ : i t 3 u I = 0, and so

2 1 - 2 c .:l&:u:, (4.21)

and expressions for the successive coefficients i3, i4, . . . in (4.16) can also be found in a straightforward way. It is seen that the post-buckling coefficient 2 , depends only on the buckling mode i i , , but the calculation of 2 , would generally require the determination of u2 as well, and so on. I t is now evident that the relations between i and < found for the bifurcated path are entirely similar to those illustrated in Fig. 2 for the k( relation of the simple model. Again, an asymmetric bifurcation corresponds to 2 , # 0, and in symmetric bifurcation the load rises or falls during buckling according to the sign of i2.

The results for I , and 2 , are, of course, not valid if &pu: = 0, but this occurrence, although possible, can only be accidental. I n fact, as will be shown later when stability is discussed, i t is generally to be expected that ;b:u: < 0. [Nevertheless, if &:u: should vanish, the post-buckling analysis can still proceed on the basis of (4.6), but nonintegral powers of < may then be needed in the expansions for 2 and i t . Furthermore, even with a unique buckling mode u l , more than one post-buckling path for each sign of 4 could conceivably then intersect the fundamental path at A c ; on the other hand, there might not be a real bifurcation at all, despite the existence of a solution to the eigenvalue problem.]

A variational principle governing u, can now be deduced from (4.19) by asserting that the coefficient of t2 must vanish for all admissible 6 u :

(4.23)

By virtue of (4.8) and (4.21) this is automatically satisfied for 6u = it,. Note that while (4.23) represents a nonhomogeneous problem for u 2 , there is an apparent lack of uniqueness in the solution since, by (4.8), any multiple of u , can be added to a particular solution of (4.23). But recall that since (F, u , ) must vanish, it follows that (u , , u l ) = 0 for M > 1. Consequently, if U2 is any particular solution of (4.23), 11, must then be

(4.24)

Once u, is found, 2, as given by (4.22) can be computed. It should be remarked that the use of (4.23) is not necessarily the best basis

on which to calculate u 2 . The original variational statement 4’[u] 6 u = 0

- u2 = u2 - ( U 2 , u,>u, .

22 Bernard Budiarz.sky

has, as its consequence, a set of field equations (be they algebraic or differential) and possibly '' natural " boundary conditions that, together with admissibility conditions on 11 (e.g., geometrical boundary conditions) determine 1 4 .

The expansions (4.16) and (4.18) could be substituted directly into these equations and a set of equations governing l i O , l i l , u 2 , . . . could be found by a standard perturbation procedure. The resulting equations for u2 must then coincide with those implied by the variational statement (4.23).

I n many (possibly most) buckling problems of practical interest, the bifurcation is symmetric, the coefficient i, vanishes, and the relations (4.22) and (4.23) for i2 and zi2 then simplify considerably to

i,, = - ($(/lYLl? + 4;u; l12)/&h:,

q5;u2 fill + +(/ l ; l l ; dl1 = 0.

(4.22')

(4.23')

In this case, since setting 6u = i i2 in (4.23') gives

q y 1 1 : L l , = -2434;

the result for i2 can be written as

2 2 = -($&VLl? - 2q5:u;)/$l;u:. (4.22")

vanishes, then the value of i2 is independent of the particular choice made for the inner-product normalization in (4.24); by (4.20) the addition to u2 of UHJ' multiple of l i l leaves 4:~; in (4.22") unaffected. All of these results would assume slightly simpler forms if the inner product in (4.12) were chosen as

I t should be noted that if

(11, r . ) = -4;LK with it1 so normalized that & ' ~ i : = - 1 ; but this is not necessarily a convenient choice from the point of view of providing a transparent physical interpretation of (, and will not be made herein.

D. INITIAL IMPERFECTIONS

If the structure under analysis is not quite perfect, in that i t contains a displacement ii before the application of load, its potential energy functional in terms of the additioiziil displacement u that occurs during loading is ex- pressible in the form

(f, = $ [ 1 1 ; I"] + $[u, u; A], (4.25)

Buckling Behuvior o f Elastic Structures 23

where 4 is the potential energy of the perfect structure, and $[u, U; A] is a functional of u and U that may also depend on A. The additional contribution $ to the energy may be always chosen to obey the conditions

$[u , 0; A] = 0 for all u (4.26)

and $[O, U; i.] = 0 for all U. (4.27)

The variational equation of equilibrium is now $'[u, U ; A] 6 u = 0, where the Frechet derivative is with respect to 1.1. Hence

@[u; A] 6u + $"u, u; A] 6u = 0, (4.28)

and it is assumed that admissibility conditions on u remain the same as they were for the perfect structure. It is convenient to let

(4.29)

where lJirl( = (i, ii)1/2 = 1, and then 5 is a good measure of the magnitude of the imperfection. A search will now be instituted for relations among the function u and the scalars A and 5 that, in the vicinity of A = A', are dictated by the equilibrium condition (4.28). Accordingly we write, in terms of the fundamental solution u,(A) of the perfect structure,

- - u = ti,

u = .,(A) + T, (4.30)

where now IlZIl can be made as small as desired by making both f and / A - Ibcl sufficiently small. No prejudgment is made concerning the uniqueness of 5 for given values of 5 and A, but the double limit

must be the same for each single-valued branch of v under consideration; it will be verified shortly that u1 = 2 1 , . For a sufficiently small 5, we can expand (4.28) into

4; 6, + +;T 6u + f&? 6 u + .. ' + $ " U " , u; A] 6u

+ $"[u, , U ; jb]F 6~ + . . . = 0. (4.31)

Now introduce the notation $*[u,, U; A] to denote the Frechet derivative of $ with respect to U, then

- $"u, , u ; I.] = $"u, , 0; A] + f$ ' * [u , , 0; A];( + '. ', $'"uo, u; A] = $"[u, , 0; 34 + @ " * [ u o , 0; A];1 + .. ' .


However, by (4.26), $'[ito, 0; I*] = $"[u, , 0; A] = . . . = 0, and &, 611 vanishes by equilibrium of the unbuckled perfect structure, so that (4.31) is

(&i76U + +@;z' 6u + ." + &I&*; 6 u + +f";**;l' n'u

+ .. . + @s*GF hu + ... + . . . = 0, (4.32)

where $;* = $ ' * [uo , 0; 21, t&** = ~,h'**[u,, 0; ,1.], and so on. Dividing by (5, V)"', letting 5 + 0, and then letting 1" -+ Ibc gives

4;u, 6u = 0, * (4.33)

and this verifies that u 1 is indeed the same as i t , . We now introduce < = (i, vl ) , as in the perfect case, and try to find a relationship among I,, 5, and t that is uniformly valid for suitably restricted small values of {, 5, and

1 /z - 2, 1 . More precisely, we will look for a dependence of R on < and along curves < = m f ? in the {-z plane (Fig. 9), where a is a scalar parameter, and the exponent 7 > 0 will be chosen to suit our convenience. (For a given ;', any combination of 5 and j' can be reached by an appropriate choice for LX.) Expanding the operators in (4.32) about /z = A,, and substituting = ut' gives

&fi 6u + (2 - I.,)&% 6u + ' ' . + +/$5' 6cr + . ' ' + & p p hu + . . ' + scy$gil 6u + a"5"'(/:**.2 isu + ... + %y$:(*.;f 611 + ."= 0,

(4.34) where

and so on.

to assume expansions in the form

and

$2 = *;* ~ l , = ~ c . If the choice of an integer value for 7 2 2 is anticipated, it is appropriate

I" = 2, + 5",t + 7 , { 2 + ' . . (4.35)

i7 = (111 f {2v2 + (3v3 + .", (4.36)

Buckling Behavior qf Elustic Structures 25

where (v,, u , ) = 0, and X I , 12, . . . , as well as the c,, will depend on a. Then (4.34) becomes

(*[q5312 + 2, &u, + fq5343 6u + (”q5pv3 + ;Z2;6pu1 + Ku:Z , ,

+ +&u; + . . . I 6u + .” + cclyl):*ir 6 u + ... = 0 (4.37)

wherein the leading term (&‘LI, 6u = 0 has been dropped, and the omitted terms in the second bracket are all proportional to 2, or 2:. Now pick 7 = 2, and set 6 u = u , ; then,

2,;b:u: + ;q!I:u; + al);*iu, = 0

A, = i, - ccpA, ,

whence [see (4.21)] -

(4.38)

p = l):*;ul/i&u:. (4.39)

where

Hence,

A = 1, + A,[ - (ccpA,)t (4.40)

and, having completed our exploitation of a = t/C2, we eliminate it from (4.40) to get

A = Ac + A,( - (Zpi,/t) (4.4 1 )

for small enough values of 5 and l. The approximation can be improved systematically, but for i1 # 0 appears adequate for asymptotic estimates of snapping loads 4 . If i1 < 0, and ( t p ) > 0, then (4.41) represents a relation that does display a local maximum near A = A,, which occurs at

5, = (-ZpA/A,)1’2 (4.42)

giving

A/?+, = 1 - 2 ( - & d / i c ) 1 ’ 2 . (4.43)

This holds also for ;., > 0 and zp < 0, and is seen to be similar to the result (2.9) for the simple model. For A, Zp > 0, (4.41) does not display a local maximum in A, but in the vicinity of 1” = A, gives A-t relations like those shown in Fig. 5b for the simple model. It must be presumed that the appropriate branches of the asymptotic result (4.41) can be matched smoothly to solutions I ( ( ; r ) for srmll A.

If A, = 0, the approximation (4.41) found is clearly inadequate (since for ( # 0 and 5 = 0, the post-buckling solution of the perfect structure is not obtained) and must be carried one step further. The easiest procedure,

26 Bernard Budiunsky

however, is to shift gears and set 11 equal to 3 instead of 2 [in (4.37)]. Then 2, = 0, and

qyv, 6u = -++:u: 6 u

so that u2 is identified as u 2 , and with 6u = u l , the terms of order g3 in (4.37) give

2, = A, + a p A c .

With a now replaced by F/t3, (4.35) gives

(4.44)

and, for A, < 0, the snapping load becomes

N" 1 - 3( -A*/Ac)1 /3(+&l)2 /3 .

tS = 2A2)1/3. at

(4.46)

(4.47)

Now the similarity to Eq. (2.10) is close, and with 2 replaced by ELI, the diagrams of Fig. 6b,c, for 3, near ic show branches of (4.45) that should match into results for small A.

Essentially the same asymptotic result would be obtained if we stayed and continued with the perturbation analysis based on y = 2; furthermore, the results for I,, # 0 could have been found with y = 3, but in a less straightforward fashion. All of the results found are susceptible to systematic improvement, but, as is in many asymptotic studies, the improvement may be illusory, or not worth the effort-or both.

The imperfection sensitivity discussed is clearly dependent (via p ) on the choice of il for the shape of the imperfection. Some general insight into the effect of this choice, and the consequent magnitude of the nondimensional parameter p, will be found later when the virtual work approach to the general theory is developed.

E. LOAD-" SHORTENING " RELATION

Returning to the perfect structure, we remark that in most problems the potential energy function 4 can be expressed as

4 [ u ; L] = 6 + B[u; A], (4.48)

where A is the struin energy of the system and B is the potential energy associated with the prescribed loading. Furthermore, it is usually true that B can be written in the form

B[u; A] = -AA[u], (4.49)


where A[O] = 0, and by analogy with the simple model A[.] can be regarded as a generalized shortening. Assume further that R does not depend explicitly on A (only implicitly via if), and it follows that

A = - ( a / d A ) 4 [ ~ ; A], (4.50)

but since

(d/dA)(b[u; A] = @[u; A](du/dA) + (d/?A)+[u; A]

and @[u; A](du/dA) must vanish by (4.1),

A = - ( d / d A ) 4 [ u ; A] = -4. (4.51)

Note that, by equilibrium along any path,

0 = ( d / d A ) 4 ' [ ~ ] 6~ = (a$ ' /aA) - 4"(du/dA) 6u = -A' 6u - 9 " ( d ~ / d A ) 6 ~ .

Now set 11 = 11, , and 6zi = u l ; since &'(du/dA)u, = 0 by (4.8), we have

A6u1 = 0, (4.52)

which says that at the critical load the shortening is stationary with respect to the buckling displacement. With the prebuckling shortening defined by

A. = - (d/dA)+[uo(A): A] = -$o

we have

A - A0 = - ( d / d A ) { 4 [ ~ ; A] - 401 = - ( d / d A ) { 4 [ ~ 0 + C ; A] - 40)

= -(d/dA){&v + +&v' + 4 y v 3 6 0 + ".} - - $ b v - l$,9 + . . . - [4; + 4gv + L V L ? + . . .]b, 2 0 -

2 0

where u = dv/&,. But the bracketed expression is the Taylor expansion of

4"uo + c; A]

about i io , and, by equilibrium @b must vanish; so does 4; L', since 4' 6u =

4; 6u = 0 all along the fundamental path. Consequently,

(4.53)

This general result has apparently not been exhibited hitherto.

28 B e r n d Builiansky

Accordingly, with A z /2, + A, 4 + A, t2; A/Ac z 1 + (Al/&)[ -(2/&'4)(A - Ao)]'12 for ( > 0 (4.54a)

z 1 - (A1/ic)[-(2/&'uf)(A - for ( < 0 (4.54b)

z 1 + { 2 ( ~ ~ 2 / ~ ~ c ) / [ - ~ ~ u f ] ~ [ A - Ao] for A 1 = 0, 2 , # 0 (4.55)

wherein, to be consistent with (4.53) and the condition &'u: < 0 previously mentioned, A - A. > 0. These results are, again, similar to those given in Eqs. (2.12) and (2.13) for the simple model (in which there was no prebuckling shortening Ao). It is worth emphasizing that they are quite general, including cases in which the prebuckling variation of A. with 2 is not linear. Some typical initial A-A curves, together with post-buckling jL-< relations to which they correspond, are sketched in Fig. 10.

I

E

P A

E A

y -

& A

E A

(C) ( d )

FIG. 10. General load-shortening relatiom. (a) i,, < 0, 5 > 0; (b) i., > 0.5 > 0; (c) i, = 0, i, < 0; (d) R , = 0, I., > 0.

For the imperfect structure Eq. (4.53) remains a valid first approximation for the relation between A-A. and 4 , and the variation of 2 with A-A. is readily found from a cross-plotting elimination of ( between (4.53) and either (4.41) or (4.45). Results similar to those given by the dotted curves in Fig. 10 would then be found.


In the case of symmetric bifurcation, it is useful to define the initial post- buckling stiffness of the perfect structure as K = d I / d A at A = Ac , and compare it to the corresponding prebuckling stiffness K O at A = I., on the fundamental path. [For asymmetric bifurcations, where

A - A. zz -(IL - Ac)’$:u:/2i;,

the two stiffnesses coincide.] We find

1/K = (VKO) + “-$34)/2A21

or, since

K / K o = [l + ((h:~:/2A,$~)]-’ . (4.56)

This implies the variety of post-buckling behaviors indicated in Fig. 11. The initial post-buckling stiffness vanishes for i, = 0, and for increasingly positive value of I,, approaches the prebuckling stiffness. For negative I , , the load must drop during bifurcation, and at a critical negative value of i, given by

2, = -&h/2$ ,

the ).-A curve becomes vertical. For 2, < I,, &/$A again approaches K O . As in the modified simple model, bifurcation with 1, < 1, implies the possibility of snap buckling (Fig. 8c) for an increasing A imposed on the structure, whereas an increasing I leads to snap buckling for all I , < 0. Values of K for I , > 0 are useful in assessing the load-carrying capacity of structures that continue to carry loads after buckling without snapping.

As in the simple model, snap buckling under imposed shortening for I , # 0 is always possible for sufficiently small z, when ?p i l < 0.

/

A FIG. 11. Initial post-buckling stiffnesseh (L, = 0).

30 Bernurd Budiansky

F. OTHER STATIONARY FUNCTIONALS

The preceding development can be applied on the basis of other stationary principles that detqmine the state of the structure. For example, the Reissner-Hellinger principle (Reissner, 1953) states that a certain functional R of stress, strain, and displacement must be stationary. If the state variable U is now used to denote an appropriate normed linear space of all the state variables contained in the argument of R, the entire analysis goes through essentially unchanged, with the starting point R' 6U = 0 as the governing variational statement.

A similar remark concerns the development in terms of stationary potential energy when there are constraints upon the displacement that are introduced by means of the method of Lagrangian multipliers. A modified functional

F [ U ; I t ] = 4 [ u ; 4 + A[u]v

involving the Lagrangian multiplier variable v is used instead of 4, where now U is in the space of state variables u and v, and the linear operator A[u] is designed to have the property that the assertion

A [ u ] 6v = 0

F' 6U = 4"u ; It] 6u + vA"u] 8u + A[u] 8v = 0

implies the constraining relations. Again, with

as the governing variational statement of the problem, it follows that all of the preceding results remain valid with the replacement of Q, by F , and u by U .

Exaniplr 1-Euler Column

Although the post-buckling behavior of an Euler column has long been understood on the basis of the exact elasticu solution, this problem usefully provides an elementary, yet nontrivial example of the application of the general theory. The usual uniform-column theory based on inextensional bending deformation gives the potential energy

where (see Fig. 12) 0 is the rotation of the column cross section. s is distance to the cross section measured along the centroidal axis, E l is the bending stiffness, and P is the vertical load. By inextensionality, sand the total curved length L are invariant under deformation.

Buckling Behai3ior Qf Elastic Structures 31

P

FIG. 12. Simply-supported Euler column

Successive Frechet differentiation of (4.57) gives . L . L

~ ' [ u I u , = E I 1 (d~/ds)(d~,/ds) ds - P j H , sin o ( i s , ' 0 ' 0

. L 1.

@'[H]O, 0 , = E l I (dO2/ds)(dO,/ds) (1s - P j0 0,0, cos 6, (is, ' 0

4""0]0, 0,8, = P ). H , O , H , sin 0 ds, ' 0

@ " [ 0 ] U l 0, u, o4 = P I' 0, H 3 0, cos 0 (1s. (4.58) ' 0

The prebuckling solution is clearly 0,(s) = 0, and the bifurcation equation (4.8) becomes

I.

@'[0]0, 60 = E l [ ( r i U , / d s ) G(riH/ds) d s - P, 8, 60 ds = 0' (4.59)

wherein 3, has been identified as P. For a simply supported column, any continuous 60 is admissible, and so (4.59) implies

' 0 r:: Ei(d2e1/ds2) + P J , = o

and the natural boundary conditions

dH,/d.s = 0 at s = 0, L.

The lowest eigenvalue is the Euler critical load P, = n2El/L2, and, with the use of the inner-product norm ~~0~~ = (tr, based on

. L

(a , p ) = (2/L) 1 a/3 ds ' 0

we have 0 , = cos(ns/L), where l l H l l ~ = 1.

32 Bernurd Budiunsky

Since fl; = 4"'[0] = 0, the post-buckling coefficient A1 given by (4.21) vanishes, as it obviously should for the symmetrically bifurcating Euler column. From (4.58)

1.

K0': = P, [ 0.: ds = $Pc L

( j y 2 - - 1 0: ds = -1

'0

and . L

1 - 2L. ' 0

Consequently, Eq. (4.22') gives = and so it is verified that the column is not imperfection sensitive. (This example is somewhat unusual in that the calculation of A 2 did not require a determination of u2 .) During the buckling deformation 8 = 5 cos(ns/L) + '":

PIP, 7z 1 + g 2 , (4.60)

but a possibly more useful representation for P/Pc would be in terms of the amplitude of displacement rather than rotation. Since the lateral deflection w is related to 0 by dwlds = sin 0 z 0 + . . . ,

w = (<L/n)sin(ns/L) + ... and introduction into (4.60) of y~ = cL/n as the amplitude of the initial buckling displacement gives

PIP, 7z 1 + (+?)(q/L)2. (4.61)

Moderately large displacements are thus produced by small excursions beyond the initial load.

The initial load-shortening relation, by (4.55), is now immediately found as

(4.62) PIP, = 1 + (A/2L), where

. 1,

A i34/2P = L - cos 8 (1s. J, The results of this analysis agree with those found from an expansion about P, of the classical exact solution in terms of elliptic integrals.

Exumplr 2-Inextmsioizal Ring

The uniform thin ring shown in Fig. 13 is presumed to be loaded by pressure q that remains normal to the surface. Inextensionality requires that

Buckling Behuvior of Elustic Structures 33

Rv

FIG. 1.3. Thin ring.

the nondimensional normal and tangential displacements shown in the figure satisfy

w + 9 + 1 [( w + ;;)2 + (2 - U ) 2 ] = 0 da 2

The rotation H of the ring cross section satisfies

sin 0 = - (dw/da) + u,

and the curvature is therefore

(4.63)

(4.64)

The strain energy is +El Jin(dt) lds)2 ds, and the potential energy of the pressure is - q(AA), where AA is the reduction of enclosed area given exactly by

AA = +R2 [ [ -2w - w2 - v2 + 2u(dw/da)] da. (4.65)

Consequently, the potential-energy functional F , modified to include a Lagrangian multiplier function ~ ( a ) to incorporate the constraining relation (4.63), is F = (EZ/R)R, where

2rr

' 0

(4.66)


where 2. = qR3/EI. With U representing the state variables w, c, and 1 1 , the variational equation 0' 6U = 0 is satisfied in the fundamental state by

vo = 2, wg = uo = 0

(as well as by rigid-body displacements, which may be suppressed). The bifurcation equation Q' U , 6u = 0 is now

(4.67)

Standard operations of the calculus of variations then give, with vC = A C , the differential equations

d4w, d 3 u ,

(4.68)

The solution (again, ignoring rigid-body motion that can occur at any load) for the lowest buckling load is

2, = 3 (4.69) with the mode

w , = cos 20, e l = -4 sin 20, v 1 = 3 cos 20. (4.70)

It i s clear that the bifurcation is symmetric, and A, = 0. For the purpose of calculating U , by means of Eq. (4.23'), the quantity 4 r U t 6U i s needed, and is found from (4.65) to be

Buckling Behuoior qf Elustic Structures 35

wherein the first integral vanishes, by (4.70). The differential equations that result from (4.71) become

d4w2 d2w, d3v2 dv, - 3

d3w2 dw, d2v2 dv, 9 . da3 dcl da

- v, = 9 cos 48, - da4 + da2 da3 dU

+ 3 + 2 --z + da = 4sin 40,

dv, 9 -w, - = ~ (1 - cos 40),

da 16 (4.72)

and the solution for U , is

w2 = -(9/16), v, = (9/64)sin 48, v , = -(27/16)cos 40. (4.73)

The terms in Eq. (4.22') for i, can now be calculated: Q r U : U , is obtained from (4.71), with 6U = U , ; K U ? is

The result for ,I,/& works out to

/12//Ic = 27/32. (4.74)

The generalized shortening A associated with the nondimensional function (4.65) is related to the actual decrease in area ( A A ) by A = A A / R 2 . Accordingly, formula (4.55) gives the load-" shortening " relation

%/,I, = 1 + ( 9 / 1 6 ) ( A A / A ) , (4.75)

where A = n R 2 is the original area enclosed by the ring. This result, together with

A/& = 1 + (27/32)t2 + ..., w = ~ ( C O S 20) - (9/16)t2 + ..., v = - (</2)(sin 20) + ((9/64)sin 40)t2 + ...,

and the equality i, = 3, sum up the buckling and initial post-buckling behavior of the ring under normal hydrostatic pressure. (For moderate values of t, the results agree with those found by Carrier's (1947) numerical evaluation of his exact, but implicit, analytical solution.)

36 Bernard Budiunskj

Similar studies can be made for rings under other kinds of normal loading, such as dead loading, constant centrally directed load, and centrally directed loading governed by an inverse-square law.

Plates were analyzed by this theory (Koiter, 1945) and an early application to a shell was executed (Koiter, 1956).

G. STABILITY

As in the study of the simple model, the general analysis is incomplete without some indication that imperfection-sensitive structures enjoy stable equilibrium states up to the snapping load; the stability of imperfection- insensitive structures should also be assessed. As discussed earlier in Section 11, a proper minimum in the potential energy implies stability of finite-dimensional systems. In turn, this minimum exists if the second variation + 4 " ( 6 ~ ) ~ is positive, in the sense that 4"(6u)' > 0 for all admissible 6ir # 0;'f for, if 4' 6 u = 0 for the system in equilibrium,

A 4 = 4 [ u + 6u] - +[.I = f+"(h~)~ + R , (4.76)

where, in finite-dimensional systems, the remainder R is dominated by @'(6u)2 for sufficiently small 6 u ; therefore A 4 > 0 in some neighborhood of u ; hence a positive second variation ensures stability. Similarly, if 4"(6u)' is negative for any particular 6u, A ~ [ c 6u] < 0 for sufficiently small c, 4 cannot have a minimum, and [with the help of a little damping (Section II)], instability is implied.

Unfortunately, this simple reliance on the sign of the second variation as a stability criterion is not easy to justify rigorously in the study of continuous systems (Koiter, 1963a, 1965b, 1966). While a non-negative second variation remains necessary for stability, a general proof of sufficiency is not available. Part of the trouble (related to distinctions between strong and weak minima in the calculus of variations) is that in continua the remainder R in Eq. (4.76) may no longer be dominated by a positive second variation during every possible approach to zero of 6u. Further, mutually contradictory decisions concerning stability can be reached on the basis of various norms adopted, in the definition of stability, for the sizes of disturbances and the responses they produce. (In finite-dimensional systems, all such norms are equivalent to each other.) It appears necessary to study each continuous system separately, with physically appropriate, as well as analytically congenial, definitions of stability introduced in each case. The outcome of several such studies (Koitcr, 1965a, 1967) has been that the second variation remains a

f If there exists a constant /I > 0 such that ~ " ( S U ) ' > /I(lcSu/l' for all Su + 0, then y1"(6u)' is md p o s i t i w definite are said to be positiiw tlqfinite with respect to the norm 11

equivalent in finite dimensional systems, but not necessarily in continua. 1 1 . Positit

Buckling Behuuior of Elastic Structures 37

sensible indicator of stability. With the viewpoint that this will probably continue to be true in general, the present general analysis of stability will be restricted to an examination of the second variation.

Consider first the equilibrium states along the fundamental path. If stability is presumed for sufficiently low A, it must follow that any transition to instability as J. increases cannot occur below J. = Ac . For below the transition load, say, 2,4: w2 > 0 for all w, and above w2 < 0 for some w's; we will assume that at A = 1, 4: w2 = 0 vanishes for at least one w = W , t remains non-negative for all other w, and, therefore. as a function of w, is stationary at w = W; thus the eigenvalue equation 4; k 6w = 0 must be satisfied at A = 1; and therefore, since 2 is the lowest possible eigenvalue, i = ,Ic. Furthermore, if u1 is unique, W is proportional to u l .

Since 4: u: is positive for ,? < Ac , and 4: u: = 0 at i = I. , , it follows that 4eu: 5 0. However, the special case &u: = 0 will not be considered explicitly, although a separate treatment can be executed without essential difficulty. ( I t is worth noting that for &'u? < 0, bifurcation at 2, implies instability on the fundamental path for i > AC, and conversely, such instability implies the bifurcation path discovered in the earlier analysis. On the other hand, if &'u: = 0 instability need not necessarily occur for J. > ,Ic, and there may not even be a real bifurcation either.) With this case disregarded, the earlier presumption that &'u: < 0 is thus justified on the basis of stability considerations.

Consider now the equilibrium paths for imperfection-sensitive structures that are traversed as J. increases from zero up to the snapping load & (Fig. 9a,c). Along such a path we can write

-

= .o(J.) + U I ( S , A), (4.77) where now y(z, A), defined for J. < &(z) is the single-valued branch of E in Eq. (4.30) that satisfies

Iim cl(<, A ) = 0. (4.78) i-0

Along the considered path

+ " ( f U ) 2 = g5"[uo + 0,; 4 ( 6 u ) 2 + $'"uO + L ' I , u; 346u)2

@'(bu)2 = 4;;(6u)2 + gi';Ul(6u)2 + + q h ; U ; ( f u ) 2 + ... + S'$;;*;l(su)' + " ' .

(4.79) which, expanded about u = uo and U = 0, provides

(4.80) t We will not worry about pathological cases in which no such iC. exists, although an

example can be contrived. As a simple example. if, for continuous w, q b " d = ji ( 1 - h ) w z t l u , we have r,$"w' positive for /I i I , and indefinite for I. > 1, but there is no M? # 0 for which 1; (1 - s)kz d x vanishes. Such situations are unlikely to arise in well-formulated problems of mechanics.


Consequently, for sufficiently small <, & ' ( S U ) ~ must have the same sign as 4 : ( 6 ~ ) ~ , which is positive for A < 4. Thus, stability up to snapping has been established, for small enoygh imperfections.

In the case of an imperfection-insensitive structure (Fig. 9b,d) the same conclusion clearly applies for A < A,. For A in some range A' > A > A,, the expression (4.77) continues to apply, but the limit in (4.78) is replaced by

lim u,(Z, A) = u(R) , (4.8 1)

where L' is the displacement increment along the bifurcated path originally defined in Eq. (4.3). The appropriate expansion of (4.79) for A > A, is now about u = uo + u and LI = 0, and this gives

;-ro

$ " ( 6 U ) Z = (b"[uo + v(A); )L](6t/)2

+ $""uo + L'(A); 149 - I~)(dU)Z + ... + ~ ~ " * [ u o + u , o ; A];l(6U)' + .. ' = 0. (4.82)

Accordingly, for A > /2,, ( b " ( 6 ~ ) ~ will share the sign of @'[uo + D ; Al(6u)' for sufficiently small z. This is twice the second variation along the bifurcated path of the perfect structure, the sign of which will be discovered by means of a search for the minimum value of

g5"[uo + u ; A](6u)2/A(6u)Z, (4.83)

where = (du, 6 u ) = I l f i ~ ( / ~ and AuT = 1, with respect to an admissible (6u). If this minimum value is positive, then the numerator is positive for all 6u.t With the assumption that A has been so chosen that the minimum, denoted by j3, actually exists for 6u = w, the consequent requirement that (4.83) be stationary with respect to variations in w implies that w and p are solutions of the eigenvalue problem

6''~ 6~ - j 3 A ~ 6~ = 0 (4.84)

on the bifurcated path. For A = A,, v = 0, and the solution is clearly f i = 0 and w = u l . Hence, for A > A,, the expansions

w = u 1 + (w2 + ('w3 + . . . , p = <PI + t2p2 + .. . (4.85)

appear appropriate, where, without loss of generality, (w,, u l ) = 0 for I I 2 2. Recalling that 1' = ( t i , + ( 'u2 + ..., expanding (4.84) about j . = 2, and I' = 0, and writing A w (h = (w, (h), gives t{&w2 + R , &u, + q5e,u: - A u , j 6w

+ <2j45:w3 + A, 4:w2 + A 2 & h l + fA:$:u, + (je,u2u,

+ ~2 + A~$;U: + $&"u: -

t In fact, d ' ' ( 6 ~ ) ~ is then positive-definite with respect to 11

Aw2 - /j2Aul) 6w + . . . = 0. (4.86)

1 1 .

Buckling Behavior qf Elastic Structures 39

Setting 6w = u 1 in the coefficient of < gives

[I1 = A 1 & h l + qyu:,

where use has been made of the fact that Au: = (u l , u l ) = /(u11)2 = 1. But since 4Tu: = -2 i1 4:~: [Eq. (4.21)] it follows that

p1 = -A1 &q. Since 2 z A, + A,( + ... on the bifurcated path, it follows from (4.85) that

p = -(A - A , ) & q + O ( A - Ac)" ( I ? 2 2) (4.87) and therefore, since $:u: < 0, p is positive for some range of A > A,, which verifies stability on the bifurcated path, as well as on the associated imperfect-structure path, for sufficiently small 1 1 .

If A1 = 0, then p1 = 0, and it follows from (4.86) that

4:w2 6w = -4:u: 6w,

and comparison with Eq. (4.23') then shows that

(4.88)

W' = 2u2. (4.89)

From the vanishing of the coefficient of j" in (4.86) with 6w = u l , and with the use of Eq. (4.22'), the result

[j, = - 2 A , 4 : U : (4.90)

is found, so that

p = -2(A - i,)&U: + O(3, - A,)" ( I? 2 2) . 3 (4.91)

Again, p > 0 for sufficiently small A - 2, > 0, and stability follows for the bifurcated path of the perfect structure, as well as for the imperfect structure with sufficiently small imperfections.

The special case A = A, has so far been exempt from the analysis, but is easily handled separately. The minimum of

~ (4.92) $"[u, + y ; & ] ( S U ) 2 - @"u, + q; n,](sU)' + $"[uo + z'1; A,](su)2 ~ ~~ -~ ~ -

A(6u)' A(&)'

is now sought. For ,I1 # 0, Eq. (4.40) shows that = O ( t 2 ) for A = A c , and, from (4.36), u, = (u l + t 2 u 2 + .... Series of the form (4.85) may again be chosen for the minimum p and the corresponding minimizing 6u = w, and then

$"w 6~ - PAW 6w = 0 expands into

({4Pw2 + K u : - p l A u l } 6w + ('{&w, + Kv,u1 1 i v 3 - + 4:u1 w, + u1 p1 Aw, - /3,Au1} 6w + O(5) + .. ' = 0. (4.93)

40 Bernard Budiarzsk y

The choice f iu = u l , now gives

= 4:u; = -2A,431:

and so

p = (-24gu:)A,s‘ + O ( y ) (n 2 2). (4.94)

Hence, at 2 = Ac stability holds, for sufficiently small r (or z), if A I 5 > 0.

z12 = u,; then (4.93) leads to If A1 = 0, Eq. (4.45) shows that 5 = O ( t 3 ) at 2 = A,, and we have flI = 0,

w, = 2u2

and p2 = 3KU: U 2 f i & U f = - 3IL2 &zZ!:.

Consequently, [j = ( - 3 i 2 $;u:)t2 + O(5“) (n 2 3) at A = ,Ic, and since )b2 > 0, we have stability for sufficiently small t > 0.

To sum up, for sufficiently small imperfections, stability, as assessed by the sign of the second variation, has been verified along imperfect-structure equilibrium paths emanating from 1 = 0, up to the snapping load in imperfection-sensitive structures; and, up to some load higher than Ac in imperfection-insensitive structures. [Similar conditions hold for the imperfection-insensitive perfect structure, except that the second variation is inadequate to determine stability at /1 = 2 , . A study of higher-order variations (Koiter, 1945) indicates the same results as for the simple model: instability for A , # 0, stability for A1 = 0, A, > 0.1

I t is now easy to show that along the post-snapping branches of paths (Fig. 10a,c) where A is decreasing from 4 , instability must hold, although this fact is not of too much practical interest. I t is only necessary to contemplate

u = UO(3L) + V,(Z, A), where y, is the branch of V that, for A < 4 < I,,, approaches the perfect- structure bifurcated solution as < + 0. The analysis following Eq. (4.81) is then applicable, the only change being that 2 is less than A, instead of being greater. The conclusions from (4.87) and (4.9 1 ) are consequently reversed, and instability is deduced for sufficiently small

A few final remarks on stability: It is tempting (and common) simply to look at a curve of load versus a deflection and conclude that there is stability if the load is “rising,” instability otherwise. But this can have pitfalls. Ob- viously, an undetected bifurcation load on a rising curve will invalidate the conclusion at loads above the critical load. This explains why the calculations just made concerning stability are limited to small t and small neigh- borhoods of 2 , . Furthermore, equilibrium on a “falling” A-t curve can be

and (A - A,) < 0.

Buckling Behucior of’ Elastic Structures 41

stable; again, a bifurcation on such a curve could conceivably cleanse the contemplated falling path of its instability. There is, however, at least one pictorial stability theorem of some generality that may be worth mentioning. It is always true that a negative slope on a load-shortening curve can only be associated with an unstable equilibrium state; that is, t/A/dA < 0 implies instability under prescribed loading. To prove this, let 4 [ u ; A] be the potential energy of the system, perfect or not, and note that along any (smooth) equilibrium path u(L) , 4 ’ [u ( l ) ; ?L](du/dA) = 0, by equilibrium; then

But 4’(d2u/d;12) = 0, again by equilibrium, and so

( ( 1 ~ ’ / l i l ) ( d U / l U ) = 0. (4.95) Next, since A[.] = -d4[u ; A ] / d A does not depend explicitly on A,

and so, with (4.95),

dA/dL = $ “ ( d ~ / d A ) ~ . (4.96) This shows that if dA/dA (or dA/dA) is negative, so is #’(6u) for 6u = du/dR, which proves instability.

V. Stress, Strain, Displacement; Virtual-Work Approach

A. STRESS-STRAIN RELATION ; VIRTUAL-WORK EQUATION

Recall the Eqs. (4.48) and (4.49) concerning the form of the potential energy of applied loads, and write the total potential energy functional as

4 = 6‘[[~] - ~ A [ L I ] (5.1) in terms of the shortening A[.], and of the strain energy B , now considered to be a functional of the generalized strains e. The strains, in turn are given by a strain-displacement function e[u] in terms of the displacements. The variational equation of equilibrium is now

4’ ~ L L = B’[E] 6~ - AA’[z~] 6~ = 0, (5.2)

42 Berwrd Budiansky

where 61: = 1:“u] 6u.

0 = &“I: ] ,

If we introduce the notation

Eq. (5.2) becomes CJ 61; - AA’[u] 6~ = 0.

(5 .3)

(5.4)

(5 .5 )

This has the form of a principle of virtual work, stating that, in an equilibrium state, the change of the potential energy of the loads associated with “virtual” displacements 6u must equal the internal virtual work of stress cr acting through strains 61: that are compatible, via (5 .3) , with the displacements 6u. [We have thus endowed the symbol (T with three meanings; it is the stress state CJ; it is a linear operator OM strains, making CJ 6e the total work of the stresses acting through 6c; and i t is a function ~ [ c ] of strain consistent with (5.4). In earlier work, the notation CJ . 61; has been used for the inner product representing internal virtual work, but it has become clear that the dot is really not needed. The appropriate interpretation for the symbol CJ will always be clear from the context in which it is used, and, with good will on the part of the reader, the fluency afforded by the notation will soon be apparent.] We can, if we wish, assert Eq. (5 .5 ) as the variational equation of cquilibrium ab initio, without reference to the total potential energy; indeed, in plasticity problems, there may be no relations of the form (5.4), and no potential energy functional. However, in the present elastic case, the link to energy is desirable, in order to derive and exploit certain symmetry properties embedded in the stress-strain relations. Thus, since

e “el 6e 1 = .“:I 61; 1

t ” [ C ] 6 C l 61:, = a’“:] (SC2 he1,

i t follows that

and since 8“ is a symmetric bilinear operator, so must (T’ be. That is,

for all I : ~ .

be collected as

d 1 . 1 c 2 = CJ11:21:1 (5.6)

The field equations governing the problem of finding (T, c , and u may now

CT (SE - AA’6u = 0 (equilibrium), (5.7a) cr = CJ[~:] (stress-strain), (5.7b)

c = ~ [ I I ] (strain-displacement), (5.7c) wherein the variational assertion of the principle of virtual work for all admissible 6 u and compatible 61; is presumed to guarantee equilibrium.


B. BIFURCATION AND POST-BUCKLING ANALYSIS

The fundamental path is now described by a,(/Z), co( i ) , and ~ ~ ( 2 ) . O n the bifurcated path emanating from the fundamental path at 3, = 3,, , we write

u = U O ( i ) + 0, (5.8a)

cr = G o ( i ) + s, i: = C,(I”) + yI,

(5.8b)

(5.8c)

where

iim (t) = 0. 1-2,

The equilibrium equation (5.7a) asserted along the bifurcated path gives

C X ’ [ U ~ + U ] 6~ - AA’[L/O + V ] 6~ = 0

G , ~ c ‘ [ u ~ ] 6~ - AA’[uO] 6~ = 0.

(5.9)

(5.10)

and on the fundamental path gives

With the notation e , = i;[uo], cb = c’[u,], .. . , and similarly for A,,, A;, . . ., subtracting (5.10) from (5.9) gives

(GI:’ - crOcb) 6u - >-(A - A;) Csu = 0. (5.11)

Now expand the operators i:’[uO + u] and A’[”, + u] into { ; I = cb + r j l 0 + ‘e”’u2 + . . .

’ 0 2 0

A’ = + Agu + “A”’u2 + . . . 2 0

and substitute into (5.1 l), together with (5.8b) for cr, to get

seb 6u + {(go + S ) E ~ - iA6)u 6u + ;((go + S ) F ~ - AA~}V’ 6u

+ ;{(G,, + S ) F ~ - iAk}u3 6u + .. . = 0.

As before, define the buckling displacement mode (assumed unique) as

(5.12)

u 1 = lim(u/lJvJ1), 1-1,

and also define the associated stress and strain modes by

crl = l im(s/J/u//) , = lim(q/ilui\). ) & - A c L-&

Then, dividing (5.12) by ~ ( u I J , and letting A + A, gives

~ 1 1 : : CSU + O,B:U~ Csu - A,A:L/I CSU = 0, (5.13)


where the subscript c denotes evaluation at A = A' on the fundamental path. The stress-strain relation (5.7b) may be expanded as

0 0 + s = OIEO + 'I] = 0 0 + ob'I + f0;;'I 2 + ... (5.14)

and so cancellation of go, division by I J u / J , and the approach to Ac gives 01 = 0;c1. (5.15)

Similarly, the strain-displacement relation ( 5 . 7 ~ ) yields 81 = &Alll. (5.16)

Equations (5.13), (5.15), and (5.16) constitute an eigenvalue problem for A, and the eigenfunctions u l , gl , and e l .

It seems like a good idea now to check the consistency of these governing equations with Eq. (4.8) for the same eigenvalue problem in terms of 4. We have

$'[.I = ~'[E]F'[U] - AA'[u] = o[~;]E'[u] - AA'[u].

Dropping explicit indication of argument of functions, taking another Frechet derivative, and evaluating it at A = Ac gives

4: = o L ( c L ) ~ + occ: - AA:.

Then 4 : ~ ~ 6 u = aLeLul&L 6~ + uc.$u1 6~ + A:u1 6 ~ .

But since & u l = .zl by (5.16), and aLcl = u1 by (5.15), we get that

To study the bifurcated path, expand Eqs. (5.8) by writing 4: 1.1 6u = 0 is precisely Eq. (5.13).

L' = u - uO(A) = i'Ul + i'2u2 + . ' . ,

'I = & - & O ( A ) = (Sl + p e 2 + ".)

s = 0 - O,(/z) = to1 + p a 2 + ' . . , (5.17)

and substitute into Eq. (5.14), as well as into a similarly expanded version of (5 .7~) . Further, introduce the expansions

0b"'(A) = 0:) + (A - Ac)o:) + +(A - /I,)%$) + ".)

eb")(A) = $1 + (2 - A c ) p + +(A - /Ic)%:) + ..., (5.18)

Ab"'(A) = A!) + (A - &)A?) + -$(/I - AC)'A:' + . . . , where the dots represent total derivatives with respect to 3,, as in Eqs. (4.14). Finally, introduce the familiar relation /I = A, + ,II 4 + A 2 t 2 + ..., and then the following results are found by systematic collection of like powers of i':

(5.19) 01 = CJLc,, 0 2 = 0;&2 + A 1 & F l + y J c E 1 1 II 2

Buckling Behavior of’ Elastic Structures 45

and ( . I = c ; u , , E 2 = c;u2 + A l i ; U , + 31;:u;. (5.20)

A similar process of substituting into the expansion (5.12) of the remaining field equation (5.7a) gives a fairly complicated expression that will not be written out in its fuIl glory, but pieces, as needed and occasionally simplified, will be extracted. The terms of order < in this expression simply reproduce the buckling equation (5.13). The terms of order <’ give the result

-{02cL + o,4‘u2 - A,A:u2 + G,&:u, + $o,~:)u: - + i , A r u :

+ A,[ir,e:u, + oc$‘ul + olEL - APu, - &A:u1]) 6u = 0. (5.21) Set 6 u = u1 and note that the first three terms of (5.21) may then be reduced to

by exploiting Eq. (5.13), with 6u = u 2 . In turn, with the help of (5.19) and (5.20), G~ and u2 may be eliminated to transform this to

0 2 E : . L / 1 - a1c;u2

1 ,I 3 i I ( G l k L U 1 - & E : ) + %CTl&;U: + zCJ,El.

Consequently, the coefficient A1 can be found as

(5.22)

where

D = {bc&;~i: + o,&‘u: + 2G1&~1/1 + &t;f - A:L/: - 2,A:~f;. (5.23)

Since none of the functions a[&], ~ [ u ] , and A[.] depend explicitly on 2, we can write o; = o;i,, = e r u , , and A: = Arii, . Hence D can be expressed in the alternative form

. .

D = C ~ ‘ E ; L / ~ + ~ , E ; U : U , + 2G1E~U1U, + o:~:b, - A:u: - 2 c A ; ~ : U , .

(5.24) Once again, as an exercise in manipulation, let us verify that (5.22) agrees with the earlier result given by Eq. (4.21) in terms of 4. We have, keeping the notation concise, and using dummy arguments u l , u 2 , . . . ,

4’ul = GE’U, - LA’u,,

( b ” U 1 U2 = G E I ’ U ~ 212 f O ’ E ’ U ~ E ’ U ~ - AA”u, ~ ; 2 ,

(5.25)

46 Bernard Budiunsky

With EL u1 = e l , this is the numerator of (5.22). A similar calculation shows that &‘uf = D, and so the desired check is accomplished. In fact, it now hardly seems necessary to continue a direct systematic derivation of additional results in terms of o, e, and u, but rather obtain them by transformation of those already found by the energy approach. Thus, if A, = 0, the result (4.22’) for A, turns to be equivalent to

A2 = - ~ / D ( ~ ~ , C : U , U , + o-,E:u: + $o~$‘u: + O : C : C ; U ~

+ oct;:al: u2 + +:c; + &Jccivu:

- Ac A:u: u2 - &Ac Aruf}.

(5.26)

In deriving this result, use was made of the field equations governing 0 2 ,

(5.27)

e,, and u2 when A 1 = 0, which, from (5.19) and (5.20), arc

c2 = c;u2 + +e:u: 0 2 = o;e2 + +o;e:,

and, from (5.21)

02eA 6 u + oce:u2 6u - ;lcA:~iz 6u + olc:ul 6u

+ Lo 2 c c c”’uf 6u - +AcA:~i: 6u = 0, (5.28)

where, as before, u2 is rendered unique by making ( u 2 , u l ) = 0. But it is expected that the variational equation (5.28) will rarely be used to set up the problem, preference usually being given to a direct perturbation expansion, via (5.17), of the governing local equations (usually nonlinear differential equations).

It may be remarked that the relation ALu, = 0 found in (4.52) has the consequence, from (5. lo), that

occ;ul = ocE] = 0,

which means that the stresses in the structure at bifurcation do no work through the initial strains of the buckling mode.

C. SPECIAL CASES

The results found are very general, far more so than one would usually require, and in various particular problems one or more of the following simplifications can be invoked:

( i ) linear stress-strain relations: dfl) = 0 for II 2 2; (ii) quadratic strain-displacement relations: c(”) = 0 for n 2 3;

(iii) linear shortening-displacement relations: A(‘) = 0 for n 2 2.


In very many problems all three of these conditions are met. Limitation to elastic strains makes (i) valid in all but rubberlike materials. A quadratic strain-displacement ( i i ) relation is, of course, obtained when the Lagrangian strain tensor, or a simplified variant thereof, is employed. (On the other hand, if a nonlinear curvature-displacement relation is invoked, as in the ring problem, it may be cubic.) The linearity of A[.] occurs in dead-loading situations (though not in hydrostatic loading). In any event, with (i)-(iii) valid we get

D = 4:~: = ~,c:u: + 20,4'~~L1, , (5.29) and the results for ,I1 and 2, simplify considerably to

A1 = -$o,c:tr:/(o,c:u: + 2014,u, L1,) (5.22')

and, if ,it = 0,

A2 = - ( ~ ~ , E : U ~ U ~ + C T , C ~ U : ) / ( ~ , E : U : + 20,4'uI i,). (5.26')

Also, o2 = O: E ~ , E~ = c: ii2 + &'u:, and the variational equation (5.28) col- lapses to

0 2 e : 6u + Ocl;:u2 6u + o,e:u, 6u = 0. (5.28') The buckling equation (5.13) also simplifies slightly. These results agree with those given by Fitch (1968).

Another simplifying assumption, independent of (i)-(iii) above, is that of a (iv) linear fundamental state.

This assumption provides the common situation associated with buckling problems in which the prebuckling displacement, stresses, and strains vary linearly with 2, and the load /z appears linearly in the bifurcation eigenvalue problem. We will define assumption (iv) to mean a special kind of linearity associated with the functions ~ [ c ] , c[u] , and A[.], and the fundamental solutions uo , G,, , and c,, , in which not only do the identities

kc,,] = ko[t.,], ~ [ k t l , ] = ks[u0] , A[ku,] = kA[u,], (5.30)

hold, for any constant k, but, in addition, the operators o$", cg), A t ) are all independent of 2 for n 2 1.7 This guarantees that the fundamental solution of the field equations (5.7) will be linear in 2, and it also makes (5.13) a linear eigenvalue problem. Furthermore, with it) = b t ) = At) = 0 for n 2 I , and the relation b, = ( l/,ic)o,, D in (5.23) reduces to

D = (~/A,)[o,E:u~ - ,I,A:U:] = - ( ~ / ~ ~ , ) O , C ~ , (5.31)

+This means that even though ~ [ u ] may be nonlinear, c [ u 0 + 1.1 = I : [ U ~ ] + I;[I.] for all c, and similarly for 0[1:] and A[u]. For example, if I:, = (?c/?xj + h(?w>/ t 'y j ' , and the fundamental state is cO = rx, w,, = 0, this condition is satisfied.


where the last equality follows from Eq. (5.13). The numerators of (5.22) and (5.26) remain unchanged under just assumption (iv), but if all the assumptions (i)-(iv) can be invoked

D = $:u: = ( l/J.,)o, c"u:, (5.32)

and the results for A, and i2 become

&/A, = -$0, 1::11~/0,C:U: = +7,e:11:/0, e , (5.22")

and, for i1 = 0,

R2/ /Z , = - 20, E P L l , 2.12 + 0 2 1::11:/0, ,:,1:

= 20, e:ul u2 + r T 2 1 : : U ~ / 0 , E l . (5.26")

The results are essentially those given by Budiansky (1965) and Budiansky and Amazigo (1968).

All of the formulas of the preceding section concerning the generalized shortening and the post-buckling stiffness [Eqs. (4.54)-(4.56)] are immediately transformable to the notation of the present section simply by replacing &'u: by D. Note that 4, = - A , = -AAii = - ( I / % , ) q C,, by (5.10). I f the assumptions (i)-(iii) apply, then Eqs. (4.56) and (5.29) give

(5.33)

Under assumption (iv) [and not necessarily any of (i)-(iii)]

3, = - ( l/A,)0, i;, = - ( 1/3,2)0, E, ,

and since we also have [Eq. (5.32)] &'u: = - ( l / i , ) ~ ~ i ; ~ for this case, we get

If all the assumptions (i)-(iv) hold,

(5.34)

(5.35)

Example 3-Flut plute

Consider a square, simply supported, flat, plate of uniform thickness compressed normally along all edges as shown in Fig. 14, by means of rigid,


lubricated loading heads. The problem will be treated on the basis of the familiar von KrirmAn equations

(5 .37)

where D2 = Et3/12(1 - v2), in terms of the normal displacement M' and the Airy function F giving stress resultants N , = d2F/?y2, N , = ? 'F /2s2 , N , , = - ? 2 F / ? ~ ?\,. The generalized stresses of this problem are the bending

F-IG. 14. Simply-supported k i t plate.

moments M,, M,, and M,, and the stress resultants N,, N , , and N,,; the displacements are u, u, and w; the generalized strain state consists of the mean in-plane strains c, , E , , and y,, and the curvatures K,, K , , and K,, . The stress-strain relation E [ U ] is represented by

(q2W K = -

ax + 2 ax ' ?Y2 ' c, =

(5 .38)

50 Brmurrl Butliansky

The stress-strain relation 41:] is

M , = D [ K x + vK,], 0, = [E/ ( 1 - V ~ ) ] [ E , + V E J ,

M , = D [ K , + VK,],

Mxy = D(1 - V)K,? 3

oy = [ E / ( 1 - V ' ) ] [ C , + VE,],

Zxy = [E/2(1 + v)]yxy,

(5.39)

and the virtual-work statement (5.7a), in the present context, is

+ M , 6 K , + M y 6 K , + 2M,, 6K,, dx djl

= [external virtual work] (5.40) 1

wherein the relations between stress resultants and stress function have been introduced. The differential equations (5.36) and (5.37) are consistent with Eqs. (5.38)-(5.40) which are the realizations, for plates, of Eqs. (5.7). This shows that the various results of the general theory can be used. In the special plate problem under consideration, the boundary conditions of simple support are

w = d2w/dx2 = O at x = 0, a,

w = F2w/?y2 = 0 at y = 0, a, (5.41)

and the conditions of zero shear and uniform normal displacement along the edges are guaranteed by

d2F/dx dy = d2F/(3x3 = 0 at x = 0, a,

d2F/6x r3y = d3FF/F~13 = 0 at y = 0, a. (5.42)

The prescribed force conditions require that

(FF/c?y)(O, U ) - (dF/c?y)(O, 0) = -AP,/a,

(i?F/dx)(u, 0) - (ZF/dx)(O, 0) = - ~ P , / u . (5.43)

The expansions for u and cr in the general theory are now represented adequately by

(5.44)

where the fundamental solution is simply

W O = 0, Fo = -(J./2u)[P,y2 + Pyx2]. (5.45)

Buckling Behauior qf Elastic Structures 51

Assuming that P, and P, are fixed in magnitude, we look for the critical value of the scalar multiplier A. The bifurcation equations, most easily found by substituting (5.44) into (5.36) and (5.37), and retaining terms of order 4, are just

DV4w, + (A,/a)[P,(d2wl/i?x2) + P,(d2w,/dy2)] = 0, V4F1 = 0. (5.46)

Since F, satisfies the boundary conditions on F , those on F , become homogeneous and the solution is

w1 = t sin(n.~/a)sin(n~~/a), F , = O,? 2, = ~TC’D/LI(P, + P,), (5.47)

where we have made ( w ~ ) , , , ~ ~ = t. The bifurcation is clearly symmetric, so A 1 = 0. To calculate A,, we will need w,, F , , and by collecting terms of order t2 in (5.36) and (5.37) we get

(5.48)

Using w, as found in (5.47) gives

V4F2 = ( ~ ~ ~ E t ~ / 2 a ~ ) [ c o s ( 2 n x / a ) + cos(2ny/a)]

and the solution to (5.48), for r r ~ j ) orthogonality condition ( i v 2 , w l ) = 0, is

w2 = 0, F, = (Et3/32)[cos(2nx/a) + cos(2ny/a)]. (5.49)

To calculate A,, we note that all of the simplifying assumptions (i)-(iv) discussed earlier are satisfied, and we can use Eqs. (5.26”). Note that c:ul u2 is, from (5.38), just a set of three in-plane strains given by

c‘w, ?w2 ?w, ?w, ?w, F w , c‘w, (7wz ?x (3x ’ dy (7y ’

and Fx ?y + c‘y ?u ’

corresponding to c,, c y , and y,,. Since w2 = 0, we need only concern ourselves with o2 r::u: and o, c:uf in (5.26”), and by (5.40) these become

7c4Et5

32a2 - -

t A possible inconsequential linear contribution to F , is ignored.


(since c?'F,/dx c?y = 0) and

The result for ,I2/& = - ( T ~ c ~ u ~ / ( T , c ~ L I ~ is simply

&/Ac = $(l - 1 (5.50)

independent of the ratio of Px and P,. Note that wmaX % ( l t ) for small l, so that we have the result

A/& % 1 + i ( 1 - v2)(wm,,/t)2 + ".. (5.51)

Finally, since

O,F, = [Az(l - V ~ ) U ~ / E ] [ P ; + P: - 2vp,P,.],

Eq. (5.35) gives

(5.52)

For uniaxial loading, P, = 0, and K I K , = 4, independent of v. For P, = P,. and v = 4, K / K , = $; this shows that, for balanced biaxial compression, the stiffness associated with change in area of the plate is reduced by a factor of 4 after buckling.

D. SHALLOW SHELLS; DONNELL-MUSHTARI-VLASOV SHELLS

In shallow-shell theory, the basic equations (5.7) assume the forms, in tensor notation, with respect to generalized coordinates ( I , t':

1 1 [MaB 6K,, + Nap jSEaa] dS = [external virtual work], . .

(5.53a) . * S

Et3 Map = [(I - V ) K " ~ + VK;.LJ~'], 12(1 - v2)

(5.5 3 b)

= 4[Ua,p + u,.a] + hap W + 4 W a Wp 5

Ka, = - W a a ' (5.53c)

Here ME" and Nap are stress couples and stress resultants; U, and Ware the surface and normal displacements; Eaa and K,, are stretching and bending

Buckling Behavior of Elastic Stvzrctures 53

strains; h,, = -z,.,~ is an appropriate curvature tensor, where z is the normal distance to a reference plane; and the metric tensor ya0 is calculated in the flat space of the reference plane. In Donnell-Mushtari-Vlasov theory, the same equations are used, but b,, and y,, are curvature and metric tensors computed on the surface. [See Sanders (1963) for further details.]

The expansions (5.17) may be written as

(5.55)

(5.56)

where now i is a scalar multiplier of a prescribed external edge and surface loading distribution. For dead loading, conditions (i)-(iii) are satisfied, but not necessarily (iv), and the applicable formulas (5.22') and (5.26'), give

(5 .58)

Here the dot, as usual, means total derivative with respect to i along the fundamental path, and the index c-placed wherever convenient-denotes evaluation at i, = 1,. Similarly, if (iv) is also satisfied, formulas (5.22"), (5.26") and (5.35) give


and, for A , = 0,

(5.62)

In the absence of tangential surface loads, the actual solution of particular problems is usually most easily executed by exploitation of the Airy stress function F giving the stress resultants as

N"b = fCo$ j 'F , oY (5.63)

where PW is the alternative tensor. The displacement U, may then be eliminated from explicit consideration and the governing differential equations for W and F become the Karman-Donne11 equations:

DV4 W + t?WPb,p F , + p = c t: 0) ' F , toy W.0

1 atu p s (l/Et)V4F - Pi?bab W,,y = - TE c Wlp Wtuy (5.64)

where p is the normal surface loading. Expansion of F into a series like (5 .55) thus leads to the equations [assuming (iv) does not necessarily hold]

(1 ) ( 1 ) ( 1 ) ( 1 )

DV4W + Eatuo""hz8 F,("? - x b ( & ) W , l p - c ~ ~ c ~ ~ F,(,,>, W.,, = 0

( 1 ) ( 1 ) ( 1 )

(1/Et)V4F - CaW&'bap W,cDy + c a m c BY W:afl(2,.)W.ooy = 0, (5.65) ( 1 ) (1 )

for the eigenvalue

for W, and F , [which provide N a p via (5.63)] are

and the buckling mode W , F . I f iI = 0, the equations (2)

( 2 ) ( 2 ) ( 2 ) ( 2 ) DV4W + PEpYb,p F,,,? - Nzp((3L,)W.ap - I; 3<" c p ; F , W y W.,p

( 1 )

F . coy w rp = f/J[;J:'

( 2 ) ( 2 ) ( 2 )

( 1/Et)V4 F - CatoCpYbn/ j W . toy + C a 0 & 8 1 ' W.,,(%,)w.

( 1 ) ( 1 ) - 1 .Z(O .or - -2" t W,to). W,lp . (5.66)

( 0 ) (0)

It is implicit here that if N Z p ( 2 ) and W,aLj(A) are nonlinear in 2, they can nevertheless be found without too much difficulty from a particular solution of the full nonlinear equations (5.64). Typically, this occurs in problems of


shells of revolution, in which a nonlinear, prebuckling axisymmetric state precedes nonaxisymmetric bifurcation. If assumption (iv) holds, and the fundamental state is linear, then Eqs. (5.65) and (5.66) are simplified to the extent that terms in We(&) drop out, and @ ( A e ) becomes ie eP, where is independent of Ae .

The initial post-buckling behavior of many shells has been studied on the basis of this theory. Examples involving linear prebuckling states are contained in Budiansky and Amazigo (1968), Budiansky (1969), Hutchinson (1967a, 1968), Danielson (1969), Hutchinson and Amazigo (1967), and Hutchinson and Frauenthal(1969), problems in which nonlinear prebuckling states are taken into consideration are contained in Fitch (1968) Hutchin- son and Amazigo (1967), Hutchinson and Frauenthal (1969), Fitch and Budiansky (1970), and Budiansky and Hutchinson (1972). Imperfection sensitivity is discovered in most of these shell problems. A much earlier post- buckling shell analysis was made by Koiter (1956) on the basis of his general formulation (Koiter, 1945) in terms of displacements. The alternative stress- strain-displacement approach appears to make shell calculations somewhat easier.

E. INITIAL IMPERFECTIONS

The results of Eqs. (4.43) and (4.46) for the effects of initial imperfections remain valid, but now some insight into the magnitude of the parameter p in these formulas will be obtained. It is appropriate, in many problems, to assert that in the presence of an initial imperfection ii = zb, the strain- displacement relations are

(5.67) giving the strain t. of the imperfect structure in terms of the additional displacement u and the old function ~ [ u ] of the perfect structure. Similarly, the generalized shortening is modified to

A = A[u + E ] - A[.]. (5.68)

The stress-strain relation is o = o[Z], and so the Frechet derivative with respect to u of the modified potential energy 4 is

- E = e[u + U ] - t;[U]

4’ = $’ + $’ = ~ [ C ] C ’ [ U + E ] - I,A’[u + El.

Since 4‘ = o[F]i:’[u] - iA‘[u], where c = c[u], this identifies $’ 6 u as $’ 6u = ~ [ F ] F ’ [ u + U ] 6 u - o[c]c’[~r] 6~ - ~ A ’ [ u + G ] + E.A[u].

Taking the Frechet derivative with respect to U then gives $’*L 621 = ~ ’ [ Z ] { E ’ [ U + Ti] - c ’ [ L ~ ] ) L E ’ [ u + L I ] 6Li

(5.69)


- and evaluation at u = LL, , u = 0, 6u = u , gives

(5.70)

if 6u = U, which is an acceptable choice for 6u if U is assumed to be an admissible displacement shape. Hence (5.70) reduces to

$;*ilul = -U1i’U

and, from (4.39), we find [ I = -oli’q)L,~:ul. . 2 (5.71)

With &‘u; = D given by (5.23) or (5.24) we can compute p directly in terms of the generalized stress, strain, and displacement quantities of the present development. Results equivalent to (5.71) were derived by Fitch (1968) under assumptions (i)-(iii), which are not necessary for its validity.

But now a substantial simplification results if we have a linear prebuckling state [assumption (iv)], and take U = u,-that is, we consider the influence of an imperfection in the shape of the buckling mode. For then, by (5.31), A, &‘u: = -ol t:l, and, since i b = 0, we have t ’ u , = c; = c,. Con- sequently, p = l ! Accordingly, we have the remarkable result that, quite generally for structures having linear fundamental paths and imperfections in the shape of the buckling mode, the results for imperfection sensitivity given by Eqs. (4.43) and (4.46) become precisely those found for the simple model in Eqs. (2.9) and (2.10).

Let us now return to (5.69) and calculate $b*U 6 u for a linear prebuckling state, with U = 2i1. Setting u = u o , ii = 0 gives -

I,//b*iil 6 ~ 1 = o:(c~ - k ’ ) i ~ , & OU + Ooe624, (SLI - A A ~ U , 611,

0 , I and since, for a linear prebuckling state, ob = u:, c = c0 = t;:,

cro = (A/A,)cr,, A; = A:, this reduces to

$b*Ul 6u = -(jL/2,)ol&: 6Ll (5.72)

with the help of (5.13). But note also that, from (5.25)

~ : U I 6 u = 6,&‘ul 6~ - A:L/~ 6u

= (l/Ac)[oC~~ul 6u - ,&A:ii, 6u] = -(l/A,)cr,&; Su. (5.73)


Consequently, I&*Ul 6u = /&u, 6u

4;v fsu + (1- - AC)&l: 6u + . ’ . + +giP 6u + ...

(5.74) and the equation of equilibrium (4.32) for the imperfect structure can be expanded into

+ +4$;z3 6 u + .. . + 624 + ” ’ = 0, (5.75) where now u = (A/A,)u, + T. It is tempting to execute an approximate solution of (5.75) for a relation among A, u, and that is uniformly valid for both 2 and ( I - A,) small. Note that for 5 and A small, linearization with respect to Z and gives

& 5 6 u + (i - nC)4;v6u + A&$:u, 6 u = 0,

and it is evident that the solution for 1: is u = ( u , , where

< = &/(A, - A). (5.76) So, for A l # 0, use V = (u l in a Galerkin solution of (5.75), setting 621 = u l , and keeping terms of order 4, <’, and z. This gives

(A, - A)< + A, t2 = i5 , (5.77) which, with p = 1 , agrees with (4.41) for A near A,, and with (5.76)for A, and hence (, small. One might therefore expect that a calculation of 34 on the basis of (5.77) would be more reliable than one based on (4.41). The maxi- mization of A by use of (5.77) leads to

[ I - (A5/AC)l2 + 4(A1/i,)(A\/ILC)~ = 0. (5.78) This is in asymptotic agreement with (4.43) for p = I and sufficiently small z, and, furthermore, provides the mathematically palatable result /2, -+ 0 for 5 + m, as opposed to the obvious breakdown of (4.43) [and (2.9)] for finite values of 5. Nevertheless, i t must be remarked that the superiority of (5.78) over (2.9) cannot be established rigorously without a study of the effects of higher-order terms missing from (5.77).

If A 1 = 0, an approximate Galerkin solution of (5.75) based on 1: = < u l + t 2 u 2 gives

(A, - A)< + A 2 5 3 = iz , (5.79) where terms of order higher than t3 and 5 are dropped. The calculation for I I , now gives, for i2 < 0,

[ i - (is/~L)]3’2 - $ ~ 5 ( - ~ ~ ~ / i , ) ~ ’ ~ ( 4 / i ~ ) 1 = 0. (5.80) Again, this result has a more reliable look about it than Eq. (2.10), with which it agrees asymptotically, but no proof is available.

58 Bernurd Budiunky

VI. Mode Interaction

A. SIMULTANEOUS BUCKLING MODES; POST-BUCKLING ANALYSIS

Suppose that the solution of the eigenvalue problem (4.8) for bifurcation yields several linearly independent eigenmodes u\'), up) , . . . , u\~), all associated with the lowest eigenvalue ;L, . Without loss of generality, these modes can be orthogonalized to each other in the form

for i # j . (6.1 1 & q ) u y ) = 0

Assume, in generalization of the earlier condition &'u: < 0, that &'(iiY))' < 0 for each i. It will, of course, also be true that

(6.2) (/guy)uy) = 0

for all i,j, since the bifurcation equation &'@ ~ S L I = 0 must hold for each i. It is appropriate now to expand the displacement along a particular post-

buckling path into N

11 = .()(A) + t 1 llizly) + i"2L12 + . . . , (6.3) i = 1

where

(u,, , C viuY)) = 0 for n 2 2, (6.4)

and the v i are fixed numbers, satisfying z= I J ~ = I , that describe the relative initial contributions of the various modes. I t continues to be convenient to normalize each uy) according to

(6.5) 1 ) .y 1 1 = (UP), L l y ) *;2 = 1,

and (,L g) may (but need not) be chosen to coincide with -4p,fi~. The magnitudes of the i~~ cannot be chosen arbitrarily; as will be shown, they must conform to requirements imposed by equilibrium. Thus, replacing u1 in Eq. (4.19) by z=, v i uy) gives

+ c 3 { . ..) 6u + = 0. (6.6)

Setting isu successively equal to u y ) (i = 1, 2, . . . , N ) i n the coefficient of c2

Buckling Behacior of Elustic Struct~ i res 59

provides the simultaneous nonlinear equations

where

and A . = -&“(u‘;”)2. (6.9)

These equations must be solved for the vi’s (subject to the constraint c = 1 ) and the associated A 1 . There may be many essentiaIly different sets of solutions (in addition to trivial duplications obtained by changing the signs of A 1 and the vi’s), and they would correspond to a variety of possible starting behaviors of different bifurcated paths.

It is easily shown that unless all the uijk’s vanish there is at least one solution with A # 0.t If uiii = 0 for each i , there are no uncoupled solutions corresponding to nonzero A l , but unless all the other u’s vanish too, there will nevertheless be one or more asymmetric bifurcations, involving two or more modes, with attendant imperfection sensitivity. This kind of mode interaction occurs in approximate solutions for buckling of cylinders under axial compression (Koiter, 1945) and of spheres under external pressure (Hutchinson, 1967b) in both of which problems many simultaneous buckling modes are available. These structures are notorious for their extreme imperfection sensitivity, having snapping loads that, in practice, are usually less than half the bifurcation load.

In the case of a nonzero A , , the generalized shortening that occurs during a coupled-mode bifurcation can still be computed on the basis of Eq. (4.53) except that u1 is, again, replaced by vi uy’. Since &‘(c vi u‘ ; : ’ )~ = -c Ai v?, this gives

N

A - A. = it2 c A,.,’, (6.10) i = 1

and since A - A, = A , 5 we get

i/Ac = 1 + (A1/Ac)[2(A - A,)/C Aiv~]’i2(sign At) (6.11)

as the proper generalization of Eq. (4.54) for the single-mode case. If the assumptions (i)-(iv) of the previous section hold, it may be conven-

ient to formulate Eqs. (6.7) in terms of stress, strain, and displacement. We

? A theorem of Bczout is quoted in Sewell (1968) and Johns and Chilver (1971) to theeffect that if (6.7) is not degenerate (with infinitely many solutions) there are at most 2N - 1 essentially different real solutions, and at least one solution.

60 Bernard Budianskv

have [see Eq. (5.25)]

( Y u u w = dE’uC)f l :W + O‘C’I)R”UW + 0’S‘””UU

so that, with c: uy) = &), a: di ) = oy), the coupling coefficient (6.8) is

U i j k = 3[hijk + bkij + b j k i ] , (6.12)

where

bijk = d;)c:uyl/p. (6.13)

Also, Eq. (6.9) for A i gives [see (5.31)]

A i = - ( I / / ~ , ) [ o , c ~ ( u ~ ) ) ~ - A,AP(uP’)’] = ( l /Ac)d ; )~y) . (6.14)

In addition to (or in the absence of) asymmetric bifurcations, there may also exist symmetric bifurcations involving several simultaneous modes. The expansion (6.3) would remain appropriate, but then the admissible combinations of vi ’s would have to be determined from equations governing u2 and A 2 that follow from Eq. (6.6). So far, the need for such calculations has not been apparent in any specific problems of practical interest, and this procedure will not be pursued here. [See Johns and Chilver (1971) and Sewell (1970) for some recent studies for finite-dimensional systems.]

B. INITIAL IMPERFECTIONS

A thoroughgoing discussion of the effects of initial imperfections in the case of simultaneous bifurcation modes would involve the exploration of a variety of imperfection shapes that provoke responses from the separate coupled modes in different ways. Indeed, the possibility exists that certain special imperfections would not lead to limit point snapping, but may instead induce bifurcations at loads less than the critical load of the perfect structure (Koiter, 1963b). It must be expected, however, that this situation, while it may sometimes be attractive analytically, would be exceptional, and limit point snapping would generally occur. Furthermore, as a demonstra- tion of imperfection sensitivity, i t will suffice here to study the simplified situation contemplated in the previous section in which assumption (iv) of a linear prebuckling state is met. In this case, let us contemplate the effect of an initial imperfection in the shape of the initial coupled bifurcation of the perfect structure. It then becomes apparent that the situation is indistinguishable from the single-mode case if only the bifurcation mode u1 is replaced by viuY) . That is, if we choose the imperfection as

N

i = 1

(6.15)

Buckling Behavior- of Elmstic Structures

the old asymptotic result (2.9) namely

)%/Ac = 1 - 2(A, Q&)1'2 ( A l < < 0)

61

remains valid. Furthermore, the modified, possibly better, equation (5.78) is also applicable.

C. NEARLY SIMULTANEOUS MODES

I t may happen that the eigenvalue problem for bifurcation has several solutions u'f) corresponding to critical loads it) ( i = 1, 2, . . . , N) that, while not coincident, are close to each other. This occurs in the sphere under external pressure (Koiter, 1969) [when simplifications rendering the modes coincident (Hutchinson, 1967b) are not made]. Much interest has also arisen in two-mode problems associated with nearly coincident eigenvalues associated with local plate buckling and overall column or panel buckling (van der Neut, 1968; Koiter and Kuiken, 1971; Tvergaard, 1972). Clearly, a single-mode analysis for post-buckling behavior and imperfection sensitivity based on the lowest critical load would not generally provide the coupled- mode results in the limit as these loads do become coincident. A reasonable way to handle the problem is to execute a Galerkin solution based on the use of the approximation 11 = uo(,4) + I:=, ti uy), obtaining algebraic equations governing the &'s, A, and initial imperfections. Thus we demand that

(6.16)

for i iu = L I ~ ) (, j = 1,2, ..., N). Set U = zii, and expand (6.16) as follows:

4; 6 u + 4; c & L l y i iu + $(f$i(C till\') I ) 2 hu . + . . '

+ 1//'[0,0; A] 6 u + z&*i 6u + ... = 0. (6.17)

The first terms of each line in (6.17) vanish, and the omitted terms are of order <:, z2, and z<. Now

(#)guy' = ($;$,y + (A - j L " ' ) i J p i p + O ( i - .(0)2, (6.18)

where

$!);i = 4;;[uo; n 3 , 4ei = (d/",4)f$ei l A = A , .

Similarly

62 Bernard Budiansliy

for any ni, and so (6.17) becomes

x @ : n l ~ ~ Y ) ~ ‘ / ’ CSU + . . . + &!/I* cm ir ~ L I + . . . = 0 (6.19)

where the vanishing terms $:i uy) du have already been dropped and the omitted terms are of order ti tjCr, (A - 2 i ) 2 t ( j ) , (i - i ( m ) ) < i t j , z2, ( 2 - A\m))z , and ztj. It is now best to drop back to the special case of a linear fundamental state, and write

i“;

thus restricting the imperfection modes. Note that by (5.25)

(6.20)

to a linear combination of the buckling

~ Y ) E ( / ) = O for i # j and

(6.23)

C (6.24) (CT;)/p)c:Lly)LI‘l” - ~ ~ t ~ ( 1 ) ~ ( / ) = $ ; l I ( l ) u ( J ’ ) = 0

Equations (6.23) and (6.24) supply various forms of the orthogonality relations that must hold among bifurcation modes associated with different critical loads when the prebuckling state is linear. Now by (5.74) and (6.20):

I 1

(6.25)

Hence, putting (su = u : ~ ) , and choosing nz = k gives the simultaneous equations

i N N

Buckling Behavior of Elastic structure.^ 63

where Ak = - & ( u \ ~ ) ) ~ is independent of A, and

Gj j k = $rk “y)L/(/’/,l\k). (6.27) [If the additional conditions r ~ ’ = E‘” = A“‘ = 0 are met----~and they will be if (i)-(iii) hold-@: becomes independent of 2, and a i j k is the same as the aijk given by (6.12).]

It is now a matter of solving (6.26), perhaps numerically, for equilibrium paths relating 2. to the tk’s for various assumed zk’s. This setup is uniformly valid for vanishingly small differences among the I$), giving results for snapping loads that approach those implied by the analysis for coincident loads.

It may sometimes be desirable to depend on a somewhat improved set of equations that incorporate individual L I ~ ) contributions associated with some of the modes. To this end, the following procedure is suggested. Calcu- late u$’ for a given mode, treating the problem as in the previous sections for isolated bifurcation. Then use the expression

N N

in a Galerkin solution of 4’ 6u + I)‘ 6 u = 0, making approximations similar to those in the above analysis, but keeping terms of order t i t i S k . Further details will not be explored here.

ACKNOWLEDGMENTS

This work was supported in part by the Air Force Office of Scientific Research under Grant No. AFOSR-73-2476, in part by thc National Aeronautics and Space Administration under Grant No. N G L 22-007-012. and by the Division o f Engineering and Applied Physics, Harvard University. Helpful cornnients from J. W. Hutchinson and W. T. Koiter are gratefully Lick nowledged.

RE I- F RENCES

BIJUIANSKY. B. (1965). Dynamic buckling of elastic structures: criteria and estimates. Proc. 1171. Cot!/: Dyfitrmic S/trhi/il!* of Strirctirrc,,s, Northwestern Univ.. Evanston, Illinois, pp. X3- 106.

BCJDIANSKY, B. (1969). Postbuckling behavior of cylinders in torsion. Proc. IL ‘TAM Symp. Theory Thiri Shrl ls . Cop<whtrgqm, 2ml pp. 2 12-233.

BCJDIANSKY. B., and AMAZIGO. J. C . (196X). Initial post-buckling behavior of cylindrical shells under external pressurc. J . ,Math. P h y ~ 47, 223 235.

BUDIANSKY. B., and HLJT(‘HINSON, J . W. (1964). Dynamic buckling of imperfection-scnsitive structurcs. Proc,. / 1 i / , Coti<qr. Appl. Mech., Mutiich, X I , pp. 636 65 I.

B ~ J D I A N S K Y . B., and HUT(.HINSON, J. W. (1972). Buckling of circular cylindrical shells under axial compression. In “Contributions to the Theory of Aircraft Structures ” (van der Neut Anniversary Volume). pp. 239-260. Delft Univ. Press, Delft.

CARKII:K. G. F. (1947). On the buckling of elastic rings. J . Mtrrh. I’hi~s. (Ctrtiihr.it/qc. hftrs\.) 26 94 103.

64 Bernurd Burliuizsky

COHEN, G. A. (1968). Effect of a nonlinear prebuckling state on the postbuckling behavior and imperfection-sensitivity of elastic structures. A I A A J . , 6, 1616-1620. See also Cohen, G. A.

DANILLSO%, D. A. (1969). Buckling and initial postbuckling behavior of spheroidal shells under

FITCH, J . R. (1968). The buckling and postbuckling behavior of spherical caps under con-

FITCH, J. R., and BUDIANSKY, B. (1970). The buckling and postbuckling of spherical caps under

H U T C H ~ N ~ O N , J. W. (l967a). Initial postbuckling behavior of toroidal shell segments. I n t . J .

HUTCHINSON, J. W. (1967b). Imperfection sensitivity of externally pressurized spherical shells.

HUTCIIINSOK, J. W. (1968). Buckling and initial postbuckling behavior of oval cylindrical shells under axial compression. J . A p p l . Mech. 35. 66 -72.

H U T ~ H I N S O N , J . W., and AMAZIGO, J . C. (1967). Imperfection sensitivity of eccentrically stiffened cylindrical shells. A l A A J . 5, 392-401.

Hu~r(.HINSON, J. W., and F K A L J ~ N IHAI., J . C. (1969). Elastic postbuckling behavior of stiffened and barreled cylindrical shells. J . App l . Mech. 36, 784-790.

HUICHINSON, J. W., and KOITIX W. T. (1970). Postbuckling theory. A p p l . Mech. Rer. 23, 1353~ 1366.

JOHNS, K. C., and CHILVEK, A. H. (1971). Multiple path gencration at coincident branching points. I n t . J . M ~ J . Sc,i. 13, X99-910.

JOLQIJET, EMILE (1930). Amortiscment des oscillations et stabilite seculaire. Proc. I n ! . Conqr. A p p l . Meek., 3rd Stockholm, Vol. 3, pp. I 1 1 - 1 13.

KOITFK, W. T. (1945). On the Stability of Elastic Equilibrium (in Dutch). Thesis, Delft Univ., H. J . Paris, Amsterdam; English trans]. (a) NASA TT-FIO, X33 (1967) (b) AFFDL-TR-70- 25 (1970).

KOITEK, W. T. (1956). Buckling and postbuckling behavior of a cylindrical panel under axial compression. NLR Rep. No. S476. Rep. Trans. Nat. Aero. Rcs. Inst., Vol. 20. Nat. Aero. Res. Inst., Amsterdam.

KOITEK, W. T. (1963a) On the concept of stability of equilibrium for continuous bodies. Proc,. K o n . Neil. A k ~ ~ i l . Wc,tensch. 666, 173-177 (1963a).

KOITCK, W. T. (1963b). The effect of axisymmetric imperfections on thc buckling of cylindrical shells under axial compression. Proc. K o n . Neil. Akad. Wetcwch. B66. 265.

KOITEK, W. T. (1965a). O n the instability of equilibrium in the absence of a minimum of the potential energy. Proc. Kon. Neil. Akad. Wefemch. 668, 107-1 13.

KOITFK, W. T. (1965b). The energy criterion of stability for continuous elastic bodies. Proc. Kon. Neil . Akuil. We/en.sch. B68, 178-202.

KOITEK, W. T. (1966). Purpose and achicvements of research in elastic stability. PJYJC. Tech. COJ$, Sot,. EJUJ. S(,i.. 4 / h , North Carolina State Univ., Raleigh, N.C.

Koi-rm, W. T. (1967). A sufficient condition for the stability of shallow shells. Proc. Kon. A ! d . Wrtenach. B70, 367 375.

KOITEK, W. T. (1969). The nonlinear buckling problem of a complete spherical shells under uniform external pressure: 1. 11. 111 and IV. Proc. KOJJ. h’cd. ,A/iorl. I ~ ’ c ~ ~ v I , w / I . B72. 40-123.

KoirkR, W. T. (1972). “Stijfheid en stcrke I-Grondslagen.” Schcltema & Holkema. Haarlem. KOITER, W. T., and KUIKEN, G . D. C. (1971). The interaction between local buckling and overall

buckling on the behavior of built-up columns. WTHD 23, Laboratorium voor Technische Mechanica, Delft.

(1969). A l A A J . 7. 1407-1408.

pressure. A I A A J . 7, 936-944.

centrated load. I ~ i t . J . Soli i ls Strue/. 4. 42 1-446.

axisymmetric load. A I A A J . 8, 686-692.

Sol ids Srrucr. 3, 97- I 15.

J . A p p l . Mech. 34, 49-55.

Buckling Behuvior qf Elastic Structures 6 5

LIUSTIXNIK, L. A., and SOHOI.EV, V. J. (1961). ’‘ Elements of Functional Analysis.” Frederich

MASUR, E. F. (1973). Buckling of shells-general introduction and review. ASCE National

RALL, LOUIS B. (1969). “Computational solution of nonlinear operator equations.” Wiley, New

REISSNER, ERIC (1953). On a variational theorem for finite elastic deformations. J . M ~ t h . Phys.

SANDI:KS. J. L. (1963). Nonlinear Theories for Thin Shells, Quart. A p p l . M o l h . 21. 21-36. SEIIX, P. (1972). A reexamination of Koiter’s theory of initial postbuckling behavior and

imperfection sensitivity of structures. Sj~jp. Thin Shell Struc‘lures. Califorilia Institute of Technology, June 1972, I n “Thin-Shell Structures” (Y, C. Fung and E. E. Sechler. eds.). Prentice-Hall, Englewood Cliffs. New Jersey, 1974.

S ~ W E I . I . . M. J . (1968). A general theory of equilibrium paths through critical poinls, I , 11. Proc,.

SEWELL, M. J. (1970). On the branching of equilibrium paths. Proc. Roy. SOC. A 315,499-517. TIIOMPSON, J . M. T. (1969). A general theory for the equilibrium and stability of discrete

conservative systems. Z. Math. Phys. 20, 797-846. See also J . M. T. TFIOMPSON A N D

G . W. H L ” , (1973). “A General Theory of Elastic Stability.” Wilcy, New York. TVEKGAARD, VIGGO (1972). Influence of post-buckling behavior on optimum design of stiffened

panels, Rep. No. 35. Danish Center for Applied Mathematics, Technical University of Denmark.

VAINIERG, M. M. (1964). I‘ Variational Methods for the Study of Nonlinear Operators.” Holden-Day, San Francisco, California.

V A N IFR NEUT, A. (1968). The interaction of local buckling and column failure of thin-walled compression members. Proc. I n / . Conqr. Appl . Mech., 12th, Siurlford Linir., 1Yh8.

Ungar, New York.

Structural Engineering Meeting, 1973, San Francisco, California. Preprint 2000.

York.

(Cunihridgqr, Muss.) 32, Nos. 2-3, 129- 135.

Ro,v. SOC. A 306, 201-223, 225-238.


Plastic Buckling

JOHN W . HUTCHINSON

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I I . Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . .

A . Discrete Shanley-Type Model . . . . . . . . . . . . . . . . . . €3 . Continuous Model . . . . . . . . . . . . . . . . . . . . . .

111 . Bifurcation Criterion . . . . . . . . . . . . . . . . . . . . . . . . A . Criterion for Three-Dimensional Solids . . . . . . . . . . . . . . B . General Bifurcation Criterion for the Donnell-Mushtari Vlasov Theory

of Plates and Shells . . . . . . . . . . . . . . . . . . . . . . C . Discussion of Bifurcation Predictions Based on the Simplest Incremental

and Deformation Theories of Plasticity . . . . . . . . . . . . . . IV . Initial Post-Bifurcation Behavior for Donnell-Mushtari Vlasov Theory . .

A . General Theory . . . . . . . . . . . . . . . . . . . . . . . . B . Two Column Problems . . . . . . . . . . . . . . . . . . . . . C . Circular Plate under Radial Compression . . . . . . . . . . . . . D . Effect of Initial Imperfections . . . . . . . . . . . . . . . . . .

A . Column under Axial Compression B . Circular Plate under Radial Compression C . Spherical and Cylindrical Shells References . . .

V . Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 70 70 76 86 86

93

97 105 106 119 123 127 132 132 135 137 141

I . Introduction

Plastic buckling has received a great deal of attention in the past and progress in the subject has been slow but steady . The column under axial compression has been studied more than any other single structure and its history is well known . The turning points in this history are marked by the work of Considere (1891) and von K a r m h (1910) to obtain the load at

61

68 John W. Hutchinson

which the straight column becomes unstable. Then almost 40 years later, Shanley (1947) explained, by way of a simple model and careful experimentation, the significance of the tangent-modulus load of Engesser (1889) at which the straight column would start to deflect laterally under increasing load. It was recognized that the tangent-modulus load is the lowest possible bifurcation load; the straight configuration loses its uniqueness at this load but not its stability (von Karman, 1947). Shortly thereafter Duberg and Wilder (1952) studied the post-buckling behavior of a column model. They showed that the inclusion of imperfections and realistic stress-strain behavior gave rise to a maximum support load that was closely approximated by the tangent-modulus load, as Shanley had conjectured and in accord with existing experimental data.

Extensive testing of plates and to some extent shells was carried out in the late 1940’s and early 1950’s and solutions for the lowest bifurcation load, analogous to the tangent-modulus load for columns, were obtained for many cases of interest [see, for example, Bijlaard (1949) and Gerard and Becker (1957)l. A major obstacle to further progress surfaced in this period. It was discovered that bifurcation loads calculated using the simplest incremental, or flow, theories of plasticity consistently overestimated buckling loads ofplates and shells obtained in tests. Calculations based on the less respectable deformation theories of plasticity gave reasonably good agreement with test results. It was found that the difficulty could be largely overcome if the smooth yield surface of the simplest flow theories was discarded. In fact, the bifurcation load predictions of a deformation theory could be justified rigorously in nearly all cases by establishing the connection between the deformation theory and a more sophisticated incremental theory which develops a sharp corner on its yield surface (Batdorf, 1949). Yet basic experiments on the multiaxial stress-strain behavior of typical metals failed to show conclusively that corners on yield surfaces do actually develop. Almost 20 years later this state of affairs remains at essentially the same impasse.

On the theoretical side, Hill (1956, 1958, 1959, 1961) placed the bifurcation criterion for elastic-plastic solids on a firm mathematical foundation which embraces solids characterized by smooth or cornered yield surfaces. His formulation applies not only to bifurcation under compressive loading-that is, problems which are loosely categorized under the heading of plastic buckling-but also to less thoroughly explored problems such as necking which can involve bifurcation in tension.

Thus, except for the difficulty in identifying an adequate plasticity theory, bifurcation theory for compressive loadings is reasonably well understood. A recent survey by Sewell (1972) includes an organized bibliography of much of the enormous amount of work on plastic buckling. Relatively less is known about post-bifurcation behavior and imperfection sensitivity in the

Plastic Buckling 69

plastic range. What is known comes largely from tests and a relatively few model studies such as that of Duberg and Wilder (1952). This can be contrasted with the situation which now prevails for elastic buckling where a great deal is known about these matters and where a general theory of initial post-buckling behavior and imperfection sensitivity is available (Koiter, 1945, 1963a).

The importance of post-bifurcation considerations in the plastic range can immediately be appreciated in noting that in almost all cases of compressive loading the lowest bifurcation occurs under increasing load in the sense of Shanley, even for a structure which bifurcates unstably in the elastic range. Furthermore, this occurs even though material nonlinearity in the form of decreasing stiffness with increasing deformation contributes an additional destabilizing influence to the geometric nonlinearity already present. By itself, the fact that the initial slope of the load-deflection relation is positive can be highly misleading since recent work has shown that the magnitude of the initial curvature of the load-deflection curve is usually infinite (Hutchin- son, 1973a), and the maximum support load of the perfect structure may be only very slightly above the lowest bifurcation load. It is also significant that even the column under axial compression, which has a fully stable post- buckling behavior in the elastic range, is appreciably affected by small imperfections in the plastic range.

The main accent in this article is placed on post-bifurcation and imperfection-sensitivity aspects of plastic buckling. A combination of model studies and analytical and numerical work has been drawn on to illustrate as wide a range of behaviors as possible.

In Section 11 a simple two degree of freedom model is used to introduce post-bifurcation behavior and a second model brings out some of the features peculiar to the behavior of continuous solids and structures. Hill’s bifurcation criterion for a class of three-dimensional solids is given in the first part of Section 111 and is then applied to a widely used theory for plates and shells, the Donnell-Mushtari-Vlasov (DMV) theory. Following specification of the bifurcation criterion, a detailed commentary is given on the extent to which bifurcation predictions for plates and shells depend on the plasticity theory used with particular focus on the differences between predictions based on the simplest deformation and incremental theories. Much of what will be said was common knowledge in the 1950’s but seems to be less widely appreciated now.

A general treatment of the initial post-bifurcation behavior of plates and shells is given in Section IV within the context of the DMV theory. The theory is illustrated by applications to several column and plate problems. A discussion of some of the effects of imperfections is also given. The article ends with a selection of numerical results for columns, plates, and shells.

70

11. Simple Models

John W . Hutchinsmi

A. DISCRETE SHANLEY-TYPE MODEL

The first model to be examined is similar in all respects to Shanley's (1947) model of plastic column buckling except that i t also is capable of illustrating the behavior of a highly imperfection-sensitive structure. In this respect it is similar to the model of von Karman et al. (1940) for the elastic buckling of imperfection-sensitive structures. The model succumbs to an elementary analysis yet still retains many of the essentials of plastic buckling. The results given in this section are taken from Hutchinson (1972).

As shown in Fig. 1, the rigid-rod model has two degrees of freedom: the

IP -=i L

FIG. 1. Rigid-rod modcl with two degrees of freedom.

downward vertical displacement u and the rotation 0. An initial rotation from the vertical in the unloaded state is identified as the imperfection and is denoted by 0 so that the total rotation from the vertical is 0 + 0. An increment in the compressive force in either of the two support springs depends on whether plastic loading or elastic unloading occurs according to

F = Eti; for F = F""" and F > 0,

P = EL for F < F""" or F = F""" and < 0, (2.1)

where e is the contraction of the spring. Initial yield occurs for F = F , and F""" is set to be F , at the start. The tangent modulus E, is taken to be constant. Geometric nonlinearity is incorporated into the model only through the nonlinear elastic spring which develops a force K ( 0 ) =

k , L202 + k2L?03 + . . . under rotation with the sign convention shown in Fig. 1. Equations of equilibrium and strain displacement are

F , + F2 = P, (2.2)

Plastic Buckling 71

( F , - F, )L + PL(0 + e ) + LK(i1) = 0, (2.3)

c 1 = u + LO and E~ = u - LO, (2.4)

where the subscripts denote springs Nos. 1 and 2.

the elastic range the load-rotation behavior of the model is found to be When the parameters of the model are such that buckling takes place in

( P , - P)H - K(f1) = PO, (2.5)

where P, = 2EL2/L is the bifurcation load of the perfect model. If k , # 0 the bifurcation point is asymmetric and for k , 8 > 0 the maximum support load of the slightly imperfect model is given by the asymptotic formula

= 1 - (2Zk , H/E)1’2 + ’ . . .

Pmax/Pc = 1 - (3)(Lk282/E)”3 + . . . .

(2.6)

(2.7)

If the load-rotation behavior is symmetric in 0 then k , = 0. If k , > 0, then

These two cases, depicted in Fig. 2, illustrate the two most common occur-

A S Y M M E T R I C B I F U R C A T I O N P O I N T S Y M M E T R I C B I F U R C A T I O N POINT ( k , > O ) ( k , = O , k , > O )

FIG. 2. Elastic post-bifurcation behavior and imperfection sensitivity of the models.

rences of imperfection sensitivity in elastic structures. When K = 0 (the Shanley model) the maximum support load of the model is always P, and in this sense approximates the small deflection behavior of an axially compressed column.

1. Belmvior of the Per f k t Model

Throughout this article a subscript or superscript c will be reserved to denote quantities associated with the lowest possible bifurcation point. The lowest bifurcation load of the model in the plastic range is given by the tangent-modulus formula P, = 2E, L?/L. Bifurcation occurs at P, under increasing load as discussed by Shanley. When the model has an asymmetric

72 John W . Hutchinson

bifurcation point with, say, k , < 0, then the load increases linearly with positive values of 8 in the neighborhood of the bifurcation point even in the elastic range. In the plastic range, one can readily show that both springs will continue to deform plastically in some finite range following bifurcation if

* k l L/(2E,) > 1 and b 5 0. (2.8) Thus, for example, if k , L/(2E,) > 1 the geometric nonlinearity itself is sufficient to ensure that P increases with negative 0 sufficiently rapidly to ensure that neither support spring unloads. Of the two bifurcation branches emanating from such a nonsymmetric bifurcation point, the above- mentioned branch is of inherently less interest than the opposite-signed deflection. On this latter branch the geometric nonlinearity has a destabilizing effect. We limit consideration to k , 2 0 and t? > 0.

For this case spring No. 1 continues to load and No. 2 undergoes elastic unloading or neutral loading (i.e., C = 0) following bifurcation. With the bifurcation load denoted by Pbif, Eqs. (2.1) to (2.4) give

In this formula P,, is the reduced-modulus load of von K a r m h (1910) where bifurcation takes place with no first-order change in load which is given by

- 2p , 4E, L2 p r m = ( 1 + E , / E ) i - (1 + EJE)

(2.11)

Bifurcation can take place at every load in the range P, I Pbif I P,, and for Phi, < P,, it occurs under increasing load as can clearly be seen from (2.10). A simple formula for the maximum support load cannot be obtained for a general function K(0) . However when K = k , L202 the maximum support load Pyx associated with bifurcation at the tangent-modulus load P , is given by the equation

(Prm - P y ) 2 - 4 k , (2.12)

If the geometric nonlinearity is strong in the sense that k , LIE, 9 1 then

Plastic Bucklirig 73

FIG. 3. Behavior of the model in the plastic range. (a) Shanley model, k , = k , = 0; (b) Model with a destabilizing geometric nonlinearity, k I > 0. Dashed-line curve corresponds to bifurcation at the reduced-modulus load P, , , . ( A ) - onset of elastic unloading: ( ' ) - maximum load. The lowest bifurcation load is P , .

When K = 0, P + P,, for large 0 as can be seen from (2.9) and Fig. 3a. This is also similar to the behavior of the model of Duberg and Wilder for their constant tangent-modulus calculations. Even when the tangent modulus is constant, the present model has a maximum load which falls below the reduced-modulus load if the geometric nonlinearity has a destabilizing effect. A similar effect is also observed in the model studies of Sewell (1969, Augusti (1968), and Batterman (1971). A strong geometric nonlinearity results in a maximum load which is only slightly above P, as can be seen from (2.13) and Fig. 3b.

2. Effect of Initial Imperfections

In the presence of an initial imperfection 0 > 0 the load-rotation behavior of the model is more complicated; however a complete analysis can still be carried out for the case of the simplest geometric nonlinearity, K = k , Lz02. There are three distinct sequences of loading and unloading which can take place depending on the magnitude of 0. We will first consider the case for which 0 is sufficiently small such that the resulting formula will be valid in the limit as 0 vanishes.

In this case it is found that there are four steps to the loading history which must be treated separately in the analysis. With the first application of load both springs are elastic. Next, spring No. 1 starts to deform plastically and is followed by No. 2 at a slightly higher load. With the load still rising spring No. 2 starts to unload at a value of P and H denoted by and 0. From this point on spring No. 1 continues to load while No. 2 responds elastically. The maximum load is attained at a point (Pmax, H * ) following the onset of

74 John W . Hutch iiwon

elastic unloading at (p, (3). The expressions for 8 and are given by

(2.14)

(2.15)

and

P / P , = 1 - 8 - [l + (k,L/E,)](e + 0) (2.16)

also.

0" = 8 + 6 + 0 + -3){[l + (2E,/k,L)]1'2 - I } (2.18) ( E + E l and

Pax = Prax - 2k, L2[e + (8 + 0)([1 + (2E,/k, L)]'" - l)], (2.19)

where Pyx is the maximum load of the perfect model given by (2.12) for bifurcation from P,.

If k , = 0 then P + P,, for all values of 0 as shown in Fig. 3a. If k , > 0 then we can show that (2.19) reduces to

= P;""/P, - (2k, iB/E,)"* + O(0). (2.20)

A small imperfection results in a reduction in the maximum support load which is proportional to the square root of its amplitude. The limit in which 8 + 0 is the bifurcation response of the perfect model for bifurcation from P,. Asymptotically the reduction in the buckling load according to (2.20) is precisely of the same form as is the analogous formula (2.6) for the elastic model, except that the effect of a small imperfection is ~?agn$fied by an amount E / E 1 compared with the elastic case.

For larger imperfections the maximum load can be reduced to a sufficiently low level such that the force spring No. 2 does not reach F,, and (2.19) ceases to be valid. In this sequence of loading spring No. 1 yields at a value of P just under 2F, and the model deflects readily under a slight increase in load, helped along by the geometric nonlinearity, until the maximum support load is reached. The maximum load is not too different from the initial yield load of the perfect model, i.e.,

Prn'lx E 2F, . (2.21)

Plastic, Buckling 75

Load-rotation curves showing the effect of initial imperfections are shown in Fig. 3b.

For even larger values of the imperfection a third possibility may arise. Suppose the bifurcation load P, = 2E, L2/z is only slightly below the associated elastic load 2EL?/2. Then if is large enough neither spring becomes plastic before the maximum load occurs and the elastic result holds. For K = k , L202 the elastic result is given exactly by the equation

( 1 - Pmax/Pc)2 = (2k1 Z a / E , ) P m a X / P , . (2.22)

Figure 4 shows the relation between P'nax and 8 for such a case. On the portion of the curve A-B the strong imperfection sensitivity associated with (2.20) is seen; on B-C (2.21) holds; and on C-D the elastic result (2.22) pertains.

I

4 012 014 016 018 Ib 1.; 114

2k1r - - 8 E, L~

FIG. 4. Ratio of maximiiin support load to maximum support load of perfect model as ii function of the imperfection amplitude. This case is an illustration of the effect of an imperfection in reducing the maximum load to the point where the structure no longer deforms plastically prior to buckling. €,I€ = 2, k , LIE, = 10, crL/L2 = 0.525. From Hutchinson (1972) J . Appl . Mrch. 39, 155-162, with permission.

The implication of the model is that if a structure is highly imperfection sensitive in the elastic range then an imperfect realization of the structure may buckle elastically even though a perfect version would buckle in the plastic range. The converse situation is also of interest. Namely, what is the effect of initial imperfections on the maximum support load of a structure designed such that a perfect realization bifurcates at a load which is slightly below initial plastic yielding? We will look at this question in connection with the next model which incorporates a more realistic stress-strain relation.

76 Johrz W. Hutclzinson

B. CONTINUOUS MODEL

According to the Shanley concept, bifurcation occurs at the lowest possible load under increasing load in such a way that no elastic unloading takes place at bifurcation. This is manifest in the simple model of Section- II,A in that one spring continues to load plastically and the other undergoes

neutral loading (C = 0) at bifurcation and then unloads elastically immediately following bifurcation. Mathematically this is expressed by the fact that the contraction rate of the unloading spring is given by i: - -0 for small 0. It is clear that elastic unloading must start at the lowest bifurcation point in this sense. To see this, suppose that no elastic unloading occurred in some finite neighborhood of the bifurcation point. Within this neighborhood the response of the model would be identical to an elastic model with moduli E, . But this leads to a contradiction since a reversal in sign of the contraction rate will occur in the elastic model at bifurcation, except when 5 k , L/(2E,) > 1 and 0 5 0 as already discussed.

FIG. 5. Continuous model

In a contiriiious elastic-plastic structure it is also generally true that elastic unloading must start at bifurcation at at least one point in the structure, as will be shown. As the bifurcation deflection increases the region of elastic unloading expands in a continuous fashion in most problems. It is this feature which makes an analytic treatment of the initial post-bifurcation behavior of a continuous elastic-plastic structure considerably more difficult to carry out than for an elastic structure. The model discussed in this section is perhaps the simplest meaningful model which is capable of illustrating some aspects of the analytical character of initial post-bifurcation behavior. It is the continuum version of the model of the previous section and is shown in Fig. 5.

Plastic Buckling 77

This model is already sufficiently complicated that closed-form formulas analogous to those for the discrete model cannot be obtained. A representation of the behavior is obtained in the form of a perturbation expansion about the lowest bifurcation point. In this study the variation of the tangent modulus with increasing plastic deformation will also be taken into account. Most of the results for this model were previously given by Hutchinson (1973a,b). Here we will indicate an alternative method of analysis of the model which is very similar to the general method applied to columns, plates, and shells later in the article.

The model of Fig. 5 differs from the previous model only in that it is supported by a continuous distribution of springs. The contraction of a spring attached at a point x along the base is given by

E = u + xu. (2.23) The rate of change of the compressive force per unit length k is related to the contraction rate at any point by

:V = E , i for s = ,smax and S 2 0,

:F = Ei for s < smax or . y = = ~ " " ' ~ and S < 0, (2.24) where E, is taken to be a smooth function of c or s. (Note that E, and E have dimensions force per length' in the continuous model while in the discrete model they have the dimensions force per length.) Equations of vertical and moment equilibrium are

. L I 1'

P = 1 S clx and [Pi(U + a)]' + Ki = 1 Sx dx. (2.25a,b) * -1- . -1,

These may be replaced by the single variational principle of virtual work L

P6u + [ P q e + TI) + Ki]" = ). S6e dx, (2.26) . -1.

where he = 6u + x60. The elastic bifurcation load is P, = 2EL?/(3L) and (2.5) continues to

apply. Asymptotic formulas for the elastic buckling load are still given by (2.6) and (2.7) if k , is replaced by 3k,/L in (2.6) and k , by 3kJL in (2.7).

1. Behavior of' Perfect Model

The lowest bifurcation load in the plastic range is given by the tangent- modulus formula P, = 2EfL?/(3i), where ,!$ is the value of the tangent modulus at P,. Just as in the case of the discrete model no elastic unloading occurs in some range of positive or negative 0 if the magnitude of k , is sufficiently large, but otherwise elastic unloading starts at bifurcation with the occurrence of neutral loading at one point.

78 Johiz W . Hutchirison

First entertain the possibility that no elastic unloading occurs so that the model behaves as a nonlinear elastic model with variable moduli E,, i.e., the unloading branch of (2.24) is suppressed. The behavior of this cornpurison model is readily analyzed. Its initial post-bifurcation expansion is found to be

PIP, = 1 + u:O + a;OZ + ..., (2.27)

where

(2.28)

and we have expanded E, about the bifurcation point according to

E, = EF + ( S - ,s,)(dE,/d~), + +(.s - , s , )2(d2E,/d.s2) , + . . . . (2.29)

The superscript e is used to distinguish the initial slope of the elastic comparison model from that of the elastic-plastic model given below; a; figures prominently in formulas for the elastic-plastic model.

The contraction rate in the comparison model just following bifurcation is given by

(2.30)

For example, if k , is sufficiently large such that u; < - 3z/L and 4 < 0, then no reversal in sign of the contraction rate will occur in some finite range of negative 0. Over this range the behaviors of the comparison model and elastic-plastic model coincide. As already mentioned, the more interesting bifurcation branch has the opposite-signed rotation. From here on we will take k , 2 0 so that 5 0 and consider bifurcation under monotonically increasing 0. In this case the comparison model does not pertain.

For the elastic-plastic model the Shanley loading condition requires C 2 0 on 1 x 1 5 L which in turn implies

(2.31)

If the ineqziu/ity holds in (2.31) then C > 0 everywhere at bifurcation and by continuity implies that no elastic unloading will occur in some finite range of positive 0. By the same argument made for the discrete model. this would imply that the comparison model pertains with its initial slope u:. But since 0'1 5 0, (2.31) is contradicted and we must conclude that the equulitj. in (2.3 I ) must hold for bifurcation at the load P , . Elastic unloading starts at bifurcation in the sense that i-: = 0 at x = -L. I t follows then that

u/L = u,/L + h,H + ..., (2.32) PIP, = 1 + u , o + ... and

Plastic Btrckliny 79

where

u 1 = 3 z / L and h , = 1. (2.33)

The instantaneous position d of the boundary between the regions of elastic unloading and plastic loading occurs where i; = 0 so that from (2.23)

(1 = - dLf/dO. (2.34)

To obtain additional terms in the expansion (2.32) write

PIP, = 1 + a , a + u20’ +” + F(f l ) ,

u / L = tiJL + b , 0 + h, 0’ + O + U(O), (2.35)

where we anticipate that 0 < [j 5 1, and require that O- l -p (F , U) = 0, as 0 + 0. Now make the identification

( . ) = d( )/do, (2.36)

so that

PIP, = a, + ( 1 + [))a, 0” + F, d / L = -ii/L = -b , - ( 1 + p)h,O” - fr. (2.37)

Substitute these expansions into the principle of virtual work (2.26) noting that one can write

. L . d

. ~ S C dx = 1 ElC6r; dx + 1 ( E - E,)86r: du, (2.38) * i L - L . - I . * - I .

and from (2.29) and (2.32)

E, = E; + O(L + x)Ef(dE, /ds) , + .... (2.39)

Using (2.31) and (2.32) to eliminate the lowest-order terms in the principle of virtual work (2.26) one obtains

P,[(l + f l )a2@ + 63 6tr + [2P,La,O + 2k, L2zU + 60

(2.40)

The last term in the above equation arises from elastic unloading and

(2.41)

must be examined closely. From (2.33) and (2.37)

C/L = 1 + (,u/L) + ( 1 + f l )b2b + 6,


and i = 0 at x = - L at bifurcation as already noted. The last term in (2.40) vanishes as 0 + 0 since d --+ - L. To evaluate the lowest-order contribution

of this integral, introduce a stretched coordinate x chosen such that the position of the elastic-plastic boundary d is independent of 8 to lowest order in this new coordinate system. With the choice

*

* ,y = 0-D (1 + x/L)/[-(l + P)b211 (2.42)

the limits of integration on the integral run from x = 0 to x = 1 + O ( f )

where CI > 0. In terms of s,

* *

*

* &/L = O D ( 1 + [1)b2( 1 - x) + h. (2.43)

To lowest order,

1 . d . 1 * *

( E - Et)i 6c dx = O2"1 + /3),h$(E - Et)L2 1 (1 - ,u)(hu - L60) du . - I 2 ' 0

= -$Oz'( 1 + B)'h:(E - E;)L2(6u - LM). (2.44)

Requiring terms of order d in (2.40) to vanish implies that P, a , = 2L2Efh,. Next, take 6u = 0 and 60 # 0 and collect terms to get

O[P,La, + k , L22 - +L4E:(dE,/ds),]

- H 2 a L 3 ( 1 + [j),b:(E - E:)/4 + ... = 0. (2.45)

If [j > $, (2.45) cannot be satisfied since the terms of order 0 will not vanish. Similarly, if [j < i, (2.45) cannot be satisfied except with h, = 0, which implies that the assumed expansion is not possible. Suppose however that B = f. Then there is a balance of terms in (2.45) and the following expressions for h, and a , are obtained:

(2.46)

where the negative root of b; was chosen consistent with (2.37) and (2.43). The expansion can be continued by the above approach or by a direct

method given by Hutchinson (1973a). One finds

(2.47)

Plastic Buckling 81

where

(2.48)

The fractional powers which appear in the expansion (2.47) are absent from the initial post-bifurcation expansions for elastic systems (Koiter, 1945, 1963a) as well as from expansions for discrete element elastic-plastic systems-see (2.10) and the studies of Sewell (1965) and Augusti (1968). Terms involving fractional powers of the bifurcation amplitude arise in connection with the continuous growth of the region of elastic unloading. The significance of the term a20312 in (2.47) is that it is negative and may become numerically significant compared to the lead positive term cil 0 at relatively small values of 0. For example, if the series is truncated after the term u2 d3I2 and is then used to find an approximate estimate of the maximum support load of the perfect model, one finds

PF""/P, = 1 + 4~:/(27~$) . (2.49)

I f the magnitude of a2 is sufficiently large the maximum load will only slightly exceed P,. Note that u 2 , given by (2.46), depends on the geometric nonlinearity through k , as well as on the material nonlinearity through (dE,/cls), .

Figures 6 and 7 show comparisons between predictions based on the expansion (2.47), up to and including the u2 03 /2 term, and accurate numerical calculations carried out for the simple model. Nonlinear material behavior was introduced by using a Ramberg-Osgood-type stress-strain relation

c/c, = S/S, + a(S/Sp)n, (2.50)

where cg and sP = Ec, are effective initial yield values. In Fig. 6 the model has a strong geometric nonlinearity with k , L/(EL) = 1 and k , = 0. The model of Fig. 7 has no geometric nonlinearity (i.e., k , = k , = 0). In both examples CI = 4, n = 3, L /L = 1, and (s$EL) = 0.1094 (corresponding to the values Ef/E = 0.46 and sJs, = 1.4, where s, is the value of s at bifurcation).

The combination of strong material and geometrical nonlinearity of the model of Fig. 6 results in a maximum load which is only slightly greater than P, and which occurs at a relatively small value of 8. In this case the first few terms in the expansion (2.47) provides an excellent approximation to the

82

e* 0.01 \ .4 11111111111

0 0.1 0.2 0 0.1 0.2 (IIIIIIIIII

8 - d e

?,,..,.,,,, 0.005 - 0.01 0 e

FIG. 6. Post-bifurcation behavior and imperfection sensitivity of continuous model for the case of a strong geometrical nonlinearity [ ( k , L’EL) = 1 and k , = 01. Solid-line curves from asymptotic formulas and dashed-line curves from numerical analysis. From Hutchinson (1973b). J . M d . Phj.5. Solids 21, 191 204 with permission.

behavior in the range of interest. In the second example with only the material nonlinearity, the maximum support load is attained further from the bifurcation point and the expansion is not accurate over the full range of interest. Nevertheless, the maximum load prediction involves rather small error.

2. Effect qf Initial Impeifections

Prior to the occurrence of any elastic unloading the behavior of the elastic-plastic model is identical to that of the nonlinear comparison model introduced in the preceding subsection. However, once strain-rate reversal starts elastic unloading must be accounted for in the analysis. Thus the analysis of the slightly imperfect model separates into two parts as discussed by Hutchinson (1973b) and the results presented here are condensed from this reference.

A Koiter-type initial post-buckling analysis can be used to obtain the behavior of the model prior to elastic unloading. The result of this analysis is

Plastic Buckling 83

0 01 0.2 0.3 9 - 6

0 0.1 0.2 0.3 9

0.81

0.005 0.0 1 s 0

FIG. 7. Post-bifurcation behavior and imperfection sensitivity of continuous model with no geometrical nonlinearity [ k I = k , = 01. Solid-line curves from asymptotic formulas and dashed-line curves from numerical analysis. Froin Hutchinsoii (1973b). J . M ~ ~ c h . P/I! . s . Solids 21, 191-204, with permission.

an exact asymptotic equation relating the load, rotation, and initial imperfection 0 at load levels in the neighborhood of P,:

( 1 - P/PC)O + a; o2 + ." = [lo, (2.5 1 ) where ~ i : is the initial slope of the perfect model (2.28) and

(2.52)

The first reversal in sign of the strain rate in the slightly imperfect model occurs when the slope of the load-rotation curve is reduced to the initial slope of the elastic-plastic model. With values marking the onset of elastic unloading topped by a wedge,

Condition (2.53) together with (2.51) gives ( P J I dP/'ro = a , + O(8, F - P , , 8). (2.53)

0 = [po/(Lil - u;)] '" + O(0) (2.54)


and

P / P c = 1 - (a , - 2a;)[pO/(a, - ae,)]'/2 + O(0). (2.55)

To obtain a uniformly valid expansion for the response following the onset of elastic unloading, which holds for small 0 and reduces to (2.47) for 0 + 0, it is necessary to introduce a new expansion parameter [ 2 0 defined for 0 2 (3 by

0 - (3 = g a 1 / 2 + p, (2.56)

where 7 is a constant parameter determined in the expansion process. The expansion has the form

P/P, = P / P c + p l i + p z i 2 + p 3 i 3 + "., d/L = - 1 + d , i + d 2 i 2 + . " ) (2.57)

where the coefficients of the expansion are given in the above-mentioned reference.

Load-rotation curves showing the effect of initial imperfections are also shown in Figs. 6 and 7. Here again the solid line curves are based on the truncated expansion and the dashed line curves are based on an accurate numerical calculation and can be regarded as essentially exact for purposes of this comparison. A wedge marks the onset of elastic unloading in these figures and a dot the maximum load. Plots of P m d x / P ~ x (where Pyx is the maximum load of the perfect model) and P/Pc as a function of 6 are also given in Figs. 6 and 7.

For model of Fig. 6 we see that

Pmdx/P~dx 2 P/Pc = 1 - (a1 - 24)[pe/(u, - a:)]"2 + O(0). (2.58)

The pivotal role of P stems from the fact that in the presence of strongly destabilizing material and geometrical nonlinearities the maximum load is attained shortly after elastic unloading starts in both the perfect and imperfect model. The significant point is that the reduction in the maximum load appears to be proportional to the square root of the imperfection amplitude for small imperfections, similar to what was established conclusively for the discrete model with k , # 0 in (2.20). For the model with no destabilizing geometrical nonlinearity in Fig. 7, (2.58) is seen to be less accurate although it appears to be qualitatively correct. I t should be possible to establish whether or not the reduction in cases in which k , = 0 is proportional to or to some other power of 0. This has not been done. Nor for that matter has an analysis of the discrete model with k , = 0 and k , > 0 been carried out even though this would be considerably simpler and also very revealing in

Plastic Buckling 85

this regard. Recall that in the elastic range the reduction is proportional to e213 when k , = 0 and k , > 0.

All the examples we have considered thus far have dealt with behavior when the parameters of the perfect model were chosen such that bifurcation occurred well into the plastic range. Equally of interest is the effect of initial imperfections on a structure whose perfect realization bifurcates in the elastic range or at least before appreciable plastic deformation occurs. Numerical calculations of the maximum support load have been made to illustrate the effect of initial imperfections under such circumstances for the continuous model with the Ramberg-Osgood stress-strain relation (2.50). In Fig. 8 we have followed Duberg (1962) and have plotted the maximum

e= 0.001 ( b ) e . @.0@1

= 0.01

nmar p m a i

FIG. 8. Effect of imperfections on the maximum support load when the bifurcation stress of the perfect model s, falls below the effective yield stress . sv . The model was taken to have no geometrical nonlinearity. (a) High-strain hardening: tz = 3 ; (b) low-strain hardening: 17 = 10. ( A , = k , = 0, Z/L = I . )

support load normalized by the tangent-modulus load of the perfect model, P”’””/P,, as a function of the bifurcation stress of the perfect model over the effective yield stress, s,/s,, for several levels of imperfection. One set of curves pertains to a high strain-hardening material (17 = 3 ) and the other to a low strain-hardening material ( H = 10). In both cases ct = 4 corresponding to the original suggestion of Ramberg and Osgood and the choice of Duberg. We defer a detailed discussion of these curves until Section V where they will be compared with analogous curves for columns and plates. Here we simply note that the curves of Fig. 8 are very similar to Duberg’s results for a two-flanged column model and that they emphasize the possibility of strong interaction between imperfections and plastic deformation when the effective yield stress is not considerably in excess of the bifurcation stress of the perfect structure.

86 John W . Hutchirison

111. Bifurcation Criterion

A. CRITERION FOR THREE-DIMENSIONAL SOLIDS

The theory given below is a specialized version of Hill’s (1958, 1959, 1961) general theory of uniqueness and bifurcation in elastic-plastic solids which was given in somewhat less detail by Hutchinson (1973a). Sewell’s (1972) survey article on plastic buckling deals at some length with Hill’s theory. Here we are principally interested in presenting a suitable background for the bifurcation criterion to be presented in Section II1,B for the most widely used theory of columns, plates, and shells. Most of the usable nonlinear theories of structures employ Lagrangian strain quantities where the undeformed configuration is usually chosen for reference, as has been discussed by Budiansky (1969). The three-dimensional approach given below is developed from the same point of view.

Let material points in the body be identified by a set of convected coordinates xi and let y i j and y’j be the metric tensor and its inverse, respectively, in the undeformed body. Denote contravariant components of a tensor by superscripts and covariant components by subscripts in the usual way. With ui and ui as the components of the displacement vector referred to the undeformed base vectors, the Lagrangian strain tensor is

(3.1) I I , qi j = t (u i . j f “ j . i ) + 7Ll.i I f k , j 9

where the comma denotes covariant differentiation with respect to the metric of the undeformed body. Let T be the surface traction vector per unit original area and let T i be its contravariant components referred to the undeformed base vectors. With dV and dS denoting volume and surface elements in the undeformed body the principle of virtual work is

T i j 6Vij dV = 1 T i Alli d S , .il ‘ S (3.2)

for all admissible variations (hi where

sqi,i = f (Au i . j + 6u.J + f(LikiAuk.,j + ukjisllk.i). (3 .3)

Body forces will be omitted for simplicity. The stress quantities ziJ which enter into this exact statement of the prin-

ciple of virtual work are the contravariant components of the symmetric Kirchhoff stress referred to base vectors in the deformed body. [See, for

P h s t i c Buckliriy 87

example, Green and Zerna (1968), Bolotin (1963), or Budiansky (1969).] Application of the divergence theorem in the usual way to (3.2) yields the connection between the stress tensor and the nominal surface traction vector

Ti = (.+.i + T k j u ; k ). j ' (3.4) where n j are the covariant components of the unit outward normal to the surface of the undeformed body referred to the undeformed base vectors. Similarly, the equilibrium equations are found to be

+ ( T k i d k ) . j = 0. (3.5)

With rates of change denoted by a dot, the incremental form of the principle of virtual work is

First we consider the general rate-constitutive relation discussed by Hill ( 19674 for the isothermal, finite deformation of elastic-plastic solids characterized by a smooth yield surface. At any stage of the deformation process denote the current elastic moduli based on the Kirchhoff stress-rates ii' by Y . If the current stress is on the yield surface, denote the components of the unit tensor normal to the elastic domain in strain-rate space by i n i i where the strain rate is given by

(3.7)

The rate-constitutive relation is

i i j - - L i j k / h k l for m k ' i ] k , 2 0, (3.8)

= y ) i j k l f i k l for m k ' h k / I 0, (3.9)

(3.10)

where L i i k l = y i j k l - g- 1 I,li,i,tlkl.

The constant 61 depends on the deformation history (as well as on the current point. on the yield surface) and determines the current level of strain hardening. I t is assumed that a pure dilatation rate gives rise to an elastic response independent of the sign of mk'4kl and this requires that m satisfy G i j r d j = 0, where Gij is the metric tensor of the deformed body. When the stress lies within the yield surface (3.9) holds for all strain rates.

Hill (1967a) has discussed the transformation of this constitutive relation in going from one choice of objective stress rate to another. For the purposes of this article it need only be noted that the rate-constitutive relation for any elastic-plastic solid with a smooth yield surface can be cast into the present

88 John W . Hutch inson

form (3.8) to (3.10).? In particular, the simple J , flow theory of strain- hardening plasticity is a special case of this relation.

Dead loads are applied to the body in proportion to a single load (or displacement) parameter A. On ST prescribe surface tractions according to T i = AT; and on S , prescribe displacements ui = AM:, where T; and are independent of lb. Attention is directed to bifurcations which occur prior to any limit point of A. All quantities associated with the fundamental solution whose uniqueness is in question are labeled by a subscript or superscript 0. It is to be understood that the fundamental solution is the solution starting at A = 0 and is associated with monotonically increasing A.

1. Bifurcation Analysis ,for Solids with Smooth Yield Suyfuces

At any stage of deformation characterized by $(A), suppose that bifurcation is possible so that for a given increment in load x (either positive or negative) there are at least two solutions LI; and $. Introduce the following differences between the two solution increments:

T ~ . ~ = i i j - ' i j 1 1 I ? "y. 1 1 = h!. 1 1 - hr i , h z u 3 u. = ir'! - u"

. . and -Ti = TL - T:, .

where from (3.7)

y i j = i ( k i , j + & j , i ) + i(uy;,,i + u : k ; k . j ) . (3.1 I ) Since Ti vanishes on ST and Ui vanishes on S, and since both solutions are

assumed to satisfy the equilibrium equations, the usual construction in uniqueness proofs gives

0 = [ f i U i dS = ' [ ? i . i f i i + $ i f i U k , j } dl/ H , (3.12) ' S .iv

where H is defined by the last equality. In the current state at load A define the moduli L, of an elmtic. coinpurisoti

solid such that L, equals L where the stress is currently on the yield surface, independent of the sign of rniJtij, and L, equals Y where the stress lies within the yield surface. To obtain Hill's (1958) sufficiency condition for uniqueness, introduce the following quadratic functional,

(3.13)

The difference between the integrands of F and H depends on n ~ ' j l j ; ~ and

F(A, L ) = (. { L : ! k ' f i i j 9 k l + T o U , i U k , j ) i j - k dV. ' V

mijlj:i according to [from (3.8) to (3.10)]

'r Later. the additional restriction Y"'" = P"' will be needed

Plustic Buckling 89

-.. for ~ d j j l ; ~ 2 0 and m i j ; l f j 2 0, or stress within the yield surface,

zVfijj - L;yfi i j f ikl = 01

= 9- l[nliJ(jlijSh - j l; j)]z

= y- l } yp( j l ; i - jl;j)mk';l;l

= 9- lmi;(;l;j - ;l;i)mk';l;'

for ~ d j j l ; ~ 5 0 and di;l!j 5 0,

for 2 0 and n&,fi 5 0,

for nzi j j l ; ; S 0 and ~ d j j l ! ~ 2 0. (3.14)

It is therefore apparent that for positive 9 the integrand of H is nowhere less than the integrand of F and H 2 F. Consequently, the condition

F(L, G ) > 0 (3.15)

for all admissible nonvanishing U i (which vanish on S,) is sufficient to ensure uniqueness of the solution increment.

Let Ac be the lowest eigenvalue with an associated eigenmode u i such that

F ( & , u ) = 0. The mode is taken to be normalized in some definite way and for simplicity it is assumed to be unique. The variational statement of the eigenvalue problem, 6 F = 0, leads to the eigenvalue equations:

( 1 )

(1)

( 1 ) ( 1 ) ( 1 ) (1) ( 1 )

y . . = ' ( u . . + 1J 2 1.1 u,jJ + +(u:;i u ki + u:: 11 : j ) , (3.16)

(3.17)

(3.18)

(3.19)

( I ) . . (1)

z I J = LYk' y k l ,

( 1 ) ( 1 ) (1 ) .

( z ' j + z + zpc 11 fk),.i = 0, ( 1 ) ( 1 ) (1) ( 1 ) .

T' = ( z ' j + z k j ~ o k + zp, u f k ) i i j = 0 on ST ,

and ( 1 )

u i = O on S , (3.20)

For bifurcation to be possible at the lowest eigenvalue of F it is necessary that H vanish when F does. However from the inequalities in (3.14) it is seen that H > F when rniJ;lYj < 0 and/or inijjl!j < 0 i n a finite portion of V. In words, both solutions are possible when F = 0 if and only if both solutions share the property that no elastic unloading occurs.

The following condition on the fundamental solution ensures that bifurcation can take place at the lowest eigenvalue A,. I t should be possible to relax

90 John W . Hutchinsoil

this condition in special cases but for many problems of interest it is clearly satisfied. Suppose there exists a A > 0 such that the fundamental solution satisfies

/ d J ( d t @ d L ) 2 A (3.21) throughout the current yielded region. As already emphasized it is to be understood that the derivative in (3.21) is the one-sided derivative in the sense that the fundamental solution is associated with monotonically increasing A.

Identify the fundamental solution increment with iry and consider a bifur-

cation solution if in the form of a linear combination of iip and 1 1 , . Let 5 u be the contribution of the eigenmode to the bifurcation solution. Define the amplitude < such that i t is positive. To obtain the opposite-signed contribu-

tion of the eigenmode change the sign of 1 1 , but not <. Regarding ( as the independent variable on the bifurcation branch, write

( 1 ) ( 1 )

( 1 )

(3.22)

so that at bifurcation

By (3.21), the above inequality can clearly be satisfied if 1, is large enough.

Unless in i i v] i j > 0 throughout the current yielded region, (3.23) implies that I,, > 0 and thus bifurcation takes place under increasing load in the sense of Shanley as generalized by Hill. One can conceive of rather extraor-

dinary problems in which it would turn out that /n i jq i j > 0 throughout the current yielded region. In such cases bifurcation may be possible under decreasing load in such a way that it is still true that no elastic unloading occurs.

( 1 )

( 1 )

2. B$irc.ution Anu1j’si.s jbr Solids with Corners on Their Y ield Sui:fuces

Generalizations of the constitutive relation (3.8)--(3.10) which account for a singular yield surface with a pyramidal corner have been given by Koiter (1953) Sanders (1954), Mandel (1965), and Hill (1966). At any stage of deformation suppose there are N potentially active deformation systems such that, with $ ( p ) denoting the plastic shear rate on the pth system, the plastic part of the strain rate is given by

; l ~ , ! ~ l s t i c - - C :, L,@) I (PI I J .

Plastic Buckling 91

The rate of change of the yield stress on the pth system is denoted by i(,,) and is assumed to be related to the shear rates by (Hill, 1966)

Z ( P ) = c ’%Nd IlW ’

where we will take the hardening matrix h(,,(,, to be symmetric. We will formulate the constitutive relation using the convected rates of the contravariant components of the Kirchhoff stress since they arise naturally in the bifurcation analysis. Other choices may be preferable depending on the application (Hill and Rice, 1972), and the development may be altered to accommodate a different choice. The conditions for plastic loading or elastic unloading of the pth system are

j(,,) 2 O if i%$) = Z(,,) (3.24a)

(3.24b) and

<> - 0 if i i j v iS) < Z ( p , .

Hill (1966) has shown that a sufficient condition for uniqueness of the stress rate given a prescribed strain rate is that the elastic moduli 9 be positive definite and that the hardening matrix h(p)(q) be positive semidefinite.

l ( l d -

Since Z i j = c i jkf ’

p (Vkf - c t(,l)”8))? the conditions for loading and unloading can be rewritten in terms of the strain rate and the shear rates as

A’ i ’ (p ) > - 0 if tn$,hij = C A ( ~ ) ( ~ ) $ ( ~ ) (3.25a)

and

$ ( p ) = 0 if ~ ~ t f j , ) hi j < C A ( p ) ( q ) 11(q) 1 (3.25b)

where , ! p ) y / i j k f (Y)

A ( ~ H q ) = ‘ r ~ “ k f + h ( p ) ( q )

and = yW ( P )

Let G i j be prescribed. Within a subdomain of strain-rate space containing G i j any solution for the shear rates associated with the systems which do not unload can be written as

” k l ’

t ( p ) = 1 B(p)(q) 4;) h i j

where B(P)(q) is a symmetric A4 x M matrix and the sum extends over the M system satisfying (3.25a). In general, the shear rates and B are not unique. I f a complete basis can be chosen within the subdomain of strain-rate space in


question, it follows from the fact that the stress rate is unique that the moduli must also be unique in this subdomain. Using the above expression in the equation for i’j, the moduli can be written as

Lijkf = y i j k f - c c Bm)mfj,)m::) 7 (3.26)

where again the sums involve only the systems satisfying (3.25a). Suppose the hardening matrix is positive dejinite. If 9 is also positive

definite it can be shown that the N x N matrix A will always be positive definite. Let B‘ be the N x N matrix given by B‘ = A ~ I. Define comparison moduli L, using B‘ in (3.26) with the sums extending over all N systems. Sewell (1972, Section 3, iv) has shown that comparison moduli defined in this way ensure that the fundamental inequality,

pjjjj - Lfikljjijjjkf 2 0, (3.27)

holds for all strain rates and ifj, where Z ‘ j = ib’ - .tiJ and 6.. = l i b . - Gu.

A central concept to our subsequent discussion is total loudiny. Suppose there exists a subdomain of strain-rate space such that no elastic unloading occurs on any system [i.e., (3.25a) holds for all systems]-this is called the total loading subdomain. Within such a domain the comparison moduli L, are the actual moduli, and it immediately follows that if ifj and efj are both within the total loading subdomain then the equality holds in (3.27). It can be shown [from Sewell’s construction of (3.27)] that, in general, the equality will not hold if no total loading domain exists or, if i t does exist, if both t j y j

and $ j do not lie within it. In the application of the comparison moduli L, to the bifurcation analysis,

suppose that the fundamental solution increment satisfies total loading at each point in the body. Then, by the same argument that was made in the analysis of the solid with a smooth yield surface, it is possible to construct a bifurcation solution as some linear combination of the fundamental solution increment and the eigenmode in such a way that the bifurcation solution also satisfies total loading at each point in the body. Consequently, H vanishes when F does as it must if bifurcation is to be possible. The initial slope 1, in (3.22) must be chosen to ensure that the bifurcation solution satisfies the total loading constraint.

The requirement that the hardening matrix be positive definite is more restrictive than one would generally wish since, for example, i t excludes the case of perfectly plastic behavior. However, for the arguments to be made later in this article it does suffice to assume a positive definite hardening matrix.

Koiter (1953) showed that the slip theory of Batdorf and Budiansky (1949) is a special case of this class of theories in the limiting sense as N -+ GO. Sanders (1954) discussed a class of theories broader than slip theory

. . . . IJ I J 1J ‘

Plastic Buckling 93

but also based on the present structure with linear loading functions which for strain-hardening materials has the property that the hardening matrix is positive definite. These theories are pertinent to the discussion of the use of deformation theories in bifurcation analyses to be given later since for total loading histories they coincide with deformation theories of plasticity.

B. GENERAL BIFURCATION CRITERION FOR THE DONNELL-MUSHTARI-VLASOV THEORY OF PLATES AND SHELLS

The Donnell-Mushtari-Vlasov ( D M V ) approximate strain measures for plates and shells apply when the strains are small and when the characteristic wavelength of deformation is large compared to the thickness of the shell yet small compared to the radii of curvatures of its middle surface. Their application is also restricted to relatively small rotations as discussed by Sanders (1963) and Koiter (1966). A modern treatment of elastic buckling using DMV theory has been given by Budiansky (1968).

Let the material points in a thin plate or shell be identified by convected coordinates x" ( M = 1,2) lying in the middle surface of the undeformed body and the coordinate x3 normal to the undeformed middle surface. The D M V approximation to the Lagrangian strain tensor in this coordinate system is

lla, = E,, + X3K,,I 3 (3.28) where Ema and K,, are called the stretching and bending strains. They are given in terrns of the displacements of the middle surface Ua and W which are tangential and normal, respectively, to the undeformed middle surface by

E n , = 3 ( U a , p + U0.m) + bas W + 3 W a Wa (3.29) and

Kap = - W a s > (3.30) where b,, is the curvature tensor of the undeformed middle surface and the comma denotes covariant differentiation with respect to a surface coordinate. Greek indices range from 1 to 2.

The approximation to the three-dimensional expression for the internal virtual work in this theory is

(3.31)

where dA is the undeformed element of area of the middle surface. The bending moment and resultant stress tensors are given by

, lj-2

- 1 / 2 Map = dx3 and W p = J zaD dx3, (3.32)

94 John W . Hutchinsoii

where t is the undeformed thickness. An exact principle of virtual work is postulated for the variables of DMV theory. Let p be the resultant force per unit original area with components pa and p referred to the base vectors of the undeformed shell. The principle of virtual work is

M@ 6K,, + N " O 6E,,) r1A = {p" 6 U , + p (S W ) dA + boundary terms. .i> i, (3.33)

Equilibrium equations in terms of the stretching force and bending moment tensors are obtained without approximation from (3.33).

To obtain the rate-constitutive relations in terms of the DMV variables we invoke the same approximations as are used in the theory of linearly elastic thin plates and shells. The state of approximate plane stress at each point through the thickness is assumed to apply. The transverse shear-rate components j 1 , 3 and t j Z 3 are taken to be zero, and it is assumed that 7 3 3 = 0 so that there is no contribution to the internal virtual work from the normal strain rate j / 3 3 . For a given strain rate denote the three-dimensional moduli by L so that

i ' j = L'ikih,, , (3.34)

p = L a l i K Y ' %)I > (3.35)

The assumption of approximate plane stress gives

where again the Greek indices range from 1 to 2 and the plane-stress moduli are given by

~ D K ; . - - L . x / J K ) ' - ~ a / 1 3 3 ~ 3 3 ~ 7 / L 3 3 3 3 . (3.36)

For the special case of the relation (3.8)-(3.10) for solids with a smooth yield surface the plane-stress elastic moduli are given in terms of the three- dimensional quantities by

(3.37) P D K i ' - y v ? f l K Y - CJWJ33yJ33KI 0 3 3 3 3 l-r . -

One can also show after some manipulation that the assumption of approximate plane stress leads to

(3.38) ,$? = (P@K;. -- 1 - Z / I - K Y '

- %I /?I 172 )VKS 3

where a = 1 for iiin8j1n, 2 0 and r = 0 otherwise, and where = &J - / 1 2 3 3 y @ 3 3 / y 3 3 3 3 and 3- 1 = c j - l p 3 3 3 3 / ~ ; 3 3 3 3 .

(3.39)

Using (3.32) and (3.35) the rate-constitutive relations involving the DMV

Plastic Buckling 95

variables are

= H$YyEti7 + HY{Y7KKy ,

@fl = H @ K Y E (2) KY + H E f l K L . ( 3 ) K K Y * ' (3.40)

where . t /2

- t / 2 H ; ~ K Y = j p f l K L . ( X 3 ) i - 1 d X 3 . (3.41)

In the linear elastic theory of shells the integrations in (3.41) can be performed once and for all. In the elastic-plastic version of the theory the moduli are stress dependent and the active branch of the moduli depends on the strain rate. Thus an essential part of a general elastic-plastic calculation using this theory is the computation of the local stress distribution using the incremental relation (3.35) and evaluation of the integrals in (3.41) by one means or another at each stage of the loading process.

Now we turn to the question of uniqueness within the context of the DMV theory. At a given stage of deformation suppose there are two possible solution rates associated with the same rate of applied (dead) load. Denote these by U,U, k, 02, W", etc. Following the uniqueness construction of Section II1.A introduce the differences = Ug - G, Tfl = izfl - Cfl, etc. Then if both solution rates are indeed possible solutions it must follow that

(3.42) H (API?r?, l j + I F E a l r + Wd' W,a f i , D ) Q'A = 0, .r, where W{ is the current resultant stress tensor and

EM, = t(kJ + K,, = - W,N, .

+ ha, i%+ t(Wpa @.a + wp, - - (3.43)

Define three-dimensional comparison moduli in the same way as in Sec- tion 1II.A and use them in (3.36) to obtain the plane-stress comparison

moduli L,. Denote the integrals in (3.41) evaluated using L, by H(i, . I f the three-dimensional constitutive relation satisfies the fundamental inequality (3.27) then one can show that the DMV quantities as they have been defined satisfy

C

C C C

ii@k no + poiafl 2 H;$'Y1?lB kKY + 2H7{;7$fl kKY + H;f;7$lj &>, . (3.44)

The equality holds if and only if both the solution rates satisfy total loading (i.e., no elastic unloading if N = 1 ) through the shell thickness.


The quadratic functional for testing for bifurcation in the DMV theory is therefore

(3.45)

For any three-dimensional relation which satisfies the inequality (3.27) the condition that F > 0 for all admissible, nonvanishing fields og, @ ensures uniqueness. Furthermore by the same argument given in Section III,A, if the fundamental solution rate satisfies total loading, then bifurcation is possible when F first vanishes with the bifurcation mode composed of a linear combination of the fundamental solution rate and the eigenmode. The eigenvalue equations associated with 6 F = 0 are listed in Section IV,A.

The Kirchhoff stress tensor arises naturally in the above formulation by virtue of the fact that the theory employs an approximation to the Lagran- gian strain tensor and the undeformed configuration as reference. Since the difference between two stress-rate measures involves terms like uj, such differences will be small if the stress is small compared to the instantaneous moduli. In the compressive buckling of columns, plates, and shells the stress level is usually a small fraction of the instantaneous moduli at buckling. For example, the compressive bifurcation stress of a column is given by the tangent-modulus formula CE,(t/L)’, where C is a constant of order unity determined by the cross section and end conditions, and t and L are the characteristic thickness and length of the column. For a practical analysis of a slender column one can therefore use a “small strain” theory of strain- hardening plasticity in which no care is paid to the specification of the stress-rate measure. On the other hand, when the stress at bifurcation is comparable in magnitude to the instantaneous moduli i t is necessary to correctly identify i? as the convected rate of change of the contravariant components of the Kirchhoff stress. The moduli in (3.34) must also be chosen consistent with this interpretation. A recent discussion of the extent to which the stress-rate choice influences bifurcation predictions is illustrated in a number of examples examined by Bazdnt (1971).

As it stands, (3.45) is referred to the original configuration as has been discussed. If it is desired to use the deformed configuration at bifurcation as the reference then (3.45) remains unchanged in form except that now the comma denotes covariant differentiation base vectors of the deformed middle surface, @ represents a deflection normal to the current middle surface, etc., and the components NOap are referred to the current base vectors. Also, W o must be set to zero in (3.43) t represents the current

Plastic Birckling 97

thickness, and dA the current element of area of the middle surface. Since the theory is restricted to small strains it is usually unnecessary to draw a distinction between the area and thickness of the shell at bifurcation and in the undeformed state.

C. DISCUSSION OF BIFURCATION PREDICTIONS BASED ON THE SIMPLEST INCREMENTAL AND DEFORMATION THEORIES OF PLASTICITY

In the examples to be discussed below the stress levels at bifurcation are a small fraction of the instantaneous moduli so that a discussion within the context of small strain theories of strain-hardening plasticity is justified. The most widely used incremental strain-hardening theory is 5, flow theory. In Cartesian coordinates the stress deviator is sij = z i j - +kkhij and J - r - , s i j s i j . The instantaneous moduli are given by

(3.46)

where E is Young’s modulus and 1’ is Poisson’s ratio. For J , = (J2)mdx, x = 1 if j , 2 0, c( = 0 if j , < 0, and c( = 0 if J , < (J2)max. The function h l ( J 2 ) is determined from the tensile stress-strain curve in terms of the tangent modulus E, (i.e., ir = E, E ) as

h , = 3[E/Et - 1] / (45 , ) . (3 .47)

The comparison moduli are given by (3 .46) with c( = 1 where J , = (J2)max. Since J 2 flow theory satisfies the inequality (3 .27) the functional F defined

in (3 .45) does form the basis of the sufficiency condition for uniqueness. Furthermore when the fundamental solution has the property that J , > 0 everywhere the yield condition is currently satisfied, then bifurcation is possible at the lowest eigenvalue of F.

The simplest total strain theory of plasticity is usually referred to as J 2 deformation theory. It is a small-strain nonlinear elasticity relation in which the total strain can be expressed as a function of the stress according to

(3 .48) = (l/E){(l + ‘ b 1 J - v g k k S r j + h 2 ( J 2 ) S 1 ~ ) 3

where

112(52) = ?(E/E, - I ) , (3 .49)

and where E, = a/t: is the secant modulus in a tension test. The instantaneous moduli are

98 John W . Hutchinsoti

(3.50)

where 11; = dh,/dJ,. Bifurcation predictions based on F in (3.45) with either (3.46) or (3.50)

reduce to results obtained by many authors. Most problems which have been worked out in detail have a fundamental pre-bifurcation solution which is a trivial uniform state of stress. Many solutions (some involving additional approximations) are presented in the well-known references by Bijlaard (1949), Stowell (1948), and Gerard and Becker (1957). More recent work includes Lee’s (1961, 1962) cylindrical shell studies, Batterman’s (1964) equations for axisymmetric shells, and Jones’ (1967) results for eccentrically stiffened shells. Sewell’s (1972) bibliography includes many more references to problems in this class.

Sanders (1954) has shown that incremental theories of plasticity based on linear loading functions such as those alluded to in Section III,A are integrable for total loading histories on which no activated loading function unloads. In particular, he has shown that there exists an incremental theory with infinitely many loading functions which coincides exactly with J, deformation theory for total loading. Put another way, for a restricted range of deformations J, deformation theory coincides with a physically acceptable incremental theory which develops a corner on its yield surface.

Slip theory of Batdorf and Budiansky (1949) is also integrable for total loading deformations but it coincides with a deformation theory involving both J, and the third invariant of the stress. We have already remarked that theories based on multiple loading functions satisfy the fundamental inequality required to establish the validity of the bifurcation criterion based on F.

It follows from Sanders’ observation that most of the results which have been obtained using J, deformation theory are 1.1’gorou.sI~~ valid bifurcation predictions based on the incremental theory mentioned above which coincides with J, deformation for total loading. This is contrary to statements made repeatedly in the literature to the effect that bifurcation predictions based on deformation theory are physically unacceptable. To be more specific, suppose the fundamental solution satisfies proportional loading everywhere, as is the case for almost all the examples which have been worked out in detail in the literature. The bifurcation solution is a linear sum of the fundamental solution increment and the eigenmode. We can always include a sufficiently large amount of the fundamental solution increment relative to the eigenmode such that the bifurcation mode satisfies the total loading restriction. It seems to be widely appreciated that J, deformation

Plastic Buckliric~ 99

theory cannot be labeled physically unacceptable for total loading histories (Budiansky, 1959). The confusion in bifurcation applications apparently stems from the misconception that when bifurcation occurs total loading will be violated. On the contrary, it is the total-loading condition itself which supplies the constraint on the combination of fundamental solution increment and eigenmode which must pertain-just as it is the condition r n i j t f j 2 0 which provides the constraint in (3.23) for the case of a smooth yield surface. The above line of reasoning is due to Batdorf (1949) who used slip theory, which was developed for this purpose, as the basis for his argument.

As long as the fundamental solution satisfies total loading there are no grounds that we have yet mentioned which favor the bifurcation-load predictions based on J 2 flow theory over those based on J , deformation theory or vice versa.

Perhaps the best example which brings out the essence of the difference between the two simple theories in bifurcation applications is the buckling of a cruciform column under axial compression shown in Fig. 9 and studied

FIG. 9. Theoretical and experimental results for the plastic buckling of a cruciform colunln. Curve a, prediction of incrcmental theory with smooth yield surfacc: curve b, prediction of any deformation theory with v = i; test data from 2024.T4 cruciform sections. From Gerard and Beckcr, (1957).


originally by Stowell (1948). If the column is not too long, it undergoes torsional buckling in which the specimen twists about its axis. Only the effective shear modulus enters into the formula for the bifurcation stress. In the elastic range the compressive stress at bifurcation is

oc = G(t/b)’, (3.51)

where G is the elastic shear modulus, t is the thickness, and b is the width of the flange plates. This result can be obtained in several ways; but, in particular, it can be obtained from the DMV theory as a limiting case for a long plate which is simply supported along one of the long edges and free on the other (Timoshenko and Gere, 1961).f

In the plastic range (3.51) still holds according to J , flow theory. Because this plasticity theory has a smooth yield surface, the increments in the relevant components of shear stress and shear strain following uniaxial compression are related by the elastic shear modulus. On the other hand, for any deformation theory for an initially isotropic material it can be shown that the relevant instantaneous shear modulus G following uniaxial compression is given by

(3.52)

where E, = o/c is the secant modulus. The deformation-theory prediction (and consequently that of slip theory too) is

oc = G(t/b)2. (3.53) Thus the ratio of the deformation-theory result to the simple flow-theory result is G/G and for Poisson’s ratio equal to 3 so G = 3E in (3.52) this ratio equals E J E .

Experimental results in the form of the buckling stress normalized by G ( ~ / L I ) ~ are plotted as a function of oc in Fig. 9. These experiments were performed on specimens of 2024-T4 aluminum and the figure was taken from Gerard and Becker (1957). The discrepancy between the two theories for the cruciform column is more dramatic than occurs in most problems. Nevertheless, it is generally agreed that bifurcation-load predictions for plates based on J , deformation theory give reasonably good agreement with experimental buckling loads while predictions based on J z flow theory are consistently high.$

The cruciform column was discussed extensively in the literature by

t For flat plates the DMV equations reduce to the von KBrmBn plate equations. $ A strong assertion to the contrary made by J. B. Newman [Inelastic column buckling of

internally pressurized tubes, Exp. Meck. 13,265-273 (1973)], stems from the use of an incorrect formula for the bifurcation load according to deformation theory.

Plastic Buckling 101

Drucker (1949), Cicala (1950), Bijlaard (1950), and Onat and Drucker (1953). In part, this discussion centered on whether or not imperfections could account for the discrepancy between the predictions of simple flow theory and deformation theory. Based on a rather approximate analysis Cicala (1950) concluded that small imperfections, which would inevitably be present in any actual specimen, would reduce the maximum support load calculated using J , flow theory to the level of the deformation-theory bifurcation load. Bijlaard (1950) refuted Cicala’s claim on the grounds that the imperfection levels Cicala was considering would by no means inevitably be present. Onat and Drucker (1953) carried out a more detailed, but still approximate, calculation of the maximum support load based on J 2 flow theory and found that extremely small imperfections did reduce the maximum load to essentially the level of the deformation-theory bifurcation load. The imperfections required to bring about this reduction were so small that they suggested that no significant scatter in the buckling loads should be expected (as is usually the case when imperfection sensitivity is involved), as the test data seems to indicate. If they are correct in asserting that the yield surface should be taken to be smooth and if their conclusion regarding the effect of small imperfections is also correct, the prospect of having to take into account initial imperfections in this manner just to calculate an effective buckling load is hardly a happy one. In any case, this example does lend further credibility to the use of bifurcation-load predictions of deformation theory for engineering purposes.

As discussed above the essential difference between the two sets of predictions revolves around the question of whether the description of the yield surface should allow for corners. Theoretical models based on single-crystal slip (such as slip theory or the more elaborate models which followed i t ) definitely indicate that corners should develop (Hill, 1967b; Hutchinson, 1970; Lin, 1971). However, experimental evidence on this question is contradictory. Adequate direct evidence in the form of measured yield surfaces is extremely difficult to obtain for these purposes since experimental probing of the yield surface tends to obliterate any potential corner. Nevertheless, many tests do show that a region of high curvature does develop at the loading point on the yield surface. Biaxial tests or tension-torsion tests which directly measure incremental stiffnesses are more likely to shed light on this matter. But here too, the experimental evidence is contradictory with some investigators finding evidence which suggests corners and others finding none. A recent survey of the history of yield surface experimentation is given by Michno and Findley (1972).

So far the evidence from basic stress-strain tests must be regarded as inconclusive with regard to whether or not adequate models of the elastic- plastic behavior of common metals should incorporate yield surfaces with


corners. Fortunately, experimental work in this area is continuing and the accuracy of the tests is improving so that there may be clearer evidence available in the future. In the meantime there seems to be little doubt that for engineering purposes bifurcation prediction based on deformation theory should be favored over those based on incremental theories with smooth yield surfaces. This should not be construed as an argument for the universal application of deformation theory. In fact where deformation histories do depart from total loading, as may be the case in the post-bifurcation regime, for example, deformation-theory predictions must obviously be regarded with suspicion.

We conclude this section with two additional examples which more typically illustrate the discrepancy between the bifurcation predictions of these two theories. Bifurcation results for these two examples will serve as the starting point for post-bifurcation and imperfection-sensitivity studies presented in Sections IV and V.

Consider a clamped circular plate of radius R and thickness t subject to a uniform radial stress r ~ . The elastic bifurcation stress for compressive loading obtained from the DMV criterion is

cC = -k2Et2/12(1 - v2)R2, (3.54)

where k 3.832 is the first zero of the Bessel function of the first kind of first order (Timoshenko and Gere, 1961).

In the plastic range the fundamental solution continues to be the uniform state of equal biaxial compression. The plane-stress comparison moduli relating the inplane stress rates and strain rates must be isotropic at bifurcation. Without any approximation we can introduce an instantaneous modulus E and contraction ratio V so that

Eta,] = ( 1 + V)i,, - VZ,.,&,~ . Also without approximation the lowest bifurcation stress according to DMV theory is given (3.54) using E and V instead of E and v, i.e.,

rJc = -kZEt2/12(1 - V2)R2. (3.56)

(3.55) _. -

For J , flow theory ,!? and V are given by

E/E = [ l + (E /E , - 1)/4], (3.57) -

\ I = ( E / E ) [ r - (E /E , - 1)/4],

where E, is the tangent modulus in simple tension which is regarded as a function of J , as in (3.47). For J , deformation theory,

E - = (EL ' + 3E,- 1)/4, V E - ' = -[2(1 - 2v)E-' + E,' - 3E,']/4, (3.58)

Plastic Buckliny 103

where E, is the secant modulus in simple tension and is also taken to be a function of J , .

An example studied by Needleman (1973) to be discussed further in Sec- tion V uses a uniaxial tensile stress-strain curve which has a definite yield stress G) and yield strain E , = oy/E and a continuous tangent modulus where

C / E , = I“”, for o 5 G ~ ,

tz-’(g/o,)” + I - / I - ’ , for G > G , . (3.59)

This curve is shown in Fig. 10 with a strain-hardening exponent of M = 12.

or

0-c 1.5-

0-Y

-

1.0 -

1.0 2.0 3.0 4.0 5.0 6.0

FIG. 10. Stress-strain curvc and bifurcation predictions for a clamped circular plate tinder radial compression. Curve a, J , How-theory predictions: curve b, J , deformation-theory predictions; curve c, u/us vs ~ / t : ~ : Eq. (3.59). I I = 12; for (a) and (b) uC,,iv vs k2(t /R)’ /12(I - ~ ~ ) t ; ~ ; v = ;.

Also shown in Fig. 10 are the predictions for the bifurcation stress from (3.56) using (3.57) and (3.58) derived from this tensile stress-strain curve. The bifurcation results are conveniently plotted as aJa, against k2(r/R)2/ [12ey(1 - v2)] so that in the elastic range the bifurcation curves plot on top of the stress-strain curve. For bifurcation stresses which are not more than about 20% in excess of the yield stress (and thus bifurcation strains not exceeding about 2f times the yield strain) the difference between the two theories is very small. For larger values of the abscissa the difference is no longer insignificant.

A second example which can be analyzed in an equally simple manner is


the thin spherical shell under uniform external pressure. The pre-bifurcation solution is again the uniform state of equal biaxial compression. In terms of E and 7 in (3.55) the stress in the spherical shell at bifurcation is

cc = -Et/[3(1 - 7 2 ) ] 1 ' 2 R , (3.60)

where now t and R are the thickness and radius of the shell. This result also comes from DMV theory as discussed by Hutchinson (1972); with E and V assuming their elastic values, (3.60) is the elastic formula. Figure 1 1 has been

t or ~ ~ ; & ( € y R l Fic;. 1 1 . Stress strain ciirvc and bifurcation predictions for a complete spherical shell

under external pressure. Curvc a, flow-theory predictions; curve b. deformation-theory predictions: curve c, >;/ i ;? = o/oy + O. l (~ /o , , )~ ; for (a) and (b): uJuv vs [3(1 - ? ) ] - ' " ( / / I : ~ R ) ; ,' = 1

plotted in the same way as was done for the corresponding plate curves. Here, however, a Ramberg-Osgood-type tensile relation has been used where E), and cy = Ec, are now an effective yield strain and yield stress, and n = 6 was chosen as illustrative of relatively high strain hardening.

Bijlaard (1949) recognized that the elastic results for the above sphere and


circular plate problems could be simply converted to give bifurcation loads in the plastic range. His paper includes predictions based on the two simple theories used above. A rederivation of Bijlaard’s result for the sphere has been given by Batterman (1969) for J , flow theory.

Sewell (1963, 1964) studied the extent to which the orientation of the normal to the smooth yield surface influenced the lowest bifurcation load. In his study of rectangular plates under uniaxial compression which are simply supported on all four sides he found that by allowing the normal to differ from that of 5, flow theory somewhat lower bifurcation loads could be obtained. Justification for the different choice of normal requires an appeal to initial plastic anisotropy. In a more recent study Sewell (1973) reexamined this same plate problem using a plasticity theory based on the two loading functions associated with the corner of the Tresca yield surface. Appreciable reductions below the predictions of J , flow theory were found. However it does not follow from these findings that the Tresca yield surface is generally suitable for bifurcation calculations. If the Tresca yield surface is used in the analysis of the cruciform column one still obtains the elastic prediction (3.51) since the corner associated with the Tresca surface does not lower the effective shear modulus in question below its elastic value.

IV. Initial Post-Bifurcation Behavior for Donnell-Mushtari-Vlasov Theory

In this section behavior immediately following bifurcation is studied within the context of DMV theory. Attention is focused on bifurcations emanating from the lowest possible bifurcation point. Growth of the region of elastic unloading is involved in an essential way in the determination of the initial post-bifurcation behavior as has been previously brought out by the continuous model of Section I1,B. The analysis will parallel that given for the continuous model as well as a treatment of three-dimensional solids given by Hutchinson (1973a).

One difficulty which must be faced immediately in the initial post- bifurcation analysis is the choice of constitutive relation. A deformation theory which incorporates elastic unloading is generally unsatisfactory in that it violates continuity requirements, as is well known. This is in addition to the loss of its justification once total loading can no longer be claimed, which will often be the case in the post-bifurcation regime. On the other hand, we have seen that where deformation theory predictions for the bifurcation load fall significantly below those of a simple incremental theory the deformation-theory predictions are in better accord with experimental data. We strike a compromise here by using the incremental theory based on a smooth yield surface and by restricting consideration to specific examples

106 John W. Hutch iiisoii

where the bifurcation load is only slightly greater than the J , deformation- theory prediction. In two column problems which will be looked at in some detail this question does not even arise since the stress-strain behavior is essentially unidirectional. However in the general situation this predicament underlines the lack of a reasonably simple yet adequate constitutive relation for common structural metals even for the restricted class of deformation histories involved in plastic buckling problems.

The general constitutive relation (3.8)-(3.10) will be used; under the assumption of approximate plane stress this relation has been rewritten in terms of the in-plane stress and strain components in (3.37) to (3.39). Although i t is not crucial to the analysis we will make the simplifying assumption that the body becomes fully plastic prior to bifurcation.

Let J. be the single load parameter and let dead loads (and/or displacements) be applied proportional to 2. Let the fundamental solution be associated with monotonically increasing J. prior to the occurrence of any limit point of J. and label the fundamental solution with a superscript or subscript 0. Attention is restricted to bifurcations which occur prior to a limit point of i. With 4 denoting the (positive) amplitude of the eigenmode associated with the lowest bifurcation load ic, we will show that the initial post-bifurcation expansion is of the form

1, = ic + A,< + i 2 ( 1 + f l + ..., (4.1 1 where 0 < p i 1. Generally, 2, > 0; in the examples examined below, p = 4 or 5 and A, < 0.

A. GENERAL THEORY

1. Equations fbr the Eigrnvalur Prohlrnz and Determination qf A I Denote the eigenmodal quantities associated with the lowest value iC for

which F vanishes by ( 1 ) ( 1 ) (1) (1) (1) ( 1 ) ( 1 ) ( 1 )

{us , W , E,, , K , , 9 ' 1 x 0 , N"", w', ~ ' " ) ~ (4.2) where the local quantities which vary through the thickness are listed along with the quantities which are functions of just the two middle surface coordinates x'. The first variation of F also vanishes at A c , i.e.,

(1 ) (1)

(A4"'6GK,, + NZ"6'E,, + N i t W,,SW,) d A = 0, (4.3a)

with


(4.3e)

and ( 1 ) ( 1 ) ( 1 )

v n p = E, , + . y 3 K Z 0 . (4.3f)

A superscript or subscript c denotes quantities evaluated at A,. The quanti-

ties L, and H(i) were defined in Section III,B and, i n general, E, may vary through the thickness as well as over the middle surface. Equations (4.3e) and (4.3f) are auxiliary to the eigenvalue problem but nonetheless are important to the initial post-bifurcation analysis.

Attention is restricted to problems in which the eigenmode associated with A, is unique. The mode is normalized in some definite way. As defined previously, [ is the amplitude of the eigenmodal contribution to the bifurcated solution. It is used as the expansion variable: 4 is defined to be positive and is increased monotonically in the initial post-bifurcation regime. To analyze the opposite-signed deflection in the eigenmode we will change the sign of the eigenmodal quantities (4.2).

(I

It is assumed that the fundamental solution satisfies

Fi:p~:p 2 A > 0,

where for the remainder of the paper (4.4)

( ) f [d( )/.'A], . (4.5) As already discussed, the bifurcated solution is of the form,

I I ( 1 )

108 John W. Hutchinson

with similar expressions for the other variables and where A 1 was introduced in (4.1). An equivalent form for the expansion is

(4.7)

where

( . ) = 4 )/& (4.8)

and, for example, wo is shorthand for (dW'/dA)(dA/d<). The Shanley condition for no elastic unloading at bifurcation requires

at every point in the body which provides a constraint on A, which clearly can be met if (4.4) holds.

Recall that two possibilities were encountered in the analysis of the models of Section 11. Either no elastic unloading occurred on the bifurcated solution branch in some finite neighborhood of A, or neutral loading occurred at one point in the body at bifurcation and elastic unloading spread from this point as the amplitude of the bifurcation deflection increased. The same is true here. In the unlikely event that the structure possesses a sufficiently large stabilizing geometric nonlinearity such that I* , is greater than the smallest value needed to ensure (4.9), then no elastic unloading will occur for some range of positive 5. The criterion for identifying this possibility will be given later, along with the associated formula for A l . The second possibility is more typical and by far the more important. Then i1 assumes the smallest value consistent with (4.9). This is the case which is analyzed below.

2. Lowest-Order Boundury-Layer Ternzs

The instantaneous neutral loading surface separating the regions of elastic unloading and plastic loading spreads from a point x: as < increases from zero as depicted in Fig. 12. The initial neutral loading point is denoted by xi and is the point where the equality in (4.9) is attained. In some problems neutral loading at bifurcation may occur at more than one isolated point or along a line of points, as for example in an axisymmetric bifurcation of a shell of revolution. We will carry out all the details of the analysis for the case where x: is an isolated point. In most problems x: will lie on one of the


FIG 12. Sketch of the initial neutral loading point, the subsequent instantaneous neutral-loading surface and the local Cartesian triad.

two surfaces of the shell as in Fig. 12 and this will be assumed to be the case here. Other variants can be analyzed with the same approach.

With A l chosen such that the equality holds at x: in (4.9), the function on the left-hand side of this inequality is greater than zero elsewhere in the body. It enters into the initial post-bifurcation analysis in an essential way. To display the behavior of this function in the neighborhood of xf introduce a set of local Cartesian coordinates zi centered at xf as shown in Fig. 12 such that z j is directed along the outward normal to the surface on which xf lies. Using a Taylor series expansion about x:, , write

(1 )

f i :qAlyl$ + yl,p) = c,z, + c,/jz,zI, + "., (4.10)

where

(4.1 1)

Since (4.10) attains its minimum in the body at x:,, C, = C, = 0 and C, 5 0 . In the subsequent analysis it is crucial to distinguish between instances where C, < 0 and where C, = 0. It will almost always be the case that C, # 0 if the eigenmode involves bending. For example, suppose the fundamental state is a uniform state of stress; then from (4.11) and (4.3f)

(1)

c, = - 1 %"K,, JXi' >

which will vanish only under exceptional circumstances. We will carry out the analysis under the assumption that C, < 0 and later comment on the case where C, = 0.


Motivated by the simple model results, an expansion is attempted in the form (4.1) so that

;z = A, + ( 1 + /))A2<” + 1, (4.12)

where the remainder satisfies (<-”I) = 0 as < --f 0. Using this relation to expand the fundamental solution about A, gives for a typical quantity such as the stress

(4.13)

As the region of elastic unloading spreads from x) and engulfs a point, the relation between the stress rate and strain rate changes from being determined by the plastic moduli L to the elastic moduli 9’. A set of

stretched boundary-layer coordinates zi will be introduced such that the equation of the neutral loading surface will be independent of < to lowest order when written in terms of these coordinates. These same coordinates are the natural choice for use in the description of behavior in the vicinity of the growing elastic unloading region.

Anticipating that the lowest-order correction to the bifurcation mode (4.7) are of order tp we can write quite generally that

*

To introduce the boundary-layer description two distinct limits of the last terms in (4.14) are considered. In the first the point x i (xi # xf) is held fixed so that the elastic-unloading region leaves x i behind as it shrinks to the point ,xp as < + 0, i.e.,

(4.15)

* Next, take the limit with ii fixed so that the point in question stays at the same position relative to the shrinking boundary-layer region as -+ 0. Define the boundary-layer quantities by

+The quantities q and t have no connection with quantities idcntilied by an (I in 11- uniqueness construction.

Plustic Buckling 1 1 1

(4.16)

Having defined these limits we now proceed to obtain the lowest-order equation for the instantaneous neutral-loading surface. The normal to the yield surface in strain-rate space is regarded as a function of stress, and for plastic loading

(4.17)

Thus the equation for the neutral-loading surface can be written as ( 1 ) I U

0 = .w&, = iI:’’(j , , r]& + r] + <”[( 1 + p)jb2 nz:q,, + ’n;pr],,]] + . . . . (4.18)

Noting the expansion (4.10) for the zeroth-order term in the above equation, define the stretched coordinates according to

* * z3 = <-[’z3/-(1 + B ) A 2 and z, = <-”2zE/[-( l + [j)jL2)’I2, (4.19)

where i t has been anticipated that A, is negative. Introduce the stretched

coordinates into (4.18) and take the limit as < + 0 with zi fixed with the result

*

* * 0 = IY1‘”jln,, = (’[( 1 + /))A, f ( z i ) + h:”(,~~.)q~,,] + . . . , (4.20)

where * , * * *

. j (z i ) ~ : ’ ~ : f i lxcc - C ~ Z , - Cz/iznz/i . (4.21)

The boundary-layer strain-rate quantities qlB are zero as a consequence of

the kinematic assumptions which have been adopted. With En8 and K,, defined analogous to the definitions in (4.14), (3.28) gives

* U U

a a a

q,p(x’) = EnB(x1) + . x ~ K ~ ~ ) ( x ~ ) . (4.22)

Since the strains are constrained to vary linearly through the thickness, the

112 John W . H utcliinson

* * * * existence of boundary-layer terms EZ8(zn) and KzD(z,) would result in the

quantities yzb varying linearly through the entire thickness. This possibility is excluded by the situation envisioned in Fig. 12 where the region of elastic unloading spreads into the shell in the z, direction as well as in the two tangential directions. To proceed we must take

Exs = K,, = qzo = 0. (4.23)

At this point one might want to question the retention of the assumption that the strains vary linearly through the thickness according to (3.28) once elastic unloading sets in. We will take this matter up again in the discussion of some specific examples. Here, however, it is possible to instill some confidence in this assumption by noting that the full boundary-layer treat-

ment for a three-dimensional solid also gives qnD = 0 (Hutchinson, 1973a, Section 6.2). Consequently, the lowest-order equation for the neutral- loading surface (4.20) reduces to

*

* * *

*

* f ( z i ) = 0

or, in terms of the unstretched local coordinates,

(4.24a)

c,z, + Ca/,Z,Z/j = - (1 + p)A25”:”q& (4.24b)

Because of (4.4) and C, < 0, (4.24b) implies that R , < 0 if the neutral- loading surface is to spread into the body as previously anticipated.

Within the region of elastic unloading,

Using (4.14), (4.3e), together with


within the elastic unloading region as -+ 0). In this limiting process make use of (4.10), (4.19), (4.21), and (4.23). The result is the lowest-order correction to the bifurcation mode for the stress rate within the elastic unloading region :

* * tap = (1 + p)I&, 'm:q,,,.f(zi). (4.26)

A similar analysis gives 8' = 0 outside the region of elastic unloading. Note that the boundary-layer stress rate is continuous across the neutral-loading surfacej'= 0.

Let N Z p and MnO be the contributions of boundary-layer stress rate to the resultant inplane stress and bending moment tensors defined in accord with (3.32) by

*

* *

(4.27)

(4.2 8 a)

(4.28b)

where the integration extends from the edge of the elastic unloading region to the shell surface in a sense consistent with that which z , = f ( x 3 f t / 2 ) holds. Also, + t/2 pertains when x," = t/2 and - t/2 when x," = - t/2.

3. Determination of p and lb2

To evaluate f l and jb2 the lowest-order boundary-layer terms are displayed explicitly in the initial post-bifurcation expansion, and in this form the expansion is substituted into the principle of virtual work. Following some further manipulation of the virtual work equation, an examination of the lowest-order nonvanishing terms in this equation permits us to identify by inspection and provides a general equation for L 2 .

114 Johii W. Hirtchinsoii

The bifurcated solution rate is written as

with similar definitions for the other quantities. Since the lowest order boundary-layer terms have been separated out it can be asserted that

b b b /I

lim < - f i [ ~ ~ , ~ ; qZp; z'p; 4 - /jN"l)j = 0, (4.30) *

< - 0 . ;, f ixcd

The incremental form of the principle of virtual work (3.33) is

{A?"6Kz,j + P"GE,p + N"'W,x6W,) CIA = E V W , (4.31)

which is satisfied by both the fundamental solution and the bifurcated solution. Eliminate the right-hand side of (4.3 I ) using the equation satisfied by the fundamental solution and rearrange the resulting equation to the following form:

/. {(hj"~ - hi"dJ)ijK,, + ( i i i "~ - i i i " $ ) ( i j ~ ~ , / , + w!:ij w,,~) + ,WJ( w , ~ - w;)iiw.O

.i,

' A

+ Ntfi(k,, - k;)SW', + (NZo - Nifi)k.m6LV'lj) ( / A = 0. (4.32)

Substitute (4.29) into (4.32) using (4.3a), (4.12), and expansions such as (4.13) to obtain

(1) I ( 1 ) ( 1 )

+ < tA[2A1 N""WP,SW, + 2 ( i 1 Nip + Nafi)W,6W,) dA + . . . = 0,

(4.33)

where 6'E is defined by (4.3b). Equation (4.33) can be regarded as the variational equation of equilibrium

for the quantities with the superscript b. We will argue that all three terms shown in (4.33) must be of order

In preparation for this argument we first digress to consider the initial post-bifurcation behavior of a nonlinear hypoelastic comparison problem in which the unloading branch of the constitutive relation is suppressed. That

and this will enable us to identify [j.


is z", = L i ( O K Y i l K i for all strain rates where L will be taken as a function of the stress. For the comparison problem the initial slope A, is not constrained by (4.9) but will be determined below. Since the boundary-layer region does not develop the associated term in (4.33) must be deleted and the superscripted b quantities are of order (. Introduce the limit

b b b h

lim ( - ' { U , ; q 2, ' .zZ"; N""' I

(2 ) (2) ( 2 ) ( 2 )

<+O.s~fixed

= 2{U,; q a B ; z"; N"'), (4.34)

where the multiplicative factor of 2 brings the notation into line with previous work on elastic post-buckling expansions to be cited later. The variational equation for the second-order quantities is

. ( 2 ) ( 2 ) (2) 2 j {M%K,, + N @ ~ C E , , + N"R, w,hw,) [ I A

' A

( 1 ) I (1) ( 1 )

+ 2 1 (IL1 N""W;hW, + (Alb;o + W)w,,Sy,J d A = 0. (4.35) " A

One can also establish the connections

and

Writing

one can show

(4.38)

where


(4.40)

where dV represents the volume element in the undeformed shell. This iden-

tity is obtained by setting 6 U , = U , and 6W = W in the variational equation for the eigenmode ( 4 3 ) and then by making use of (4.3b), (4.3c), (4.36), (4.40), and the property LzBKY = c;'aB. Eliminate the common term in (4.41) and (4.42) to obtain the following equation for A, :

( 2 ) ( 2 )

a? + A@ = 0, (4.43)

where ( 1 ) ( 1 ) (1)

.d = 1 3NUD W," W,, d A + 1 ' A ' V

. I (1) ( 1 ) (1 ) ( 1 )

& = 1 {2NG0 W,@ W., + 4NuD W ; W,,} dA ' A

(4.44b)

The expression (4.43) for lbl in terms of the eigenmode specializes to formulas given by Budiansky and Hutchinson (1964), Budiansky (1969), Cohen (1968), and Fitch (1968) for constant elastic moduli. When the variable moduli are derivable from a strain energy density the result can be

regarded as deriving from Koiter's (1945,1963a) general approach to conservative elastic systems. The formula is more general, however, in that it applies to the nonlinear hypoelastic comparison problem for which the moduli are given by the loading branch of the elastic-plastic constitutive relation and are not, in general, derivable from a strain-energy-density function. The initial slope of the comparison problem is denoted by A:. to distinguish it from the slope of the elastic-plastic problem A, and in this notation (4.43) is replaced by

.d + ;I:..# = 0.

The symmetry of the eigenmode in many problems implies that s2 = 0 and thus ;Ite = 0. But if as given above is sufficiently large such that

( 1 )

fi;q3,h,'y$ + y a p ) > 0 (4.45)

throughout the body, then no reversal in sign of the strain rate will occur in some finite range of positive 5. The behavior of the comparison problem will coincide with that of the elastic-plastic problem in this range and, in particular, I,, = Ate. However as has been indicated, the typical and more interesting situation occurs when strain-rate reversal occurs at bifurcation in the comparison problem. In this case A , is the minimum value consistent with (4.9) and E L , > ),h,".

We return to the analysis of the elastic-plastic problem in which elastic unloading must be taken into account. The contribution of the boundary- layer terms in (4.33) can be rewritten using the stretched coordinates and (4.28) as

. * tD !A(MXY6KX,, + ';fFPii'E,,} d A

* * = (3q1 + p)3in;[j; 1 m 3 ~ ~ ~ l ~ , ~ (.+ r ' (ZI ) d v + . . . . (4.46)

The volume V is the region of elastic unloading expressed in terms of the stretched coordinates. It is enclosed between the two surfaces

' V

*

* * * * f ( z I ) = 0 and k 2 z 3 + h,,] (x ,zXzg = 0, (4.47)

where the plus (+) holds if x," = t/2 and the minus ( - ) if .x: = -t/2. By considering the three possibilities, [I > 3, f i < 4, and p = 4, we will

show that f i = 3 must hold implying a balance between the three terms listed in (4.33). First, suppose p > 3. Then by (4.46) the boundary-layer contribution to (4.33) would be of order larger than 5 . A balance between the first and third terms in (4.33) would require the first term to be of order 5. With the limit forjixed xi defined as in (4.34), the subsequent analysis leading to (4.43)

118 John W . Hutch iizsor i

would pertain. That is, with /I > 3 the effect of elastic unloading would drop out to this order requiring 2, to satisfy (4.43). But this is not possible because i1 > A t e by assumption.

If it is supposed that < 3 then there must be a balance between the first and second terms in (4.33), and in the limit 5 + 0 for fixed xi the quantities

W and LIE must be of order t3/’. Proceeding in a manner similar to that outlined i n Eqs. (3.34)-(4.43) one can show that the only solution for these limit quantities is some multiple of the eigenmode. Furthermore, /I < 3 requires the boundary-layer contribution (4.46) to be identically zero.

As already suggested, the proper choice is /j = 3. Then all three terms in (4.33) are of order ( and (4.34) still pertains.? The boundary-layer contribu-

tion [(4.46) with Csq = q ] must be added to the left-hand side of (4.35) as well as to (4.41). Equations (4.36)-(4.40) and (4.42) remain unchanged. Equation (4.43) is replaced by

h h

( 1 )

( 1 ) . * * (4.48)

where .c/ and A9 are still given by (4.44). This equation is the general equation for L2 . Note that it involves the eigenmode, the initial slope R I , and the derivatives of the instantaneous moduli evaluated at bifurcation.

The integral J f’dV can be evaluated in closed form for most cases of

3 -- I -a/) ( 4 ~ - 2 ) [ ~ c m c ~ u / j l x ~ l l % , f ( z i ) ~ V J = - ( , v l + ~ld ) ,

‘ V

*

interest. With C , , = 0 and h , , = 0 the general expression is

(x = 1, 2 ; no summation implied), where the minus ( - ) holds if .Y;‘ = t /2 and the plus (+ ) if x,” = -t/2.

The analog of (4.48) for three-dimensional solids derived by Hutchinson (1973a) can be specialized directly to (4.48) when the assumptions of DMV

I, I ,

TEigenmodal contributions to ( L i e , W ) or order lower than < are possible: however, i t is readily shown that such contributions do not influence the term iz<l+‘J in (4.1). To carry the expansion beyond this term it does become necessary to require the higher-order terms iii (4.29) to be orthogonal to the eigciimodc in some specific way.

Plastic Bucklirzg I19

theory are invoked. A n example will be given in the next subsection in which neutral loading at bifurcation occurs along a line of points. I n such cases two coordinates rather than three are stretched and one finds an expression similar to (4.48) but with /j = +.

When C, # 0, the unloading region is a thin sliver whose penetration in the z3 direction normal to the shell surface is O ( t U ) and whose extent along the surface is O(t” ’). Thus, as will be seen more clearly in examples which follow, the lateral extent of the unloading region will in general be on the order of the lateral dimensions of the plate or shell before the normal penetration is more than a small fraction of the thickness. I t is therefore not unreasonable to retain the assumption of approximate plane stress even after elastic unloading sets in. For an unusual problem i n which C, = 0 one cannot necessarily still make this claim since a different choice of stretched coordinates will have to be made. Such cases must be examined individually.

B. Two COLUMN PROBLEMS

1. Sinzplj~ Slipporter/ Colirriin with LI Solirl Circular. Cross Sect iori

Although the theory given above was derived for plates and shells i t can be converted by inspection to apply to one-dimensional column theory. Tensor quantities become their counterpart scalars. For a coluni~i undcr axial compression Y’ = E , FI = - 1, i - I = E - El, and rlL/rlr = rlEl/rl~. I t is well known that the approximate strain-displacement rclations of DMV theory are not adequate for an accurate analysis of the post-buckling behavior of linearly elastic columns. In the plastic range, however, the nonlinearity associated with the material behavior dominates the initial post-bifurcation behavior and more accurate strain-displacement relations are not needed.

The tangent-modulus load for a solid cylindrical column of radius R and length L is

P, = I?E:‘I/L‘, (4.50)

where I = $.rrR4. The eigenniode and its associated stress and strain fields are

( 1 ) ( 1 )

U , = 0, W = R COS(TCS,/L). ( 1 ) ( 1 ) ( 1 1

y~ = ( ~ ‘ R . ~ , / L ~ ) C O S ( ~ . ~ , ~ L ) , and r = E : V . (4.5 I )

where the Cartesian coordinates xi are orientated as shown in Fig. 13a. With

120 Johiz W. Hutchinson

FIG. 13. (a) Conventions for analysis of columns of circular and rectangular cross sections. (b) Approximate maximum support load from (4.61) for the column with a circular cross section and comparison with the reduced-modulus load for a column with a constant tangent modulus equal to E:; 4 = (nR/2L)*(dE,/tls),.

( 1 )

this normalization an eigenmodal contribution 5 W corresponds to a one- radius lateral deflection when ( = 1.

Let 1” = PIP,. (4.52)

I

Then qo = - (7&/2L)’ and, with fi, = - 1, 1 ( 1 )

ikc(/z, qo + q ) = (TcR/L)’[$L, - ( xJR)cos ( z~ I /L ) ] . (4.53) The requirement that L1 be the smallest value such that (4.53) is non- negative gives

2, = 4 (4.54)


and x: = (0, 0, R). From (4.11)

c, = - ( T c R / L ) ~ R - ~ , c,, = + ( 7 1 ~ / ~ ) 2 ( 7 1 / ~ ) 2 , c,, = c,, = 0, so that from (4.21)

* * * f ( z i ) = ( Z R / L ) ~ [ + + ( z J R ) - ~ ( T c z ~ / L ) ' ] . (4.55)

The term .d of (4.44) is zero by symmetry as can easily be verified; and for the one-dimensional column theory

= - E : ( ~ R / L ) ~ ~ R ~ L ( ~ + q), (4.56b) where q = ( ~ T R / ~ L ) ~ ( ~ E , / L / T ) , . In the stretched coordinates the lowest-order

equation for the surface of the column is 2z , + z$/R = 0 ; and using the general formula (4.49) one obtains

* *

( 1 ) * * (4&/3)'[[y; ' kc qIx,, (.* f ( z , ) a'V = - ($/22)3(E - E ; ) ( T C R / L ) ~ R ' L / ~ ~ ~

' V

(4.57) Combining (4.56) and (4.57) according to (4.48) gives

A, = - 3 p q i + 4) / (E - ~ 3 1 1 ' 3 . (4.58)

Thus the initial post-bifurcation expansion is

;1 = P/P, = 1 + 4< + ;1,(413 + ..., (4.59)

where < = 1 corresponds to a lateral deflection of one radius at the center of the column. From (4.24b) the lowest-order equation for the instantaneous neutral-loading surface is

(x, - R)/R - ~ ( T C ~ ~ / L ) ~ = & < 1 ' 3 . (4.60)

An alternative expression for A, was obtained by Hutchinson (1973a) within the context of the full three-dimensional theory by using the Euler- Bernoulli approximations to obtain approximate eigenmode fields. That expression agrees exactly with (4.58) when Poisson's ratio is f.

If the truncated three-term expansion (4.59) is used to estimate the maximum support load one finds

&/d< = 0 * t1l3 = -3/&

and

/2"'"" = 1 + ( E - E;)/~TcE;(~ + q ) = 1 + 0.106[(E - EF)/Et(l + q)]. (4.61)

I22 Jolzii W . Hutchiri.soii

From (4.60) the elastic unloading region has penetrated to the middle of the column and along the length of the column to y i = +J2 L/x. Figure 13b shows the approximate support load (4.61) a s a function of E;/E for several values of q. Included in this figure is the von Karmlin (1910) reduced- modulus load calculated for a circular cross-sectional column with constant tangent modulus E ; . I f the tangent modulus is constant the load should approach the reduced-modulus load (neglecting nonlinear geometry effects) and (4.61) is not accurate for this case. However, q is typically of order unity or larger for relatively slender columns with a stress-strain curve representative of common structural metals and thus enters into (4.61) in a significant way.

A further comparison of these approximate results with predictions based on full numerical calculations will be made in Section V,A. The main point to be made here is that the initial slope in (4.59) is reduced after extremely small lateral deflections due to the term i,, t4 which gives rise to an infinite curvature to the 1-5 relation at bifurcation.

2. Simplj, Supported ColLimz with a Solid R ~ T ~ L I I I ~ ~ N I . Cross S r c t i o ~ ~

Let h and c be the half widths of the cross section of the column as in Fig. 13a. The tangent-modulus load (4.50) still applies with I = $c3h. Now we choose the following normalization for the eigenmode:

( 1 ) w = c cos(n.L.,/L). (4.62)

Proceeding as in the previous example one finds:

i., = 3, (4.63)

(4.64) * * *

f ( z J = (xc/L)2[+ + ( z 3 / c ) - + z , / L ) 2 ] ,

and

ii.8 = - 4 E f ( ~ c / L ) ~ c h L ( I + q), (4.65)

where here

q = $ ( x ~ / L ) ~ ( d E , / d t ) , . . (4.66)

The main difference between this example and the previous one is that neutral loading occurs along a line (uiz x, = 0, x3 = c as depicted in Fig. 13a) rather than at an isolated point. To evaluate the lowest-order boundary-layer contribution to the principle of virtual work one now intro-

duces only two stretched coordinates z i and z3 defined as before in (4.19). * *

Plastic Birckliriy 123

Instead of (4.46) one obtains

(/' (. (Lao 6K,,) + I&' cS'E,,,i d A

. * * * - - -p- ( I + / ~ ) n , 1 ~ 1 ~ [ ~ ; lm;o 6p7ao]x,l ) * ,I'(~J [ I z , dz, dZ3 .

' V

By the same argument made when the neutral-loading surface is isolated, i t follows that /j = f . The equation for i, becomes

( 1 ) . * * * (-$,) 5 / 2 [yc -~ 1 wi, -a/J qaalx, , ! * , f ( z i ) d z , d z , d z , = .d + il,8. (4.67)

' V

Evaluating the left-hand side for the present case gives

- ( - { % 2 ) s ' 2 [ 16J2/( 135n,/3)](E - E : ) ( ~ c / L ) ~ ~ c L .

Combining the above and (4.65) according to (4.67) gives

A, = -(15/7)[15nEf(l + q)/4J2(E - E 3 ] 2 / 5 .

Thus the initial post-bifurcation expansion for the column with the rectan-

(4.68)

gular cross section is

1" = I + 3< + / 1 2 p + .. ' . The maximum value of A as predicted by the truncated series is

i."'"" = 1 + 8$(E - E:)/35nE;( 1 + 4 ) = 1 + 0.103[(E - Ef)/E:( 1 + y)],

(4.69)

which is almost identical to the cylindrical column result (4.61) except that y is defined by (4.66).

C. CIRCULAR PLATE UNDER RADIAL COMPRESSION

Bifurcation of a clamped circular plate of thickness t and radius R and subject to compressive loading by a uniform radial stress CJ was discussed in Section II1,C. The formula (3.56) for the lowest bifurcation stress gC involves the instantaneous modulus E,. and contraction ratio TC in (3.55); i.e.,

0,. = -k2ECt2/[12(1 - T:)R2]. (4.70)

The difference between the predictions of J , flow theory and J , deformation theory is illustrated by Fig. 10. I n this section the coefficients in the initial post-bifurcation expansion will be evaluated using J , flow theory which, of course, is a special case of the plasticity theory employed in the general

development of Section IV,A. We have already remarked that numerical results will be presented only for the range of parameters where the flow- theory predictions for the bifurcation load are very close to the deformation- theory predictions.

The fundamental solution is characterized by the uniform state of stress and strain (in Cartesian coordinates xi with x3 normal to the undeformed plate):

= ohzo and q$ = i;(SZo . (4.71)

It is convenient to introduce the moduli T and 7; for equal biaxial compression where ir = T i for an elastic response and 6 = 7;i for a plastic response where from (3.55)

T = E/(1 - v ) and 7; = E/(1 - i). (4.72)

In the notation of Section IV,A ~

m:, = -dEo and Zj, = +(T - Tf). (4.73)

U , = 0 and W = trr[J,(kr/R) - J,(k)], (4.74)

(4.75)

The eigenmode associated with the lowest bifurcation stress is ( 1 ) ( 1 )

where

u- ' = 1 - J , ( k ) 2 1.4027,

and r2 = .x: + xi. Here J , is the Bessel function of the first kind of nth order and k z 3.8317 is the first root of Jl(L) = 0. The normalization in (4.74) implies that the maximum eigenmodal deflection occurs at r = 0 and is equal to the thickness t. Define the load parameter according to

/I = O/ol. (4.76)

The function for determining I., is found to be

I I

where ( ) = [ I / ( ) /dA] , as previously defined and 1: = a,/TF. The minimum of this function occurs at x: = (0, 0, t/2) and choosing A, such that its value is zero at this point gives

A 1 = 3(1 + i , ) L I . (4.77)

and thus 1 ( 1 )

r%~a(IL1 q$ + q X a ) = (-20,i.,/Tf)[l - 2(.~,/t)J,(kr./R)]. (4.78)

Plustic Buckling 125

Using the definitions (4.1 1 ) and (4.21) one obtains * *

(4.79)

* * * where r = (x: + ~ 1 ) ' ' ~ = <-f l 'zr / { - (1 + @A2}1 /2 . The boundary-layer contribution to (4.48) can be evaluated using the general formula (4.49) for

J f t l V and is found to be *

The term .d in (4.48) is zero as a consequence of the symmetry properties

= (12x .Jt3))hilXll, ,?I ( 1 ) ( 1 ) ( 1 ) ( 1 1

of the eigenmode. Using the facts y~ aB = .Y 3 K afland z defined by (4.44b) can be written as

A fairly lengthy calculation using J , flow theory gives the derivatives of the instantaneous moduli evaluated at the state of stress zzB = CJ,~,,:

where

and

$2 = -3(T - T:)(l - v)/[(l + v)ac].

Here dT,/do denotes the derivative of the biaxial tangent modulus (4.72) with respect to the biaxial stress CJ introduced in (4.71).

Using standard identities for Bessel functions, d9 can be evaluated without

126 J o h n W . H utdi i n s o t i

approximation. The result is T : [ a k 2 J , ( k ) l 2 ( f / R ) ~ n R 2 f

6(1 + T,) .# = -

3(1 - v) (1 - 7,) T - T [ k 2 t (17; 4( 1 + v ) ( Ti ) + 24( l7) (lo l j l '

Thus (4.48) gives

4( I + 1,) +

where

const = 3 1 2 [ ~ ~ k J , ( k ) ] ~ ) ' 2 4.3808

(4.83)

(4.84)

(4.85)l-

The initial post-bifurcation expansion for the clamped circular plate is

A = 1 + l" , t + A 2 ( 4 3 + f . . , (4.86)

where A 1 is given by (4.77) and recall that ;" = 1 corresponds to a deflection at the center of the plate of one thickness. As an illustration. Fig. 14 shows a plot of A as a function of < as given by (4.86) for an example for which a full

N E U T R A L L O A D I N G S U R F A C E A T C = 0 001 1 / - A T C = O O I 6 ( M A X L O A D )

-_- __-' ._ a _ - -

-- ~ R - O - I ' - * -

L I P - I I 1 - 1 0 0.01 0.02 0.03 004 0.05

c FIG. 14. Initial post-bifurcation behavior of ii clamped circular plate under I-adinl

compression [see (4.X7) for a specifcation of parm~eters]. The thickness is exaggerated i n the insert. Dashed line: 1. = I + 1.730:: solid line with dot: 1. = I + 1.730: - 5.143<' '.

.I I am indebted to A. Needleman of MIT who independently derived these results and made them available to me to provide a check on my own analysis.


numerical analysis will be reported in Section V,B. The stress-strain curve (3.59) for uniaxial tension-? is used in this analysis with a strain-hardening exponent I I = 12 so that the bifurcation results of Fig. 10 for J , flow theory apply. The abscissa in Fig. 10 was chosen to have the value li2(t/R)’/

. [ 1 2 4 1 - v2)] = 2 and Poisson’s ratio 1’ was taken to be f. The associated valu-es of the quantities which enter into the evaluation of i1 and %, are

oJa, = 1.12, Tc = -0.191, E J E = 0.607,

T;/E = 0.510, and ~,,(dT,/do)~ = 4.58. (4.87)

All the terms in i1 and i2 can be expressed in terms of these nondimensional values and one finds

i,, = 1.730 and i2 = -5.14, (4.88)

and these were the values used in (4.86) to plot Fig. 14. Also shown there is the neutral-loading surface at two values of <, where the larger v’ ‘i 1 ue corresponds to the maximum load point. According to these results the slope has been reduced substantially from its initial value at < = 0.01 corresponding to a normal deflection at the center of the plate of only one-hundredth of its thickness. The three term asymptotic expansion indicates the maximum load is less than 1 higher than the bifurcation load and is attained when the normal deflection is approximately 1/50 the plate thickness. We will see in Section V,B that these values are underestimated by the truncated initial post-bifurcation expansion, although not by much. The rapid transition from stable to unstable behavior under dead load (i.e., the transition from increasing to decreasing load) in the plastic range as typified by this example can be contrasted with the highly stable post-bifurcation of the plate in the elastic range.

D. EFFECT O F I N I T I A I , IMPERFECTIONS

N o general treatment, analogous to Koiter’s (1945, 1963a) theory for conservative elastic systems, is available for the effect of initial imperfections on the maximum support load of a structure compressed into the plastic range. The importance of accounting for the interaction of imperfections and plastic deformation has long been recognized i n the design of columns against buckling where approximate ways for estimating the effect of imperfections, such ;IS the Pert-). first-yield formula, are uscd [see, for c\aniple. the

+ In (3.59) CJ denotca a uniaxial stress wliilc in this section o has bcen reserved for thc applied radial stress and thc pin-bifurcation biaxial strcss (4.7 I ) .

128 Jolin W . Hut ~ 1 1 iiison

monograph by Johnson (1966) and the recent paper by Calladine (1973)l. In addition to the results already presented for the models of Section 11, some selected numerical results for columns, plates, and shells will be given in the next section. In this section two theoretical points will be made which have a bearing on imperfection sensitivity in the plastic range.

1. Effbct q f m 1niper:fl.ction OM the Behacior iii the Plustic Range Prior t o the Oiiset of Elustic Unloading

Assume the imperfection is an initial stress-free deflection of the middle surface from the perfect configuration in the form zw(x'), where is the amplitude of the imperfection. For simplicity we will restrict consideration to problems in which the stress histories are proportional at every point (as in the column problem) or in which a deformation theory is used. In either case the constitutive relation can be regarded as being nonlinear elastic prior to elastic unloading and Koiter's general theory for elastic systems can be applied. The general case where the loading is not proportional is somewhat more complicated than that which will be presented below but may still be worked out along the lines given by Hutchinson (1973b) for the three- dimensional theory.

The result of applying the Koiter approach is an asymptotically exact relation involving the load parameter 1, the amplitude [ of the eigenmodal contribution to the deflection, and the lowest-order contribution of the imperfection 5:

(4.89)

where ( 1 )

p = ()A$)- (. {Nt'L w , ~ w./J + ' A

The superscript e is used to emphasize that

( 1 )

N a p W z W , p ] rlA. (4.90)

this result holds for nonlinear elastic solids but not for hypoelastic solids; /i: = - d / M , where x/ and A are still given by (4.44). Associated expansions for other quantities such as the strains are of the form

( 1 ) ( 2 )

q z p = q $ ( 4 + i; q '/r + t2 q a/? + ' ' ' (4.91) ( 1 ) ( 2 )

= q:i + (i - lc)qt/l f t q a / j + t Z q a / J + " ' > (4.92)

where terms of order (3, - %J2 and higher-order terms involving are not needed in the subsequent analysis and are not shown.

The above results apply to the elastic-plastic structure (under the restrictions mentioned) prior to the onset of elastic unloading. Reversal in sign of


the strain rate occurs for the first time when i7zGl"& = 0. Using (4.17) (4.92) and ( ' ) = d ( )/Lit we find

(4.93)

The initial slope A 1 of the perfect structure, required for bifurcation to occur at &, is the smallest value consistent with (4.9). From (4.93) it follows that in the slightly imperfect structure the onset of elastic unloading occurs when L ~ A / I I < is reduced to the value A1, i.e.,

(4.94)

where ( A ) identifies quantities evaluated at the onset of strain-rate reversal. Using (4.89) and the condition (4.94) one finds

= [ A ' p z / ( A , - 2 3 ] l ' 2 + " ' , (4.95)

;z/i, = 1 - [(Al - 24)/i'][1.'pz/(Al - + " ' ) (4.96)

generalizing the simple-model results (2.54) and (2.55). The maximum load occurs following the onset of elastic unloading when

dA/d( = 0. If the destabilizing nonlinearities are large the maximum load will be attained shortly after the onset of elastic unloading; and (4.96) is suggestive of the effect of very small imperfections on the maximum load, as discussed in conjunction with the simple model. I n general, however, it is not possible to predict the effect of an imperfection on the maximum support load by developing a perturbation expansion about the bifurcation point, as in the elastic range. For the discrete model of Section II,A it was possible to obtain a closed-form formula, (2.19) or (2.20), for the effect of an imperfection. The reduction in the maximum load was proportional to the square root of the imperfection. Although no explicit formula could be obtained for the continuous model, the expansion (2.57) as well as the approximate formula (2.58) suggests that the imperfection enters through the square root of its amplitude in this case too.

2. Ejjkct of un Inipeiyfkction on First Yielding MfIien Bifiircut ioii oftlie Prryfict Strircture Occurs in the Elustic Runye

We turn now to the complementary situation where the bifurcation load of the perfect plate or shell occurs in the elastic range at stress levels only slightly below the initial yield stress. Using the asymptotic initial post- bifurcation expansions for the response prior to plastic yielding, we examine the condition for the first occurrence of plastic yielding when imperfections are present. I t is assumed that there is a smooth initial yield surface

130 Jolm W. Hutch in son

3(@) = 0 such that for stresses within the surface ( i .e . ,9 < 0) the material response is linearly elastic. If at any point the bifurcation stress of the perfect structure is just within or on the yield surface then the first-yield condition can be approximated by

.9, + (T*P - T : ' J ) ( ? T / ? T q = 0, (4.97)

where .Fc = .F(T:") I 0. Prior to first yielding the expansion (4.89) to (4.92) applies with the

moduli taken to be the constant elastic moduli. The solid-line curves in

I ( a ) i ( b )

I YIELDING I & &

FIG. 15. Skctchcs of the initial poat-bifurcation behaviol- and imperfcction sensitiL ity where thc bifiircatioii stress of the perfect structure falls j u s t below the initial yield stress. (a) Synimctric bifurcation point. (b) Asymmetric bifurcation point.

Fig. 15 depict the elastic load-deflection response; the slash indicates first yield and thereafter the curve is dashed to indicate further yielding. Using the expansion for the stresses which is similar to (4.92) the first-yield condition can be rewritten as

I ( 1 ) ( 2 )

.p, f [(IL - + < T "' + t2 5 "' + ' ' ' ] ( ? , p / ? T " " ) , = 0. (4.98)

Multiply (4.98) by ( and use (4.89) to eliminate (A - A,) thereby obtaining the first-yield condition in terms of ( and <:

( 1 ) , (.Fc + p ( . F + r; 30) - ?piFo + . . ' = 0, (4.99)

where ( 1 ) (1 )

3 = T *~J(M/?~3LB)c and .Po = ~~; ' ( (c? . ' i ' / c7~"~) ) , . (4.100)

Equation (4.99), together with (4.89), provides a set of equations for obtaining the values of i, and ( at which yielding will first occur.


To simplify the discussion assume that the fundamental solution is a uniform state of stress and that for r = 0 bifurcation occurs on the verge of yield so that Fc = 0. Thus the fundamental solution has the property that

9, is a positive constant; in general, F will be positive over part of the structure and negative over the rest. Let

I ( 1 )

( I )

to = maximum value of (3 + 2:.Fo). (4.101)

I f w > 0, then from (4.99) first yield occurs at the point (or points) in the structure where the maximum in (4.101) is attained and

I

tfi,.;, yjc,d = ( ~ p i c F o / ~ / p 2 + ’ ’ ‘ . (4.102)

Here 2‘ has been substituted for i by neglecting terms of order Jz(2 - i,). Note also from (4.89) that z p > 0 implies 5 > 0; the opposite-signed deflection in the eigenmode corresponding to sp < 0 can be treated similarly. Equation (4.102) is asymptotically exact for small imperfections. If the bifurcation point is symmetric so that )$ = 0, it is readily verified from (4.89) that a maximum load for the elastic. structure occurs in the neighborhood of %, only if 1”; < 0. It is attained at a value of < = 0(11/3), which is inherently larger than (4.102) for sufficiently small imperfections. Thus if a perfect structure has a symmetric bifurcation point and undergoes bifurcation just on the verge of yield, the maximum load of a slightly imperfect version of the structure will always be attained ufter- plastic yielding has occurred, whether or not the bifurcation point of the elastic structure is stable or unstable. Using (4.89) and (4.102), the asymptotic equation for the effect of small imperfections on the first yield of a structure with a symmetric bifurcation point is

,Ilirst = i,, - (p<iw,w/-Fo)1’2 + . . . . (4.103)

Thus when 3, = 0 the imperfection reduces the first-yield load in proportion to the square root of the imperfection as has been noted for columns on the basis of the Perry first-yield formula by Thompson and Hunt (1973) and Calladine (1973).

If the bifurcation point is asymmetric it can be possible for the maximum load to be attained hejor-e first yielding occurs in the imperfect structure even when the perfect structure buckles just at initial yield. If 2; < 0 the maximum load of the elastic structure calculated from (4.89) occurs when t 2 = @</( -2:). Substituting this value into (4.99) (still with 3, = 0) gives

prlL[w + 4 .To] for the point where the stress is nearest the yield surface. If this quantity decreases with increasing <, then the maximum load will occur

132 John W . Hutch insoii

before first yield for sufficiently small imperfections. The significance of this, of course, is that the maximum load can be obtained from the elastic analysis. Using (4.10 l), the condition that yielding occur following attain- ment of the maximum load in the slightly imperfect structure, when the perfect structure bifurcates just at the yield stress, is that 2.; be sufficiently negative such that

( 1 1 , maximum value of (9 + 2/2:F0) I 0. (4.104)

The long cylindrical shell under axial compression is one of the best known examples of a structure with a highly asymmetric bifurcation point. This problem has a multiplicity of eigenmodes associated with its bifurcation stress and is not described by the one-mode expansion (4.89). However the generalization of (4.104) for the multiple-mode case can easily be obtained. Using Koiter’s (1945, 1963a) solution to this problem we have found that for the case of an axisymmetric imperfection in the shape of the axisymmetric eigenmode the condition analogous to (4.104) is not satisfied. In other words, if the perfect shell buckles just at the initial yield stress, the buckling load of the slightly imperfect shell will nut be given by the elastic analysis but undoubtedly will be somewhat lower. Further results on this problem will be given in Section V,C. There it will be shown that, if the yield stress is just 5 % above the bifurcation stress of the perfect cylinder, buckling of the imperfect shell will be elastic over the entire range of imperfection levels of interest, at least for this particular imperfection.

V. Numerical Examples

A. COLUMN UNDER AXIAL COMPRESSION

Consider a simply supported column of length L with a solid circular cross section of radius R. The elastic-plastic material comprising the column is taken to have a Ramberg-Osgood uniaxial stress-strain curve, as in Section IV,B, i.e.,

C / E , = ,c/o, + +(O/(T?)n, (5.1) where c, is the effective yield strain and (T, = E q is the effective yield stress. The tangent-modulus load P , is given by (4.50) and the bifurcation stress (T,

is given in terms of the single geometric parameter ( x R / ~ L ) ~ c ~ : by

(~R/~L) ’E , = ((T,/(T,) + Sn((T,/oy)n. (5 .2)

This relation is plotted in the form most often used for displaying column-


buckling results as a solid-line curve in Fig. 16 for M = 10, where P, = nR2cr, and P, = nR2u,..

or

p max - PY

1.2 -

1.0 -

.a -

.6 -

.4 -

.2 -

FIG. 16. Bifurcation stress of perfect column and maximum support load for two imperfection levcls for a circular cylindrical column under axial compression. WIMP - = <K cos(nX/L). (Ramberg-Osgood strcss-strain curve with I I = 10.)

Included in Fig. 16 are the results of an accurate numerical analysis for the maximum support load (normalized by P,) of the column with two levels of imperfection, where the imperfection is in the shape of the buckling mode

Wimp = ZR cos(nx/L). (5.3)

The results are based on a column theory which is a one-dimensional version of the plate and shell theory used in Sections I11 and IV; details of the numerical calculations are similar to those reported for the axisymmetric deformation of shells by Hutchinson (1972) and to those employed in a recent paper on column buckling by Huang (1973). The influence of imperfections is most marked in the region where the elastic predictions break down. Results such as these are well known and have been incorporated into column design procedures. The monograph by Johnson ( 1966) reviews many of the column-buckling studies, both theoretical and experimental, and discusses design criteria. A recent contribution by Calladine (1973) considers the application of a Perry first-yield-type formula for imperfect columns comprised of strain-hardening metals. The papers by Malvick and Lee (1965) and Huang (1973) discuss additional aspects of plastic column buckling.

Figure 17 displays these same results replotted in the manner of Duberg

134 J ohri W . Hutch iris on

pmax p, .9-

8 -

.7

(1962) and the simple-model results of Fig. 8. Hcre the maximum support load is normalized by the tangent-modulus load P, of the perfect column and is plotted as a function of the bifurcation stress of perfect column (T,

normalized by (T,.. These curves bring out the fact that even when the bifurcation stress is substantially below the effective yield stress the effect of a small imperfection can be appreciable due to plastic deformation. These

-

prnax -

Pc .9-

8 -

.7 -

1 I 0 .2 4 .6 8 10 1 2

WUY

u =O.l

L b 1

0 .2 4 .6 .8 1.0 1.2 5 c /U

FIG. 17. Effect of impcrfcctio~is on tlic I ~ ~ L X I I ~ U I I I support lond its a function of thc bifurcation stress of the perfect column (i, normalized by the effective yield stress (iv. Dashcd curve is the approximatc maximuin load (P"'." ' P , ) for the perfect column as predicted by the initial post-bifui-cation aii~ilysis [Eq. (4.6I)l. (a) I I = 3 : (b) I I = 10. W""' = :K cos(?rX'L).

curves are quantitatively similar to Duberg's (1962) curves for a two-flanged column model, as well as to those for the continuous model of Fig. 8. Also shown in Fig. 17 are the predictions of PmdX/Pc for the perfect column calculated using the approximate formula (4.61 ) of the initial post-bifurcation analysis. The values of EE/E and q needed in this evaluation can be expressed in terms of the tangent modulus and its derivative evaluated at ( T ~ together with the geometric parameter of (5.2). The approximate predictions appear

Plustic Budl ing 135

to underestimate the maximum support of the perfect column slightly as would be expected from the discussion given previously in connection with Fig. 13. Recall that the asymptotic equation (4.60) for the neutral-loading surface predicts that the elastic unloading region penetrates to the middle of the column at its midpoint at maximum load and along the column to a distance 3 L/n on either side of its midpoint. The numerical analysis indicates that asymptotic results at maximum load somewhat overestimate the penetration on the midplane and underestimate the distance attained along the column.

B. CIRCULAR PLATE UNDER RADIAL COMPRESSION

Bifurcation results for a clamped circular plate under radial compression were discussed in Section III,C and the initial post-bifurcation expansion was given in Section IV,C. Needleman (1973) has carried out a full numerical analysis of the post-bifurcation behavior and imperfection sensitivity of clamped and simply supported circular plates using a finite element method. A few of his examples will be presented here.

Needleman (1973) used J 2 flow theory together with the tensile stress- strain relation (3.59) which has a distinct yield stress and a continuous tangent modulus. The first example is the one considered in Section IV,C. The strain-hardening exponent is taken to be / I = 12 with 11 = i, and the bifurcation curves of Fig. 10 apply. The geometric parameter is chosen to be

k2(t/R)2/[ 12( 1 - v 2 ) 4 = 2, (5.4)

A t this value the flow-theory and deformation-theory predictions are essentially identical: CT,/(T, = 1.12 and other parameters are given in (4.87). In Fig.18 curves of the applied edge stress u normalized by the bifurcation stress of the perfect plate u, are plotted as a function of = W(o)/ t , where W ( 0 ) is the additional deflection at the center of the plate. The imperfection is taken as an initial stress-free normal deflection in the shape of the buckling mode and its amplitude at the center of the plate is denoted by w(0). Responses for two slightly imperfect plates are shown ; the curve for the " perfect " plate was calculated for a plate with an extremely small imperfection, = w(o)/t = lop4.

The maximum support load of the " perfect " plate is only about 1.5 higher than the bifurcation load and the maximum load is attained at a lateral deflection of about 0.065t. Comparing these values with the corresponding values from the initial post-bifurcation analysis shown in Fig. 14

I36

/

LO,? -

John W . Hurchinsori

FIG. I X . Post-bifurcation behavior and imperfection sensitivity of a circular plate radially compressed into the plastic range. Solid line ct initial yield; ++ maximuin load: A t* onset of strain-rate reversal. (From Needleman, 1973.)

indicates that the initial post-bifurcation analysis in this case underestimates both the increase above the bifurcation load and the value of the deflection at which the maximum load is attained. At deflections beyond the maximum load the initial post-bifurcation curve turns down rapidly while the actual curve remains quite flat. This is characteristic of the behavior of a truncated (asymptotic) series beyond its range of validity. The importance of the initial post-bifurcation expansion is that i t explains how the stable region (for dead loading) can be as extremely small as it is in spite of the substantial initial slope of the load-deflection curve. Coupled with this is a definite imperfection sensitivity which is completely absent in the elastic range. The size of the elastic unloading region at maximum load predicted by the asymptotic analysis of Section IV,C is in reasonably good agreement with the numerical predictions (Needleman, 1973), although the penetration of the region into the plate at its center falls slightly short of the asymptotic prediction.

Figure 19a shows load-deflection curves for a simply supported plate which bifurcates just outside the elastic range = 1.02, n = 12; the stress-strain curve (3.59) is still used]. As the first-yield analysis of Section IV,D would suggest, the imperfection sensitivity is about as pronounced as for a compressed column. The significant difference between the column and the plate shows up when the bifurcation stress of the perfect plate is somewhat below the yield stress. Then, as can be seen from the curves of Fig. 19b, the imperfection has relatively less effect in reducing the maximum load below the bifurcation load. This is related to the fact that in the elastic range

137

0.8 l . o h T RFECT

n Y

1 0 1 i b '

0.8 1

o.2 t nL

"0 05 .10 .I5 2 0 2 5 .30 .35 "0 0.2 0.4 0.6 0.8 10 E - W i O ) / t u c /uy

FIG. 19. (a) Post-bifurcation behavior and imperfection sensitivity of ii circular plate undei radial stress. lcvels

compression where thc bifurcation stress of the perfect plate is nearly equal to the yield Solid line u initial yield; u maximum load. (b) Maximum support stress for scvcral of imperfection as a function of cr)cr,. (From Needleman. 1973.)

the plate has a highly stable post-bifurcation behavior and undergoes much smaller lateral deflections at loads above the bifurcation load than does the column. Consequently it is subject to far smaller bending stresses.

Graves Smith (1971) has studied the effect of imperfections on the buckling of thin-walled box columns which are built up of flat plates. He also has shown that the effect of imperfections on plate buckling can be significant when the yield stress is not sufficiently in excess of the bifurcation stress. Tests and calculations of Dwight and Moxham (1969) and Dwight (1971) definitely show that imperfections of various kinds can have an important influence on the buckling of flat plates in the plastic range.

C. SPHERICAL AND CYLINDRICAL SHELLS

The two-shell structures discussed below are characterized by highly imperfection-sensitive behavior in the elastic range as opposed to the columns and plates discussed above which are relatively insensitive to small imperfections in the elastic range.

The bifurcation stress for a thin, perfect spherical shell under external pressure is given by (3.60), and predictions for a shell whose material is characterized by a Ramberg-Osgood-type tensile stress-strain relation (2.50) with c( = & and I I = 6 are shown in Fig. 11. Full numerical calculations for the axisymmetric post-buckling behavior of the sphere were reported by Hutchinson (1972) for J , flow theory and a J , deformation theory with elastic unloading incorporated. I n Fig. 20a curves of applied

138 Johii W . Hutchiii.soii

INITIAL SLOPE

t I/ )'PERFECT SHELL"

O ~ $ l I I , I

OO .I .2 .3 .4 .5

J, FLOW THEORY PREDICTIONS

0.6

THEORY PREDICTIONS 0.2

E = Wpole/t r FIG. 20. (a) Plastic post-bifurcation bcliavior and iniperl'ection sensitivity of a spherical

shell undcr exiernal pressure. (b) Imperfection scnsitivity of spherical slicll. From Hutchinson (1972) J . A / J ~ / . Mech. 39, 155 162, with permission.

pressure are shown as a function of the amplitude of the buckling defection (corresponding to the inward dimpling at the poles of the sphere nor-

malized by the shell thickness t) . The geometric parameter of the sphere is

[3 ( 1 - ?)I- '12t/c:,R = 3, (5 .5 ) and J , fow theory was used for the calculations of Fig. 20a. From Fig. 1 1 it is seen that the bifurcation stress of the perfect shell is 1.5 times the effective yield stress cry and is about 7% above the prediction of J 2 deformation theory. The imperfection was taken in the shape of the eigenmode and its amplitude is denoted by g, corresponding to the inward initial deflection of the dimples at the poles of the sphere normalized by t. Additional details are given in the above-mentioned reference.

Imperfection-sensitivity curves are shown i n Fig. 20b in the form of the maximum pressure normalized by the maximum pressure of the perfect sphere. Results for flow theory and deformation theory are shown; the maximum load of perfect shell calculated using J 2 flow theory was used for the normalization in both cases. Significant imperfection sensitivity is indicated. Very small imperfections reduce the discrepancy between the two sets of predictions. Similar behavior has been noted for the cruciform column by Cicala (1950) and Onat and Drucker (1953) where, as discussed in Section- lII,C, the disparity between simple flow and deformation theories for the perfect structure is considerably larger.

Highly unstable post-buckling behavior in the plastic range was observed by Leckie (1969) in a series of tests on hemispherical shells subject to concentrated loads applied through rigid bosses of various diameters. A large variation in the maximum support load as a function of the boss size was found and was correlated approximately with a rigid-plastic post- bifurcation analysis.

Plastic Biicliliiig I39

The thin monocoque cylindrical shell under axial compression provides an interesting illustration of the phenomena brought out by the simple model in Fig. 4 where, due to high imperfection sensitivity, the buckling of an imperfect version of a structure may be less influenced by plasticity than is the perfect structure. Recall from Section IV,D,2 that when the perfect cylindrical shell bifurcates just at initial yield, the asymptotic analysis indicates that a slightly imperfect shell will start to yield plastically before the maximum load is attained. A closer look at this problem shows that for even larger imperfections the opposite will happen. We make use of Koiter's (1963b) special solution for the effect of an axisymmetric imperfection on the elastic buckling of a long cylindrical shell under axial compression. The imperfection is in the shape of the axisymmetric eigenmode associated with the bifurcation stress, and an exact, relatively simple nonlinear solution for the axisymmetric pre-buckling deformation is available. Koiter's (1963b) upper bound to the buckling load of the imperfect shell P* is plotted in Fig.2 la as P*/Pc as a function of z, where zt is the amplitude of the imperfection and t is the shell thickness.t

Using the axisymmetric pre-buckling solution for the elastic shell the maximum value of the effective stress, oefl = occurring in the shell can be calculated. With P denoting the axial load and 3, = P/P, , this value is given by

((T,~~/(T,.)~ = IL2 + yL2(l - % - '(6 + c - 311)

+ z 2 i 2 ( 1 - it)- '[9( I - 11 + v 2 ) + cz + 3c - 6 ~ ~ 1 , (5.6) where c = [3( I - v ' ) ] ' : ~ and oc is the bifurcation stress of the perfect cylinder. Furthermore. the maximum value of (5.6) to be attained prior to buckling does occur at P *. Figure 2 I b shows a plot of ( r~~~ .~ /o<)* as a function of 4 calculated using i,* = P*/P, from Koiter's upper bound with 13 = 4. For small 5 one can use the asymptotically exact result A* = 1 - ( ~ C Z ) ' ~ ~ + . . . in (5.6) to obtain (with 11 = 4)

(o,.ct./oc)* = 1 f 0.55z"2 + ..., (5.7)

consistent with the result mentioned in Section IV,D,2. However (5.7) holds only for very small t; and in an intermediate range o f t , (crrff/oc)* drops below unity as seen in Fig. 21b.

If the perfect shell bifurcates just at yield (i.e., crJ/o, = l), then for P*/P, greater than about 0.5 the maximum load is attained after plastic yield

+The buckling load P* in this context is dcfned to be the load at which bifurcation from the axisgmmetric statc occurs. Budiansky and Hutchinson (1972) have shown that this bifurcation load is actually the maximum support load of the elastic shell for all values of P* ' P c greater than about 4.

140

RANGE OF ELASTIC BUCKLING FOR Uy/rc = 1.05

p" PC

0.2

R A N G E OF ELASTIC BUCKLING FOR r y / q = 1

OO .i .i . t - d 1.: 1.; 1.; E

I \

p" PC

R A N G E OF ELASTIC BUCKLING FOR

'0 .2 .4 .6-.8 1.0 1.2 1.4 E

0 .2 4 .6 . 8 - 1 . 0 1.2 14 1.6 E

Flc;. 21. (a) Imperfection-sensitivity curvc fo r tlic elastic buckling of a long circular cylindrical shell with an axisymmctric imperfection and subject to axial compression. k~roni Koiter (I963b). Also shown is the range of validity of the elastic results for two values of oJoc . (b) Maximum effective stress occurring prior to buckling as predicted by the clastic analysis 01' the imperfect shell.

occurs, assuming the Mises yield condition applies. But in the intermediate range 0.2 < P*/P, < 0.5 the buckling load is attained bejow plastic yielding sets in and thus the elastic analysis is strictly valid in this range. For thin cylindrical shells with typical imperfection levels this is the range in which many shells buckle. Since the elastic analysis is valid for the perfect shell and in the intermediate range, presumably i t cannot be far off for 0.5 < P*/P, < I . In fact if ciy/cic L 1.05, the elastic analysis holds over essentially the entire range of interest as indicated in Fig. 21a.

Although some experimentalists have drawn attention to the possibility of interaction of plastic deformation and imperfections in their buckling tests on thin cylindrical shells, most thin monocoque cylinders of structural metals are reported as having buckled elastically. The present observations suggest that this is not to be unexpected as long as the yield stress is somewhat above the bifurcation stress of the perfect shell, but other imperfection shapes and boundary effects may alter this conclusion to a certain extent. Mayers and Wesenberg (1969) and Wesenberg and Mayers (1969). have carried out detailed numerical calculations for the interaction of imperfections and plastic deformation in stiffened and unstiffened cylindrical shells under axial compression. They have delineated the range of thickness to radius in which this interaction will be important for several stress-strain curves of metals used in cylindrical shell construction which have rather ill-defined yield stresses.


ACKNOWLEDGMENTS

This work was supported in part by the Air Force Office of Scientific Research under Grant No. AFOSR-73-2476, in part by the National Aeronautics and Space Administration under Grant No. NGL 22-007-012, and by the Division of Enginccring and Applied Physics, Harvard University.

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Multistructured Boundary Layers on Flat Plates and Related Bodies

K. STEWARTSON Depurtment of Mathernutics G'riicersity Colkgqe Loritlori

Lorirlon, Eiigqltrnd

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 A. Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . 146 B. Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . 152

11. The Basic Flow . . . . . . . . . . . . . . . . . . . . . . . . . I56 111. The Triple Deck . . . . . . . . . . . . . . . . . . . . . . . . . 158

A. The Main Deck . . . . . . . . . . . . . . . . . . . . . . . . 15X B. The Upper Deck . . . . . . . . . . . . . . . . . . . . . . . 162 C. The Lower Deck . . . . . . . . . . . . . . . . . . . . . . . 163

167 V. Transonic Free Interaction . . . . . . . . . . . . . . . . . . . . 169

VI. Free Interactions i n Supersonic Flow . . . . . . . . . . . . . . . . 171

IV. The Fundarnental Equation or the Triple Deck . . . . . . . . . . . .

VII . Expansive Free Intcractions . . . . . . . . . . . . . . . . . . . . 174 VI I I . Compressive Free Interactions . . . . . . . . . . . . . . . . . . . 176

A. Lower Deck. Sublayer ( i i i ) . . . . . . . . . . . . . . . . . . I84 B. Lower Deck, Sublayer ( i i ) . . . . . . . . . . . . . . . . . . 184 C. Lower Deck, Sublayer ( i ) . . . . . . . . . . . . . . . . . . 185 D. MainDeck . . . . . . . . . . . . . . . . . . . . . . . 185 E. The Upper Dcck . . . . . . . . . . . . . . . . . . . . . . . 186

X. Comparison with Experiment . . . . . . . . . . . . . . . . . . . 1 X X

A. Slot Iiijection . . . . . . . . . . . . . . . . . . . . . . . . 203 B. Plate Injection . . . . . . . . . . . . . . . . . . . . . . . . 206

IX. The Plateau . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

XI. Convex Corners in Supersonic Flow . . . . . . . . . . . . . . . 194 XII. Injection into the Supersonic Boundary Layer . . . . . . . . . . . . 200

XII I . The Trailing Edge ol'a Symmetrically Disposed Flat Plate . . . . . . . 213 XIV. Trailing-Edge Flows for Bodies with Finite Thickness 222 XV. Viscous Correction to Lift . . . . . . . . . . . . . . . . . . . . 225

XVI. Catastrophic Separation . . . . . . . . . . . . . . . . . . . . . 229 231

References . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

. . . . . . . . .

XVII . Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . .

I45

146

1. Introduction

K . Stewartsoti

A. INCOMPKESSIRL.E FLUIDS

I t is remarkable how little progress there has been in the development of a proper mathematical description of the nature of incompressible fluid motions at high Reynolds number based on the properties of the Navier-Stokes equations, when one considers that the equations have been known for over 100 years and are generally regarded as providing a correct description of the macroscopic properties of the motions in a wide variety of circumstances. The two features of these flows which proved to be too difficult for mathematical analysis to date and have probably been the main obstacles to successful theories are, of course, turbulence and separation. Of these, turbulence arises from the increasing proneness of fluid motions to instability as the Reynolds number increases, and the outlook for a proper mathematical understanding of this phenomenon is, at present, not encouraging. Since we shall restrict attention in this review to steady flows, or at least well-ordered unsteady flows, we shall not consider it further even though we recognize it as the central problem of present-day fluid mechanics. The other feature, separation, seems to be present in many, if not most, steady flows at high Reynolds numbers, and must be regarded as a highly significant factor in the mathematical theory describing them. In this phenomenon the main body of fluid streaming past the obstacle appears to break away from it at some point on its surface; in experiments an unsteady eddying wake is observed to occur to the rear of the body. Theoretically no solutions are available at Reynolds number R sufficiently high to enable us to make confident predictions about the nature of separated flows in the limit as R --+ x, even by numerical means. Reliable solutions are, however, available at values of R - 100 (e.g., Dennis and Chang, 1970; Takami and Keller, 1969) which supply convincing evidence that the separated part of the fluid flow occupies a large region when R S 1 and may extend to infinite distances downstream as R --+ sc. The theory of the laminar separation has recently been reviewed by Brown and Stewartson (1969) who bring out some of the formidable problems facing the mathematician endeavoring to set up asymptotic theory for flow with separation, including a possible pathological structure of the limit solution.

Provided separation does not occur, the procedure for setting up the asymptotic structure of the solution was, until recently, generally believed to be as follows. First compute the inviscid flow according to irrotational theory. Then deduce the boundary-layer structure near the body, needed to

reduce to zero the slip flow on the body, which is a consequence of the irrotational theory. Third, modify the irrotational solution by taking note of the displacement thickness of the boundary layer whose effect is equivalent to a small normal velocity at the body. Fourth, find the perturbation to the boundary layer which is induced by the change in slip velocity at the body according to the irrotational theory. And so on. A comprehensive review of the higher approximations to boundary-layer theory has recently been prepared by Van Dyke (1969). As a successful application of these ideas one may cite the flow past a symmetrically disposed parabolic body which has also been extensively studied numerically by Dennis and Walsh (1971). Davis (1972), and others. The agreement between the numerical solutions and the asymptotic expansion is good, and we can expect that as a result the flow structure is understood at all values of R. The asymptotic expansion of the local skin friction C,. for example can be expected to take the form of a series of descending powers of R, starting with R-"2. Thus near the nose of the parabola

Cf = (1.2326R-'" - 3.03R-').s + O ( . S ~ , R - ~ ' ~ 1 1 (1.1)

where the Reynolds number R is based on the nose radius and s is the nondimensional distance from the nose.

The limit, as the nose radius tends to zero, of a parabola, is the semi- infinite plate, which can also be regarded as fully understood. There is now no question of an expansion in descending powers of R, since R = 0. Rather, we need expansions valid near the leading edge, and at large distances downstream, together with a numerical solution spanning the two series. If we assume the plate coincides with the positive x axis, formal expansions, valid near x = 0, are provided by Carrier and Lin (1948), but their attempt to determine one of the unknown constants by reference to the Blasius boundary layer turns out to be valid only in an approximate sense. The structure they propose for the flow near the nose is dominated by a Stokes' flow, in which the pressure is singular like (-x)- ' I 2 on the x axis as x -+ 0- while it is bounded as x -+ 0 +, i.e., on the plate. This structure has been criticized by Lugt and Schwiderski (1965) but a full and careful numerical solution by Van de Vooren and Dijkstra (1970) confirms the correctness of their general structure and determines the unknown constants. This numerical solution also joins on to the asymptotic structure when s 4 1 (reviewed by Goldstein, 1960). However, this expansion is formally incomplete since it omits all the eigenfunctions of the Blasius solution after the first, together with the series of functions which depend on them and arise from the nonlinearity of the Navier-Stokes equations. Each of these eigenfunctions has associated with it an arbitrary constant which cannot be determined solely from the asymptotic expansion and needs a reference to the full numerical solution before it

148 K. Stcwrrtson

can be fixed. The reader is referred to papers by Stewartson (1957), Libby and Fox (1963), and Brown (1968) for a full discussion of the properties of these functions. It is important for the student of asymptotic theory to appreciate that whenever the procedure adopted thoroughly neglects one of the boundary conditions (in the present instance, as x + - a ) difficulties with these eigenfunctions, or their equivalent, can be expected to arise.

Other examples of bodies whose flow structure is of a similar type include paraboloids (Davis and Werle, 1972), cruciform semi-infinite flat plates (Rubin and Grossman 1971), and the quarter-infinite flat plate (incompletely studied by Stewartson, 1961).

All these studies are for bodies with the essential features that they extend to infinity downstream and the associated pressure gradients are everywhere favorable. Once we turn to finite bodies, which is the main topic of this review, the straightforward procedure explained above, for obtaining the asymptotic expansion, breaks down. The most usual cause is that the body has finite thickness. This means that the pressure gradient, derived from irrotational inviscid theory, is adverse near the rear-stagnation point, and all boundary layers separate before such a point is reached. Again, as the study by Proudman and Johnson (1962) shows, a steady boundary layer of the conventional type is impossible near a rear-stagnation point. Instead, the unsteady boundary layer sets up a region of reversed flow which grows, exponentially rapidly. with time. Further, once the external pressure gradient is prescribed, as is the case here, the solution to the boundary-layer equations becomes singular at separation, and it is impossible to continue the solution downstream within the same framework. The structure of the solution near separation was first elucidated by Goldstein (1948), and a representative calculation, illustrating the singularity, has been carried out by Terrill (1960) for a circular cylinder. Thus, the only reasonable conclusions we can draw are that once separation occurs we may not use the irrotational inviscid solution as a basis for setting up a uniformly valid first approximation to the flow as R -+ a , and that the correct form is quite different. A possible body shape, which might permit separation and yet be responsive to the methods presently available for studying flows at high Reynolds numbers, is one that depends on R , and we shall discuss this further below.

Separation also occurs on bodies with wedge-shaped trailing edges, again because there is a stagnation point there. In order to avoid separation the body must have a sufficiently sharp trailing edge-for example, a cusp of the type y .x x", I I 2 4, and the inviscid flow must come off smoothly on both sides of the cusp. The simplest example is a finite flat plate at zero incidence, and for such a body i t is possible to set up a consistent asymptotic expansion of the flow field in descending powers of R. The expansion is not, however, of

the relatively simple kind that proved satisfactory for semi-infinite bodies. The first approximation to the irrotational flow i s simple enough-uniform flow-and the first approximation to the boundary layer on the plate is given by Blasius' equation, as one might intuitively expect. The solution is not a uniformly valid first approximation to the flow, however, since it fails at the leading edge and also just downstream of the trailing edge. The first of these nonuniformities can be removed by adding the Van de Vooren- Dijkstra (1970) solution for the semi-infinite flat plate, and the continuation of the boundary layer into the wake downstream of the trailing edge may be effected by means of Goldstein's wake boundary layer (1930). However, the joining of the Blasius boundary layer with the Goldstein wake is not smooth. For as Goldstein showed, the change of boundary condition at the trailing edge means that the component of fluid velocity normal to the plate is discontinuous, being finite on the upstream side and infinite on the downstream side. In consequence, the streamlines in the boundary layer develop a right-angled corner at the trailing edge. Physically the termination of the plate leads to a rapid acceleration of the fluid at the bottom of the boundary layer, since its restraining influence is removed, and this draws fluid abruptly in toward the center from the edges. A complicated multistructured asymptotic expansion must be set up for the flow in the neighborhood of the trailing edge, in order to smooth out the discontinuity in the streamline slopes, and we shall explain how this is done below. I t is worth emphasizing here, however, that this multistructure is not just a detail, for i t must be determined before one can find the term following the Blasius contribution in the asymptotic expansion of the drag on the plate. We have in fact,

where d , is a constant yet to be determined from a numerical solution of the fundamental equation of this multistructured part of the boundary layer. A comparison with numerical results of Dennis and Chang (1969) suggests that d , 3.T In (1.2) L* is the plate length, CJ: the undisturbed fluid velocity, R is the Reynolds number based on L*, and D is the total drag on one side of the plate.

Generally throughout this review we shall adopt the definition

= R - 1/8,

and i t will always be supposed that I: Q 1. Then the classical boundary-layer

-1- C. E. Jobe and 0. R. Burggraf (private communication) find that t l , 2 2.69

150 K. Strrvartson

thickness is i:"L*, and the multistructure near the trailing edge is principally in the form of a triple deck of length r:'L* in the x direction. Normal to the plate the upper deck is of width r?L* and is controlled by the inviscid irrotational equation. The main deck lies immediately below and has thickness c4L*, just like the boundary layer and wake on either side, and plays a relatively passive role in the mechanism. The controlling equations here are also inviscid, but the pressure variation across the deck is small. The lower deck of thickness csL* is next to the wall and is controlled by the conventional boundary-layer equations, but with novel boundary conditions. Broadly the upper deck provides a pressure gradient which helps drive the lower deck. In turn the lower deck produces changes in the displacement thickness of the boundary layer, and these generate the pressure gradient in the upper deck.

Even with the addition of the triple deck to the first approximation to the flow field it is still not uniformly valid. First, the pressure gradient ip* / i . x* is discontinuous in the main deck as x = (,x* - L*)/r:"L* + 0, being finite as x + 0- and O(p:, UZ,'i:-- 'L*-'.x-- 1'3) as .x + O + . An inner layer of width O(c4L*) is needed to smooth it out, but the over-all effect is weak, for example, making a contribution to the drag coefficient CD of order R-'". One might argue, in fact, that this correction should be regarded as higher order since the relative changes in the velocity components are small. The second failure occurs in the lower deck as x + 0 since the boundary-layer assumptions then become invalid i n the same way as occurs at the leading edge. Since thc fluid velocities are smaller here, the breakdown region is, however, thicker, extending a distance i:"L* i n all directions from the trailing edge. The governing equations are the full Navier-Stokes equations and the boundary conditions needed to match with the triple deck outside are those for a uniform shear. The correction to the drag is also relatively weak and of order R - ' '.

With the understanding provided by this study there is a wide class of problems open to attack on similar lines. These include suction and injection, convex and concave corners, plates at incidence and bodies of nonzero thickness, the essential requirement being that catastrophic separation does not occur, and in turn, this means that the representative parameters defining the departure of the problem from that of the basic finite flat plate at zero incidence must all be Reynolds-number dependent. Of these the most interesting are those i n which the skin friction falls to very small values so that the flow is on the verge of separation. Thc key qucstion as yet unanswered is whether separation always occurs catastrophically in multistructured boundary layers, or whether it can be regular so that renewed fows can be computcd within thc framework of the asymptotic thcory. We remind the reader that, i n principle, separation can either bc catastrophic or

M u 1tistruc.t ciretl BoLindarj, Layers 15 I

regular (Brown and Stewartson, 1969), but that when the pressure gradient is prescribed externally it always appears to be catastrophic. Here, however, the pressure gradient is related to the properties of the lower deck, and so i t is conceivable that regularity might be preserved at separation in another context. An example is already known (Jones, 1973); in that problem the local pressure gradient is effectively provided by a temperature variation across the boundary layer, which depends in turn on the velocity distribution. The mutual interaction between the thermal and velocity boundary layers is apparently sufficient to prevent catastrophic separation. In order to settle the question of regular or catastrophic separation in our context, however, i t seems necessary to perform an explicit numerical integration of the fundamental equations [(4.5)-(4.9)], and this has not yet been done. Three aspects are involved: the inversion of the Hilbert integral, the solution of the boundary-layer equations, and the iteration procedure. The first of these determines the pressure, in terms of the displacement thickness, from the upper deck, and the second likewise from the lower deck, while the iterations are necessary to ensure that the two pressure distributions are identical. At the moment there are a number of adequate numerical procedures for integrating the boundary-layer equations. The inversion of the Hilbert integral still presents difficulties, however, owing to the singulari- ties that are liable to occur at interior points and the rather slow algebraic decay of the pressure gradient as Is 1 -+ 'XI . The development of an adequate numerical procedure for finding the pressure distribution in the triple deck is perhaps the outstanding problem at the moment.?

Failing that, progress has primarily been in the directions of evaluating the structural details in the various problems, solving linearized forms of the equations when the disturbances are weak in some sense, and obtaining approximate solutions. One noteworthy feature has been to bring out in explicit terms the interaction between an adverse pressure gradient, due to an external source, which threatens separation near the trailing edge, and the favorable pressure gradient engendered by the trailing edge, in particular the acceleration of the fluid associated with the end of the restraining influence of the plate. This phenomenon is of special interest in connection with the generation of lift on an inclined plate, and it is possible to quantify the role played here by viscosity. Furthermore i t is important that separation does not occur outside the triple deck. I f it does, i t is likely to be catastrophic and possibly cause the occurrence of stall. The condition on the angle a* of incidence is a* = O ( C ' ' ~ ) = O ( R - a result which emphasises the importance of viscous effects in the phenomenon, but also explains why it appears experimentally to be practically inviscid in character.

Burggraf (1974). t Considerable progress i n solving this problem has rcceiitly bccn made b y Jobe and

152 K. Stewartson

B. COMPRESSIBLE FLUIDS

When we turn to consider the effects of compressibility as well as those of viscosity on fluid motions, the present status of the asymptotic theory is even more primitive. The reasons are not hard to find. Not only is the pressure related to the density, but i t also depends on the temperature, and so we need to consider the energy equation with the additional complication of thermal conductivity. The occurrence of shocks in supersonic flow also necessitates special procedures to elucidate their structure, and more difficulties arise when they intersect each other and the boundary layers.

On the other hand, there is clearly no point in pursuing the goal of uniformly valid first approximations to its logical conclusion as has been described for incompressible flows in Section ],A, for at some stage the validity of the Navier-Stokes equations themselves must be called into question. Thus, in shock-structure studies there is no value, as far as a precise theory for real fluids is concerned, in considering anything other than very weak shocks. If the shocks are any stronger, the Navier-Stokes equations are not strictly adequate to describe the details of the flow; to be sure, of course, they are still very useful as approximations, and the shock studies are of considerable interest. Again, at distances from the leading edge of a flat plate of the order of the mean free path of the molecules the continuum hypothesis fails and new governing equations are needed. A n extreme case is provided by hypersonic flow when the breakdown of the continuum hypothesis and the boundary-layer equations occurs simultaneously as the leading edge is approached.

A more sensible program seems to be to develop an asymptotic theory of the flow structures up to the limitations of the basic assumptions underlying the equations, but to require that it remain physically realistic, especially with regard to the order of any singularity as the last interior limit is reached. Within this framework most progress seems to have been made with the semi-infinite flat plate. Even here, however, if the Mach number M, of the undisturbed stream is finite i t is necessary to consider the analog of the leading-edge study by Van de Vooren and Dijkstra (1970), and this has not yet been done. Consequently, within our framework, we do not yet possess a uniformly valid first approximation to the flow, although again, to be sure, it is possible to compute further terms in the expansion if the neighborhood of the leading edge is excluded. This expansion follows the conventional lines of the mainstream-boundary-layer interaction procedure already familiar in studies of incompressible flows. The presence of a leading-edge shock if M, > 1 must also be taken into account, and its interior must be excluded from the domain of applicability of the expansion

Multistructured Boundary Layers 153

after the leading term of its structure has been obtained. I n the double limit M , + a, R + cc, with a certain function of M, and of R finite ( R is a characteristic Reynolds number), an asymptotic expansion can also be set up which includes the neighborhood of the leading edge so long as the Navier-Stokes equations are valid, but excludes the interior of the leading- edge shock. The necessary functional relationship between M , and R depends on the way the viscosity varies with temperature and takes the form M;/R'' ' if this variation is linear. The general features of the solution are extremely complicated, many separate regions needing to be considered and interrelated, but follow the lines originally suggested by Lees (1955). The reader is referred to an article by Bush (1966) for the fullest account of the flow structure. A similar structure has been worked out for a slender cone (Stewartson, 1964a; Cross and Bush, 1969). A somewhat related problem is the flow set up by an infinite flat plate set in uniform motion after an impulsive start (first considered by Howarth, 1951 and reviewed by Stewart- son, 1964b).

We shall not discuss the detailed properties of these solutions, partly because they should more properly be considered within the framework of hypersonic viscous flows, and partly because the boundary layers are rather conventional after the appropriate transformations have been made. The unifying theme of this review is the subdivisions of the boundary layers that become necessary under the impact of sudden streamwise changes. The examples already given for incompressible boundary layers clearly have analogs when the fluid is compressible-trailing edges, corners, injection and suction, finite plates at incidence-and we shall explain the modifications necessary to allow for this effect in the review.

If the boundary layer is supersonic ( M = > l ) , however, a new phenomenon occurs which appears to have no counterpart in subsonic flow. Furthermore, i t leads to a greater ease of study of the flow properties and helps to overcome, in part, the barrier of separation which appears to hinder progress in the incompressible studies. The phenomenon is, of course, the ,free-inttvwction houndury layer first observed by Ackeret et at.? (1947) and independently by Liepmann (1946) in their study of the interaction between a shock wave and a boundary layer and more extensively studied subsequently by Liepmann et a1. (1954), Chapman et ul. (1958), Hakkinen et al. (1959), and many others. These studies show that when a shock, sufficiently strong to provoke separation, strikes a laminar boundary layer, the boundary layer actually separates ahead of the foot of the shcok and, moreover, the flow features of the separation region are independent of the characteristics of the shock and depend only on the local properties of the flow. Further-

* The original German version, by Ackeret r t ((/., was published in 1946.

154 K . Stewartson

more, the separation which takes place is rial catastrophic and the boundary- layer thickness changes smoothly as the reversed flow sets in. Downstream of separation the boundary layer detaches from the wall as a free shear layer, and a region of slowly moving fluid is set up between it and the wall. The pressure is constant, provided the flow remains stable, until the shock is reached. Complications arise if transition takes place, or if the wall is curved as is usually the case in practice (e.g., an airfoil).

To a theoretician the most challenging question is to explain why the boundary layer separates so far ahead of the main-shock impingement point. There is no obvious a priori means of provoking separation because the boundary-layer equations are parabolic and so, unless separation has already occurred, disturbances can only propagate downstream. Again, the mainstream is supersonic so that here all disturbances travel downstream. How then, can the changes occur in the boundary layer which lead i t to separate?

From a physical standpoint a simple explanation has been given by Chap- man rt a/. (1958), following a suggestion by Oswatitsch and Wieghardt (1946). A boundary layer of displacement thickness 6*(x*) induces a pressure rise K dd*/d.x* in the mainstream. Consequently, if d 2 6 * / d ~ * 2 > 0 an adverse pressure gradient occurs in the boundary layer which provokes further thickening of the boundary layer, thus opening the way to an evolution of the boundary layer toward separation. (Notice that, although 6” is an increasing function of .Y* in the Blasius boundary layer, d2d*/rIx*2 < 0 and so the induced pressure gradient is favorable.) This physical argument leads to a prediction of the pressure rise at separation which has a qualitatively correct dependence on Reynolds number and Mach number when compared with experiment. Subsequently, a number of approximate methods have been devised for predicting the flow properties in the shocked boundary layer and related problems, the most successful of them being, perhaps, that originated by Lees and Reeves (1964) using integral relations. We shall not discuss them in detail here because of our interest in asymptotic expansions, and it is not possible to fit them into a rational scheme of this sort. Moreover, they can give rise to phenomena such as “supercritical jumps” (Grange et a/., 1967) which are inexplicable on the basis of asymptotic theory and which have not been observed in practice. The reader is referred to reviews of interaction studies by Brown and Stewartson (1969), Fitzhugh (1969), and Georgeff (1972) for further details of these methods.

The first step in the development of a consistent theory to explain the phenomenon of free interaction was taken by Howarth (1948), who supposed that it is inviscid in character, propagation upstream taking place in the subsonic part of the boundary layer. However, it was established by Lighthill (1950) that a purely inviscid theory is inadequate. The second step

Multistructured Boundary Luyers 155

was initiated by Stewartson (1951), who considered the effect of a large pressure gradient acting over a short distance so that the change of pressure is small. He showed that an inner boundary layer is developed very close to the wall, which may separate before the main part of the layer has significantly altered. Lighthill (1953) in a later paper added the notion of an inner boundary layer to his earlier inviscid study and produced a coherent self-consistent theory of an interaction between a weak shock and the boundary layer. Furthermore, his study indicated how a spontaneous change occurs in the boundary layer which might lead to separation. This paper is the key to the rational theory of free interactions and indeed to the whole of this review. From it developed the theory of trailing-edge flows, viscous corrections to lift, and other aspects considered here. Thus the ideas developed by Lighthill are consequently of great importance to the mathematical theory of viscous flows at high Reynolds number. Immediate exploitation of his theory was, however, hampered by an unnecessarily complicated treatment of the inviscid part of the boundary layer. He assumed that the x-wise scaling factor is c4L* which is the same as the boundary-layer thickness, whereas the length of the interaction when finally computed turned out to be e3L*. The generalization of linear theory to include nonlinear disturbances on the scale c4L* involves the solution of a complicated nonlinear equation of mixed hyperbolic-elliptic type for which the appropriate numerical procedures have only recently been developed. Once it is clearly recognized that the crucial streamwise scale is E~L*, however, the governing equations can be simplified without any loss of accuracy, and the nonlinear equation of mixed type reduces to a quadrature. The nonlinearity, in fact, is primarily contained within the subboundary layer near the wall for which numerical techniques are readily available. The necessary reformulation of Lighthill’s approach was carried out simultaneously by Messiter (1970) in the related problem of trailing-edge flow, discussed in Section I,A, by Stewartson (1969) for the same problem, by Stewartson and Williams (1969) for the free-interaction problem, and by Neiland (1969)f It is emphasized that Lighthill’s main results are correct, but can be obtained in a simpler way, and the generalization is immediate.

The nonlinear formulation of the free-interaction regions which results is of the triple-deck type, just as in the trailing-edge problem discussed in Section I,A, and is also essentially controlled by the lower deck, which is of the classic boundary-layer type but with unusual boundary conditions. No- table among these is that the pressure rise is proportional to the rate of increase of the streamwise component of velocity in the outer part of the lower deck [see (4.7) and (4.8)]. Since the pressure is determined by local

The au thor is indebted to Professor Mcssitcr for the reference to Neiland.

156 K. Stewirtson

flow properties, forward integration is possible right up to separation which is, and must be, passed in a regular manner. The catastrophic singularity which bedevils boundary layers with a prescribed pressure gradient has disappeared, for if i t were present there would be an irregularity in the pressure too, and this contradicts the choice of separation point. There are considerable difficulties about continuing the integration into a region of reversed flow, not the least of which is the fact that the problem is no longer well posed. An extra downstream condition is needed over that part of the velocity profile which is negative. Nevertheless, numerical experiments have suggested a form for the flow structure well downstream of separation, but within the free-interaction zone, and it has been possible to develop a credible asymptotic expansion terminating the free interaction and to match it to the flow further downstream. The pressure plateau which is a feature of experimental measurements in the region between the free interaction and the shock incidence is an essential feature of this expansion.

It also appears that the free interaction is not the only self-induced flow that can occur in a supersonic boundary layer. I t is produced numerically by making an initial positive kick ( - in the pressure, and the boundary layer then evolves in a thoroughly stable way to produce the free interaction. Suppose, however, that the initial kick to the pressure is negative. Then a conjugate solution occurs with the pressure now falling instead of rising and the skin friction increasing. Again, the evolution is stable, but this time i t terminates in a singularity reminiscent of that in the flow between converging plane walls. The lower-deck boundary layer subdivides once more, the outer part becoming inviscid with a velocity determined by Bernoulli’s equation. Whereas the free-interaction solution is relevant to interactions with shock waves, concave corners, and injection problems, this new type of solution is relevant to convex corners and suction problems. I t can be regarded as providing a means of enabling the boundary layer to speed up in anticipation of an expansion even though it cannot permit upstream disturbances. The two types of free interaction will be referred to subsequently as comprrssirr and rxpcinsiue.

In view of the greater effort that has been expended on supersonic boundary layers and the much deeper progress that has been made, than has previously been possible, in elucidating fow structure in subsonic boundary layers, we shall reverse the usual procedure and begin with a study of free- interaction boundary layers in supersonic flow, deferring the consideration of incompressible flow problems until Section XIII.

11. The Basic Flow

We consider a compressible viscous fluid flowing past a flat plate of length L*, whose leading edge is at the origin O* of a set of Cartesian coordinates O*x*y*, and which occupies part of the positive x* axis. We shall denote

physical quantities by a superscript *, conditions at the wall by a subscript w, and the uniform conditions at infinite distances upstream by a subscript a;. The properties of the fluid are limited only by the requirements that i t be Newtonian and that the viscosity p* and the thermal conductivity k* be functions of the temperature T* only. Furthermore, it is supposed to begin with that upstream of the plate the motion is uniform with velocity components ( U z , 0) and that the plate is maintained at a constant temperature T:. We shall be particularly interested in the neighborhood of one point (x,*, 0) of the plate when considering multistructured boundary layers, and shall take this point to be the separation point (xt,O) in compressive free- interaction problems, or (L*, 0) in trailing-edge problems, or the singularity (x:, 0) in expansive free interactions. The Reynolds number is then defined to be

R = U: x ,* /v~ , = c-’, (2.1)

where v* is the kinematic viscosity, and R is assumed to be large. It is then justifiable to assume that the boundary layer is fully developed at x,* and that the velocity profile is determined by the Blasius equation, suitably generalized. Denoting the velocity components parallel to the (x*, J’*) axes by (u*, v*), respectively, we then have that u* = U,*(J’*) in the boundary layer at .xa, where

U,*(y*) = ug 47/Llij, (2.2a)

. Y*

7 = ~ R ” / X , * ( ~ C ‘ ) ’ ’ ~ , - j = 1 (p* /p: ) dy*, (2.2b) ‘ 0

and

(2.2c)

Here, p * denotes the fluid density, C‘ is a constant, and fsatisfies the further conditions f = clfldij = 0 at i j = 0, dfldij ---f 1 as i j -+ a. Strictly speaking there should be another equation, for the temperature (see Stewartson, 1964b, p. 35. for example), but we shall omit i t because our explicit demands on u,*(y*) are so limited. I t is usually sufficient to assume that the Chapman viscosity law

p*/p*, = C(T*/TZ) , C = &TTY,/pY, T: (2.3) holds, when we may set C’ = C, and then

158 K . Strwartson

independently of the energy equation. However, there is no formal difficulty about solving (2 .2~) and the corresponding energy equation, whatever be the dependence of p* and k* on T*, and determining the corresponding value of d@/dy* at y* = 0. This is the only property of the basic flow that we shall need subsequently and any modification that might be thought necessary can easily be handled with an appropriate definition of C.

Il l . The Triple Deck

A. THE MAIN DECK

In order to simplify the presentation of the triple deck, the reader is asked to accept provisionally that streamwise changes in the boundary layer from the Blasius profile over a distance O ( E ~ L * ) in the neighborhood of x; can induce pressure gradients which have a significant effect on the way the boundary layer behaves in this region. The justification for this remark will be seen when the whole triple deck is put together. Meanwhile, the expansion procedures are consistent within the framework set up for them so that only the motivation is unclear.

We begin by considering the main deck which comprises the majority of the boundary layer itself, and we write

y* = c4x;y,, x* = x; + c"x,*x, (3.1) where X and Y, are the new independent variables and are both supposed O( 1 ) in the main deck. Simultaneously we write

ai*/Ug = u , ( Y M ) + F U ~ ( X , Y M ) + c2u2(X, Y,) + . u*/u: = c ~ u , ( x , Y ~ ) + e30,(X, Y,) + .

where

U,( YM) = UX(J*)/U*, ;

(P* - P*,)/PZ UZ2 = c p l ( X , YM) + 2 p 2 + c 3 p 3 + p*/p: = R,(Y,) + c p , ( X , YM) + c2p2 + " ' ,

, (3.2a)

, (3.2b)

.., (3 .2~)

. (3.2d)

where p* is the pressure and p*, R , is the undisturbed density profile. This profile may be determined from the velocity profile or may have to be found simultaneously with it. A t the present stage in the development of the theory our principal need is for R,(O), which is obviously known if the wall temper-

Multist,.uctirr.c.tl Boundary Layers 159

ature is prescribed, or is given by [ 1 + $ ( I > - l )M;>]- ' if the wall is adiabatic and the Prandtl number is unity, 1' being the ratio of the specific heats of the fluid.

On substituting these expansions into the equations of motion we find that the viscous terms are only important when terms of relative order c3 are considered. Thus, the equations which govern u1 and u2 are entirely inviscid in character, while the equation governing u3 must involve the viscous terms since the basic fldw is a function of x*. Furthermore, i t immediately follows that p I is independent of Y, and indeed this is also true in the upper deck which we shall consider below. Hence, if p 1 is nonzero, there must be an external agency outside the boundary-layer region driving the triple deck, and we specifically wish to exclude this. A crucial feature of the triple deck is that it is intrinsic to the boundary layer and wall geometry, and hence we write

Pl(X, YM) = 0. (3 .3)

Since we may treat the determination of u1 and u2 as an inviscid problem the governing equations reduce to

p * u * ( ~ u * / ~ x * ) + p*r*(?1r*/?y*) = - c'p*/?s*, (3.4a)

(3.4c)

p*u*((?c*/(?\-*) + p*"*(2""/y*) = - r?p*/?y*, (3.4b)

(c?/ix*)(p*Lr*) + ( ( ? / c ' ! . * ) ( / ) * U * ) = 0,

I)*("* ip*,!?x* + I>* ?p*/?y*) = i'p*(Li* ?p*/?x* + P* ?p*/?!,*), (3.4d)

terms O( 1 ) in (3.4) being neglected. The expansion (3.2) is now substituted into (3.4), and we obtain for the leading terms

U , r?u1/ZX + 01 dUO/dYM = 0, U , iIp,/c?X + 1 ' 1 dR,/tlYM = 0,

(3.5) ap,/(?Y, = 0, and R , 2ul/?X + U o ?p,/i?X + d(R,c,)/?YM = 0.

We have in mind that the disturbance, characteristic of the triple deck, dies out upstream of it, the flow then reducing to the Blasius solution. Con- sequently, the appropriate solution of (3.5) is

111 = A , ( X ) dU,/tlY, ,

P 1 = A , ( X ) m # ~ M >

rl= - ( ( / A J d X ) U , ) ( Y,),

P2 = P2(X) (3.6)

A , being at present an arbitrary function of X except that A , + 0 as X -, -m. The physical significance of this simple solution is that the streamlines in the boundary layer have been displaced, possibly by an effective change in the position of the wall, and it could have been obtained by replacing Yzf by Y l f + A , ( X ) in the boundary-layer equations.

160 K . Stewartson

Turning now to the equations for 1 i 2 , c 2 , p 2 , and p 3 , we have first of all

RoU, (a~ l /dX) = -?p -JdYM, (3.7) so that

where M is the Mach number of the flow so that M i = R, U i U*,"/*pz, and p3(X, 0) is a function of X at present indeterminate. The equations for i f 2 , c 2 , and p2 are

and

?p2 dR, . (3.9c) dP2 - i P 1 ?P 1

- 1'1 c?x + " 1 dX (1 Y M d Y M M:R,U, + Uo (lx + " 2

Hence, we find that

(3.10)

where A 2 ( X ) is also indeterminate at present,

(3.1 I ) and

Of particular interest is the behavior of 1 1 2 , c 2 , and p 2 as YM + cx, and as YM + 0. It immediately follows from the above expressions that, as Y,, -+ x,

and dp2 d A 2 dX

c'2 + (Mz, - 1)YM (lx -+ - (3.13b)


The behavior near YM = 0 is more difficult to evaluate, and we must distinguish two cases.

1. Therrnully Insulated Wull

Here

Rb(0) = 0, U;;(O) = 0, R;;(O) = crb) - l)M? [Ub(O)]2[Ro(0)]2

and hence

(3.14a)

2. Constant Wull Temperature

Here Rb(0) # 0 but RO(0) is prescribed, and if we assume that p* M _ T*, it

Ub(O)Rb(O) = Ro(0)Ug(O). (3.15) follows that

Thus after some algebra it may be established that as Y, + 0

and

are all bounded. The actual value of the limits can be computed if required, but are not needed in this review. An additional useful result is that

(3.17)

is also bounded as Y, + 0. The expansion can be continued to include higher powers of E , but viscous

terms in the Navier-Stokes equations now enter the perturbation equations and the formulas rapidly become unwieldy. I t is worth noting that the

162 K . Stewartson

leading terms ( u l , u , , etc.) in the expansion satisfy the conventional boundary-layer equations in some sense, and so do terms of the next order, except for p 3 , which varies across the boundary layer.

B. THE UPPER DECK

In the upper deck we write y* = & 3 X ; x , s* = x; + c 3 x ; x , (3.18)

so that the scaling is the same in both directions and the appropriate expansions are

lL*/U*, = 1 + &%,(X, Y,) + C3i3(X, Y,) + ..., c*/u: = C2û2(X, Y,) + i.”%, + ” ‘ ,

e 2 p , ( x , Y,) + c3i3 [ > * / [ I * , = 1 + e’fi2 + 1:3;)3 + ....

(p* - p*,)/p: u*,’ = + ..., (3.19)

On substitution into the full equations of motion it is found that the terms O(F’) and the terms O(c3) all satisfy the equation

( M : - 1) F 2 P / i X 2 - i?’P/;IY; = 0, (3.20) and that

Y, = -G,l/i?x, I 1 = 2, 3. (3.21)

The triple-deck structure under investigation is supposed to be induced by some feature inside the boundary layer, and hence we must require that C, -+ 0 as Y, + cc for finite X and also as X + - cc. Consequently we have

b,(X, 0 ) = (M: - 1)-’I2in(X, 0) if M , > 1, (3.22) and

1 i n ( X , , O ) d X 1 n.-, x - x , Fn(X, 0 ) = - ( I - M2,)-”2 ~ ~~ ~ if M , < 1.

(3.23) In (3.23) an additional assumption is made about C, as X -+ cc so that the Cauchy integral may converge and this must be justified in each problem considered. Difficulties also arise in the double limit M r( -+ 1, c + 0, and this case is given special consideration below (Section V, and Messiter et al., 1971).

Mtiltistructirred Boundary Lajvrs 163

It now remains to ascertain the values of in and V,, on & = 0 from the main deck. We rewrite the formulas, obtained for v2 and u3 when YM is large, in terms of % and obtain

Similarly

Hence

. .

(3.244

(3.24b)

(3.25)

i . , (X, 0 ) = -dA, / t lX,

and so we have two conditions connecting the four arbitrary functions appearing in the expansions to date. Similar forms for u, and p,, can be written down and the match between the two decks shown to be consistent.

C. THE LOWER DECK

The lower deck of the system is required because the solution in the main deck implies a slip velocity over the plate as YM --+ 0, and one of its functions is to reduce that to zero. Its thickness is a fraction I: of the Blasius boundary- layer thickness.

We write

J,*= C~.X,*Y, s*= .x,* + c~x,*x, (3.26)

and take the velocity components (u , I , ) to be of order r:U*, and e 3 U z , respectively, while the pressure variation is O(c2pz ). Accordingly, the Navier-Stokes equations reduce to conventional boundary-layer equations, the relative error being O ( E ~ ) . It is convenient to write

u* = 3- u* = i;iiUz , 1: vu; , p* - p* = .2 * U*2" p* = R, I 8, P a 01 P..

(3.27) -

1 64 K . Stewur-tson

where U, 5, R", and ?, are functions of X and and also of F , but their limiting values as F -+ 0 are all finite. Further, we shall neglect all terms of relative order 2, since we shall not carry the expansion that far, in which case T* = T: p*, /p* , from the equation of state. Finally, we shall suppose that the Chapman viscosity law holds. Then, to the order of accuracy we are interested in.

(3.28b)

and

the contributions to the energy equations, from the dissipation and the rate of working of the pressure, being both of relative order c2. The specification of the lower deck is completed by assigning appropriate boundary conditions, determined by the match with the main deck and the undisturbed upstream conditions. The discussion of the behavior of the main deck as YM + 0 strongly suggests that we deal with the cases of insulated walls and heat transfer separately.

1 . Thermally Iizsulated Wulls

Here we write

u = U , ( X , y,) + CU2(X, y,) + ..., (3.29a)

i) = U , ( X , Y,) + ~ 5 2 + .", (3.29b)

p = FZ(X , y,) + FFJ + ' . ' , (3.29~)

R" = b , ( X , y,) + c52 + . ' . , (3.29d)

the t: dependence being separated out in a power series in each case. When X is large and negative (3.29) must join smoothly to the undisturbed flow for which

u*/u*, = Y, Lib(0) + O(Y$) ,

R = R,(O) + O( Y L ) .

p* = p: , (3.30)

Multistructured Boundary Layers

Hence, as X -+ -E,

165

(3.31)

When (3.6) and (3.14), we see that in terms of r, the solution takes the form

is large, (3.29) must join on smoothly with the main deck and, using

u = Ub(0)K f A,(X)ub(o) + EU2(X, 0) f O(Ez),

i; = R,(O) + c2(fY:Rg(0) + Y,A,(X)Rg(O) + pZ(X, 0)) + O(C'), (3.32)

F = p z ( X ) + t:p3(X) + ..., when YM 4 1 and Y, 9 1. The appropriate boundary conditions for the functions in (3.29) follow immediately, and we see that compatible solutions are

P 2 = p z ( X ) , P3 = p 3 ( X ) , and i ; , = R,(O), ?l2 = 0, (3.33) while the matching condition on U is redundant. The governing equations now reduce to

together with the boundary conditions (3.3 I), and

as Y, + cx), where p 2 , p 3 are related to A , , A , through (3.22), (3.23), and (3.25). In addition there are boundary conditions to be satisfied at the wall, e.g., ii = U = 0 at r, = 0, but since it is these conditions which provoke the triple-deck phenomenon we shall discuss them in detail for each type of problem separately.

2. Constant Wall Tenzperuture

In this case it is convenient to write down first the matching conditions with the main deck in the double limit r, -+ cx), YM + 0. Writing the solution

166 K . Stewartson

in the main deck when YM < 1 in terms of r,, using (3.6) and (3.16), we have

(3.36)

p = p 2 ( X ) + t : p 3 ( X ) + ... . Thus the appropriate expansion in the lower deck is more complicated than for a thermally insulated wall and proceeds in powers of E and log E,

beginning

U = UI(X, Y,) + E l o g ~ i ~ , , + & L I Z I + ..., + .”,

s i j 2 + (3.37) 6 = 6,(X, r,) + E l o g e ~ z o

R = R,(O) + ?, = p 2 ( X ) + F p 3 ( X ) + ’. . .

c2 log C1I3, + . . . ,

The differential equations satisfied by the new functions in (3.37) are easily obtained by substitution into (3.28) and the boundary conditions as Y , -+ cc followed by comparison with (3.36). So far as U , and are concerned, these are identical with the equations and boundary conditions for the corresponding functions in (3.29). As X -+ - CE the requirement that the flow is undisturbed by the triple deck means that

L1, + Ub(O)r, , u,, -+ 0, 6 2 ] -+ +Y:U;;(o),

c’, + 0, v2, - -+ 0, L 2 , -+ 0, (3.38)

p 2 -+ R’,(O)Y,, p 3 o + 0, p 2 -+ 0, p3 + 0. The scaling laws of the triple deck are now seen to be complete and the

procedure clear for working out more terms as desired. In each deck the expansion proceeds in powers of c and eventually in powers of loge [probably even when Rb(0) = 01 whose coefficients are functions of the scaled variables only and explicitly independent of E. The boundary conditions are either imposed by the wall, by the necessity of the triple-deck

Multistriictured Boundary Layers 167

phenomenon dying away upstream of x$ and at large distances from the boundary layer, or by matching one deck against another. They also are explicitly independent of E.

Two special features of the triple deck are worth noting. First, the leading term in the expansion is the same whether there is heat transfer across the plate or not. Thus, the qualitative features of any phenomenon controlled by the triple deck are independent of thermal conditions at the wall although, of course, they will control the amplitudes of the variations. For example, self-induced separation can occur in a supersonic boundary layer at all wall temperatures. Second, the governing equations for the first two terms of the expansion are the boundary-layer equations in both the main and lower decks except for the pressure variation across the main deck described by (3.7). Unfortunately, the resulting effect is serious, especially at high Mach number, and may limit the practical value of the expansion.

IV. The Fundamental Equation of the Triple Deck

It is convenient to set out, in as simple a form as possible, the fundamental equation (3.34) controlling the solution in the lower deck, together with the associated boundary conditions derived from the upper deck (3.25) and the main deck (3.35). We shall assume here that the plate is thermally insulated, that 0 = 1, and that Chapman's viscosity law holds: the results may also be applied directly to plates maintained at any constant temperature, so far as the leading terms are concerned.

In this case,

RO(0) = T*,/T: = (1 + i(1) - l ) M $ ) - ' ,

Ub(0) = AC-'"T:/T,Y, (4.1)

A being defined in (2.4). If we take C' = C in (2.2c), f satisfies Blasius' equation, and then we have

" 1 1 - 1) dYM = **jo"( I - I ) d Y M I0 ( R , Ug - R o ( O X U ~ ( 0 ) ~ ) 2 RO u:,

168 K . Stewurtson

where . r

A I = 1 A2 = **) [( 1 - 72)/f’2] dv] = -3.663,

( 1 - f ” ) d q = 1.686... , - 0

. r

* n

(4.3)

and the upstream boundary condition to

where

p = -A’(.Y) - r;B’(x) + c2112A1 C1’8E,5’4(M2, - l)’I8

x (;;I - I i 2 A”(.Y) + O(E‘) if M , > 1, (4.8)

and

Multistriicturrd Boun(1ary Layers 169

where the principal value of the improper integral is to be taken. Finally, there are conditions to be imposed at y = 0 of which u = c = 0 is typical, but we shall discuss them in the context of particular problems below.

Of special interest in triple-deck studies is the value of the skin friction at the wall. On inserting the appropriate scaling laws we find that

(4.10)

and, unlike most of the properties in the triple deck, the variation of skin friction is of the same order as its undisturbed value in the Blasius solution. For convenience we shall define

z(.) = (?u/?y) 1) = o , (4.11)

so that z = 1 is the undisturbed value and separation occurs at z = 0. With the scaling laws (4.4) of the triple deck known i t is now possible to

form a more precise notion of the conditions for the validity of the expansion procedure adopted. I t is necessary that c 3 X 4 1, so that the streamwise variation of the basic flow over the length of the triple deck may be neglected, that r;G 4 1 so that the relative variation of the velocity is small, and that

I:& B (T,*/T:)C''2 (4.12)

so that the thickness of the lower deck is small compared with that of the original boundary layer. All these conditions may be combined into one, namely

I;C'" I M: - 1 1 - 'IX((T:/TZ B 1. (4.13)

When M , = 0(1) but M , # 1 this condition is equivalent to I: 4 1, but when M , 9 I it may be rewritten as

111s4 1 where x = M: C"2/R1i2 (4.14)

and is the well-known viscous hypersonic parameter. It is noted that when M , 9 1, this condition implies that the higher-order terms of (4.8) are relatively small.

V. Transonic Free Interaction

The scaling laws (4.4) fail when M , = I , and the theory needs reconsideration (Messiter rt a/., 1971). The main modification occurs in the upper deck where (3.20) must be replaced by the transonic small disturbance equation. Since the only source for the factor (M:, - I ) in (4.4) is (3.20) it is clear that,

170 K . Strwartson

so far as the main and lower decks are concerned, ( M : - 1 ) may be replaced by the symbol 6 without otherwise altering their structure and properties. If M2, # 1, we can replace 6 by - 1) at any stage, but by using 6 we d o allow ourselves some flexibility when M , = 1. Then, as YM + a,

L'* + 2u: c1/41t1/261/4 ( d A / d X T ) ,

p* - p*, + "2pz U y / 4 ; 1 1 / 2 ( 5 - 1/4p(Q.),

(5.1)

(5.2)

x7 = C-3/8p463/8(T;/T*, )-3/2x, (5.3)

where

and is the appropriate streamwise coordinate for transonic triple decks. Only the leading terms of the expansions have been retained in (5.1) and (5.2). In the upper deck we need a transonic coordinate yT which, when used in conjunction with x T , reduces the governing equation to a parametrically simple kind. Define

and, since the motion in the upper deck is irrotational and inviscid to leading order, we may also define a velocity potential

with the properties that

Then on substitution into the inviscid equation of motion and on neglect of c3 , i t follows that

Midtistructured Boundary Lujvrs 171

The appropriate choice for 6 is now clear. If h1: - 1 is significantly different from zero, choose

6 = p I - 1 1 (5.9)

6 = 1.”sJ2/sc1/s (5.10)

and we recover the scaling laws (4.4). Otherwise choose

and then & satisfies

(5.1 1 )

where

K 0 - - ( ~ 2 , - l)c-8/5J2/5c1/’ (5.12)

and is constant. The scaling laws analogous to (4.4) are easily found by the use of (5.10) [replace (A42 - 1) by E ~ / ~ ~ . ~ / ~ C ’ ’ ~ throughout (4.4)]. We especially note that

P(.KT)? p* = p z + p* ~ * 2 c ’ / 2 0 ] L ” ’ O R - ’ / 5

I, .YT . (5.13)

3 ( x

and .x* = + X + ~ 3 i 1 0 ] , - 7 / 5 ( ~ ,* / T: ) 3 / 2 ~ - ” l 0

A start in elucidating the properties of the Eqs. (4.5) and (5.1 1) governing this free interaction has been made by Brilliant and Adamson (1973).

VI. Free Interactions in Supersonic Flow

After catastrophic separation and transition, the most interesting property of laminar boundary layers is their capability, in supersonic flow, of spontaneously undergoing a radical change without the apparent help of any locally operating external agency. This phenomenon can be identified in the experimental investigations of Ackeret rf ul. (1947) on the interactions between shock waves and boundary layers and was studied in detail by Chapman et al. (1958). It is observed that the boundary layer can separate some distance ahead of the point of incidence of the main shock and in so doing generates compressive waves which can coalesce to form a secondary shock. Dowp- stream of separation the boundary layer detaches from the wall as a free shear layer, and a region of stagnant or slowly eddying fluid is set up below it. The main shock then impinges on the free shear layer, which reflects it as

172 K . Stewartson

an expansion fan and simultaneously is turned back to the wall. The reattachment of the free shear layer to the wall, on the assumption that it occurs, which is not certain, generates another shock. A sketch of the chief features of the flow is given in Fig. 1. Similar phenomena occur when the separation

FIG. 1. Sketch of main properties of flow when a shock wave interacts with a laminar boundary layer.

is provoked by a concave corner and may well occur if sufficient fluid is injected into the boundary layer.

One special feature of interest to the theoretician is that small disturbances can travel downstream only (increasing x*), both in a supersonic inviscid stream and in an unseparated boundary layer. Yet the boundary layer evolves away from its undisturbed Blasius form toward and through separation, well upstream of the impinging shock. A simple physical explanation of how this might occur in physical terms was given by Chapman et a/. (1958) roughly on the lines that a cycle can be set up in which the growth of the boundary layer produces an adverse pressure gradient in the main stream which, in turn, promotes the further growth of the boundary layer. The triple-deck structure described in this review enables their idea to be quantified (not for the first time) and also to be embedded in an asymptotic theory of the solutions of the Navier-Stokes equations.

I f it is possible for the boundary layer to evolve in such a way that the pressure rises, it must also be possible for it to evolve so that the pressure ,fulls. Again, by analogy, we might expect this situation to arise in flow near a coiirex corner or when suction is taking place. (The analogy to a shock would be an expansion fan which, however, is not a discontinuity.) Although the result is not so spectacular as when separation occurs, it is essential to include it in any asymptotic theory of such flows. We shall distinguish the two types of evolutionary boundary layers by referring to them as compressive ,free interactioris and expaiisizv jree interactions.

Multistructureil Boundary Layers 173

On confining attention to the leading terms only and assuming that x114 4 1, the fundamental equations of the triple deck (Section IV) reduce to

u = - '* (6.1) i l l ?u rip (12u '* u + l > = - 11 = , ? Y c'y d x + Jy2' ?Y i x

$ being the stream function. The corresponding boundary conditions for a free interaction are

11 - I'+ 0 as .Y + -K,,

u - L' + A ( x ) as 1' -+ K'; A ' ( s ) = - p ( x ) , (6.2) together with the no-slip conditions on the plate,

11 = 1' = 0.

The problem thus posed is homogeneous and has the simple solution (6.3)

u f J', Z' = 0, p = 0, (6.4) which corresponds to the continuation of the Blasius solution, unchanged, through the triple deck. This is as it should be since there are supposed to be no local agencies, geometric or physical, to change the flow. This solution is not, however, unique, and it is possible for two other solutions to be generated spontaneously. In each, the origin of x is arbitrary and their properties are quite different and remarkable in their own way. Their existence was first established by Lighthill (1953) who considered a linearized form of (6.1) (and of the triple deck). He sought solutions in the form

u = y - 11, C~"~~"~(J~), I' = 11, ~iPf;(!.), p = u,o"", (6.5)

where u1 and K are constants, and,fl is a function of 11 only, satisfying the boundary conditions

f I ( O ) = f > ( O ) = 0, . j" ;(cc) = ! < - I . (6.6) On substitution into (6.1) and with neglect of irt, we recover Lighthill's results, namely

.Y

, f ; ( y ) = - [ ~ ' / ~ / A i ( 0 ) ] 1 A i ( z ~ " ~ ) dz, (6.7)

(6.8)

' 0

I< = [( -+)! 37'6/2~]3'4 = 0.8272 . . . ,

where Ai is Airy's function and u1 arbitrary. The expansion can be continued and we have

p = a, ehx - 0.459afe2"" + ..., z = 1 - 2 . 3 0 9 ~ ~ e"" f 0.659~: eZKx + .. ..

174 K . Stewui-tson

Further progress in elucidating the properties of the solution depends on numerical studies.

VII. Expansive Free Interactions

It is clear from (6.9) that only the sign of u , is significant since I a , I can be absorbed into the definition of the origin of x. I f we take u1 = - 1, the pressure falls initially while the skin friction z rises as ,Y increases, and the corresponding solution will be referred to as an expansive free interaction. From (6.2) and (6.9) the effect of a decreasing pressure is to speed up the fluid near the wall which increases z and A , thus accelerating the process. One might expect that in the absence of external stimuli such a solution can only be terminated by a singularity with p + - ir, since there is no change in the broad properties of the boundary layer to brake the process. Assuming that this singularity occurs at x, we can write down an asymptotic expansion for the solution valid near .Y = .Y, . Let

on substitution into (6.1), it is found that (7.1) is an exact solution in the limit x + xj provided

F b ( 0 ) = 0 and F h ( x ) = 1. Thus the form (7.1) is identical with that occurring in the theory of viscous flow between converging plane walls, and the appropriate form for Fb is

F"' 0 - 2F: = -2. (7.2)

Fb = 3 tanh2(t7 + [j) - 2 where tanh' \j = 3. (7.3) The forms assumed by u, u, and p in (7.1) fall short of being an exact

solution because 11 is bounded as J' + cx: for all .Y < x,. The correct boundary condition as y + cc; and the differential equation can be satisfied formally if we write

II/ = 2 I(., - X ) ~ " F , ~ ( V ) . n = O

(7.4)

where F , satisfies (7.2) and F , satisfy linear equations. I t should be noted that (7.4) cannot be complete because all the F,, are uniquely determinate, given x r , whereas the correct solution must contain a measure of arbitrar- iness, since the condition ii - y + 0 as x + - ir,, has been ignored. Eigen-

Multistructured Bouridary Layers

TABLE 1

VAKIATIOK or: - p . z, A N D T~ wITi1 Y I N

A N EXPANSIVE FREE I N 1'1:~,4("1'10N

(x, 2 14.78)

Y T - I' T ?

8 9

10 1 1 12 13 13.2 13.4 13.6 13.8 14.0 14.2 14.4 14.5

0.008 0.0 18 0.041 0.097 0.242 0.699 0.897 1.180 1.604 2.294 3.538 6.223

14.13 25.91

1.009 1.021 1.050 1.119 1.301 1.905 2. I78 2.577 3.193 4.219 6.134

10.38 23.12 42.16

I .00 I 1.002 1.006 1.012 I .027 1.049 1.053 1.054 1.052 1.045 1.032 1.015

175

FIG. 2. Some profilch of I I in an expansive frce interaction: .Y, = 0.41

176 K . Stewartson

functions must be added to the expansion (7.4) in some way, possibly as suggested by Stewartson (1970b).

A numerical integration of (6.1) with a , = - 1 has been carried out by P. G. Williams, who initiated the solution by arbitrarily decreasing p at x = 0 from zero to - lo-'. The solution then evolves in a quite stable way and was integrated step by step using a method described in Stewartson and Williams (1969). In Table 1 we give the values p and z as functions of x together with that of

z2 = ( d u / i y ) l Y Z 2 . The variation of z2 with x brings out the tendency of the viscous effects to be concentrated in the neighborhood of the wall as x -+ x,- so that one can almost think of there being a slip velocity at the wall. In Fig. 2 we display some velocity profiles which show the same tendency.

VIII. Compressive Free Interactions

We now turn to the third independent solution of the fundamental equation of the triple deck in which the pressure rises and separation can occur. The numerical problem of integrating (6.1) and (6.2) and extending the analytic solution (6.5) with a , = 1 is straightforward so long as 5 > 0. As in the expansive case, the solution (Stewartson and Williams, 1969) was initiated by increasing p slightly from zero, to either LOp3 or lo-', whereupon the solution evolved naturally until separation was reached. Although the separation points in the two integrations are different, the two solutions differ by very little once the origins of x are moved so that the separation points coincide. It appears in fact that, if we fix the origin of x by requiring z(0) = 0, there is a stable unique solution of (6.1) and (6.2) in x < 0. These conclusions are confirmed by independent numerical studies due to F. T. Smith, R. Melnik, and P. G. Daniels (private communications). In the rest of this review we shall adopt the convention that in compressive free interactions

z(0) = 0. (8.1)

Given the pressure gradient, that the boundary layer develops a singularity at separation is, in practice, apparently a universal rule. Here, however, the pressure gradient is related to the over-all boundary-layer properties, and we can establish that that kind of singularity c ~ i i i i o t now occur. The structure of

Multistructured Bouizdury Layers 177

the solution near the separation singularity was first established by Gold- stein (1948). He assumed that the pressure gradient is finite and that the separation profile us(y ) is smooth [this last assumption is not quite justified (Brown and Stewartson, 1969), but the modification is unimportant to the basic argument]. Then two regions can be distinguished:

(i) J’ - ( -x)’l4. Here

a being a constant. (ii) y - 1. Here

Goldstein also assumed, and Terrill (1960) convincingly demonstrated its correctness by a numerical integration, that these two regions overlap. Hence we are able to determine a in terms of a;

u = a/Uf‘(O) = a/p’(O), (8.4)

and it follows that

A ( x ) = A ( 0 ) + [a/p’(O)](-x)’ /2 (8.5)

near x = 0. Thus we obtain a contradiction because p(x) = --A’(x) and must be bounded at x = 0 for Goldstein’s argument to be applicable. The only other possibility available to us at present is that p’(0) is finite, but ct = 0 and the solution is regular at x = 0. In broad terms the argument is that any singularity in z at z = 0 provokes a worse singularity in p which implies that separation must have occurred at some negative value of x, contradicting the assumption (8.1). The numerical solution is quite regular at s = 0.

In Table 2 we give a set of values of p and z as functions of .Y. mainly for .Y < 0, but including a few values i n x > 0. As explained earlier i t is believed that the solutions in .x I 0 are correct, but we shall see that as x increases from zero the values of p and z become less certain. In Fig. 3 these results are displayed graphically.

In fact, once x > 0 the computational problem becomes much more difficult. Catherall and Mangler (1966) who were the first to penetrate into

178 K . Stewartson

TABLE 2

VARIATION OF /J A N D T I N A COM- P I ~ I S S I V E FREE INII:RACTION

\- T P

- 13 - 12 - 1 1 - 10

-9 -8 -7 -6 -5 -4 -3 -2 - 1

0 1 2

0.9999 0.9998 0.9996 0.9989 0.9976 0.9947 0.9876 0.9720 0.9375 0.8642 0.7226 0.4932 0.2 I83 0.0000

-0.1117 -0.1465

0.0001 0.0002 0.0004 0.0008 0.0020 0.0045 0.0102 0.0232 0.0521 0.1 139 0.2372 0.4491 0.7366 1.0260 1.2500 1.3997

1.6) I /

1

I I I I

I I I I I I ‘i! I

X

FIG. 3. Plots of / J and T as functions of x in a compressive frec interaction

Multistructured Boundary Luyers 179

the region of reversed flow, albeit for an artificial incompressible boundary layer with the pressure related to the displacement thickness, found that instabilities soon set in. The reasons are not hard to find. Stewartson (1958) had already shown that when the solution is regular at separation, an infinite set of undetermined constants appear in the solution just downstream of separation (x > 0) and these are probably associated with propagation of small disturbances upstream through the reversed flow region. Indeed, if u = u l (y ) [< 01 when Y = x1 [> 01 and 0 < y < y l , while 24 2 0 when x = x1 and j’ 2 J’, we might expect, on physical grounds, that u,(J.) can be arbitrarily prescribed. Numerical computations carried out by Belcher et a/. (1972) for a boundary layer induced by a generalized vortex also point to the truth of this statement. Finally Friedrichs (1958) has made a rigorous study of a class of differential equations which have a property equivalent to reversed flow, and his conclusions strongly suggest that ul(y) may be chosen arbitrarily. I t so happened that Williams’ computations did not at first reveal any instabilities or nonuniqueness. He found that he could continue his solutions into x > 0 and they remained stable. Further investigations over a number of years showed, however, that the stability was illusory: any change in the program in x > 0 leads to instabilities in the flow field.

The present state of knowledge about the solution properties in x > 0 has been reported in Stewartson and Williams (1973). In view of the high degree of nonuniqueness the most sensible procedure is to look for solutions of physical significance, by which we mean that they, the compressive free- interaction solutions, can be matched to another solution valid downstream in which the x scale is considerably larger, possibly O(L*). Such a solution might hold in the plateau region of the shock-wave boundary-layer interaction flow field. In the absence of detailed knowledge of such a solution, the best way to achieve the match would seem to be to look for solutions of the compressive free-interaction equations which achieve some sort of self- similarity as x + m. One likely possibility is that u 1 ( j > ) -+ 0- as x, + i~ so that the reversed-flow velocity rises to zero as x + cc. Consequently the negative values of u are quite possibly small for all x, and the approximate method devised by Reyhner and Flugge-Lotz (1968) in which the term u(c?u/c?x) is neglected in (6.1) whenever u < 0 has some merit. The most successful of Williams’ experiments made use of this approximation. He finds first that the pressure curve, which is concave upward in x < 0, now becomes convex upward, and p shows every sign of approaching a finite limit as x + m, which we shall denote by Po.? Its numerical value is about

t P. G. Williams (private communication) has substantially improved his method of numerically integrating the equations when .Y > 0. The latest results are in close agreement with the asymptotic structure proposcd here, and his value for P , is now 1.80.

180 K . Stewwtson

1.8. Second, the skin friction decreases to a minimum negative value of about -0.15 at x = 2, subsequently increases, and shows every sign of approaching zero in the limit x + x. Third, the region of reversed flow steadily increases in width with x, being % P o x when x B 1. Fourth, the minimum value of u itself reaches a minimum value of about -0.28 at x % 30 and then slowly increases, quite possibly tending to zero as x + zc. In Table 3 we set out the variation of p, z, and I!,,,,, with x, as given by a representative numerical integration, and in Table 4 give the corresponding velocity profile at x = 40.25.

TABLE 3

VARIATION OF 11 W I T H y AT s = 40.25, W I I I ~ R E p = 1.803.

v 0 8 16 24 32 40 48 56 11 0 -0.1 10 -0.219 -0.251 -0.244 -0.213 -0 162 -0.071

1‘ 64 72 80 88 I1 +0.323 t 2 . 7 1 +9.57 + 17.54

I1 - y + p.u 8.5 3.4 2.2 2.1

TABLE 4

VARIATION OF p , T, A N D id,,,,,,, T H ~ LEAST VAILJE OF ii R ~ G A R D F I I AS A FUNCTION OF )’, WITH X.

Y 10 20 30 40 50 P 1.719 1.779 1.796 1.803 1.807

- 7 0.083 0.041 0.027 0.0 I9 0.0 15 - T-‘ 12.1 24.2 31.6 52. I 67.6

lo5 t / p / t / u 1268 274 109 56 33 (t/p/d.u) ~ 14 34 60 90 I22

“,,,,,I -0.258 -0.277 -0.267 -0.253 -0.241

The variation of u,,,~,, with x is not encouraging for the Flugge-Lotz approximation, and indeed it turns out that the method has a fatal flaw from the point of view of self-similarity in the reversed flow region. It is not possible to set up an asymptotic expansion in this region using their approximation. Nevertheless the broad results are encouraging and strongly suggest that a possible self-similar asymptotic form for the solution of (6.1) is

p + P , , u + 0 - if J>/X < P o ,


in the limit x -+ a, and we shall refer to this as the Williams asymptotic profile.

The description of the flow field in x 3 1 provided by the numerical solution is not unreasonable on physical grounds for compressive free interactions which are not dominated by external agencies. Of course, some external agency does provoke the free interaction, but often it is far downstream and separated by slowly moving fluid so that it is hard to imagine that the agency controls anything more than the position of the free interaction. Suppose, instead of p + P o , p reached a maximum and then decreased; the reversed flow would have to contend with an adverse pressure gradient and could never increase from zero at infinity. Hence there would have to be a local external agency. Again, if p + a with x, Bernoulli's equation would give a contradiction and although it is not necessarily strictly relevant, it raises doubts about this possibility.

We now demonstrate that the Williams profile (8.6) may be used to set up a consistent description of the behavior of the solution of (6.1) when Y S 1. It is believed, although we cannot prove it, that the structure is unique apart from a set of arbitrary constants. The description we have in mind is itself multistructured : the lower deck, subdividing into three distinct sublayers, is illustrated in Fig. 4. The structure may be thought of, in physical terms, as consisting of ( i ) a vorticity layer of thickness cx .x1/3 near y = P o x in which the vorticity ?u/i'j. increases from zero to unity and above which the vorticity remains constant, (ii) a slow inviscid reversed flow below y = P o x , which supplies the fluid needed for entrainment into the vorticity layer, and (iii) a subboundary layer of thickness cx x2/3 below (ii), which enables the no-slip condition to be satisfied on the plate. The idea is that if the reversed flow is small, as (8.6) suggests, then it may be neglected to a first

0

FIG. 4. Subdivisions of thc lower deck

182 K . Stewurtson

approximation in considering the vorticity layer (i). In that event an amount of fluid cx x2l3 must have been entrained into (i) from below when x % 1. Consequently the velocity of the reversed flow in ( i i ) must be O(x-’I3) since its thickness is O(x). It now turns out that the flow in (ii) is inviscic the usual scaling arguments establish the boundary layer in (iii).

For the leading term of the solution in ( i ) we write

< = (y - Pox)/xl’3, 0 = ,y2’3 Go(<),

and take 4 = O(1). Then, on substituting into the basic equations (6. find that

G; + +G, c; - gc;2 = 0,

where, as usual, primes denote differentiation with respect to the independent variable, in this case t. From (8.6) or the boundary condition (8.2) as j’ + x, it follows that C;(t) + 1 as ( -+ a. In addition, since the velocity in ( i i ) is reckoned to be small, Gb(() + 0 as ( + - z. Apart from the origin of (, the solution of (8.8) subject to these conditions is unique, the limit as < + - ~8 is approached exponentially, and

Go(- a) = -C,,, where C, = 1.2521. (8.9)

I t is convenient to consider regions (ii) and (iii) together, taking for the leading term of the solution

$ = ~ ‘ / ~ q ~ ( r ] ) , where r] = y/x213. (8.10)

In view of our earlier remarks, we can expect gd(r]) to be constant and O( 1 ) when r] 4 I but ( is large and negative. The motion in these regions must be driven by a pressure, and the appropriate form is

p = Po - 3 ~ / , / 2 . ~ ~ ’ ~ , (8.1 1 )

where d o and Po are constants to be found. As a result of (8.1 1 ) we see that A z - P o x + z d , ~ ’ ! ~ + . . . when x is large. On substitution of (8.11) into (6.1) we have

3yl;; + qb’ + goy; = 3t/, , (8.12)

and qo(0) = &(0) = 0, from the no-slip condition. The third condition which fixes go uniquely is that

qL/b(q) + - (3t/”)”2 as r ] + x. (8.13) The form for q, then corresponds to inviscid flow when r] 4 I and has the properties

qG(0) = 31’4 d;’4ao , q, + ( 3 r I o ) ” 2 r ] + 33’4 r/;/4ao as r] + a, (8.14)

where a, = - 1.087 (Mills, 1938; Burggraf rt ul., 1971). In region (ii), therefore (y + I),

u % -(3d0)’?-t’3, I// = -(3d0)1’2yx-”3 + O( .Y”~) (8.15) and, to effect a smooth match with region (i) when J’ z P o . Y , the most important requirement is the continuity of I//. Hence, from (8.9) and (8.15)

c,, = ( 3 d o ) ” ’ ~ o . (8.16)

The match is incomplete because u is exponentially small when ( is large and negative [see (8.7)], while 2 1 . x ~ ’ ~ is finite for all y > x2I3 [see (8.15)]. The apparent contradiction is resolved by taking more terms in the expansions of I// in descending powers of x, and eventually in ascending powers of log x, of which (8.10) is the leading term, and we note that this form leads to an expression for A which is consistent with our basic assumption A z - P o x. Furthermore, details of the expansions, including comments on the eigenfunctions that eventually appear, are supplied in Stewartson and Williams (1973).?

A comparison between the predictions of this asymptotic theory and the numerical experiments can be made. With reference to Table 4, there are strong indications that p is approaching a limit as s + K . Also ( d p / d . ~ ) - ~ : ‘ is almost linear in .Y when Y g I . which is consistent with the asymptotic form (X. 1 1 ) for p. Again 7- is almost linear in x, which is consistent with the result

T z .Y- ‘(/:(0), (8.17)

derived from (8.10). The comparison shows up in a worse light, however, when we consider the reversed velocity profile. Numerically, as Table 4 indicates, there is no sign of an inviscid structure even at x = 40 and u,,,~,, is increasing only very slowly. I t is believed that the fault lies in the neglect of ii du/?x, when u < 0, in the numerical solution of (6.1) which implies that a structure of the kind sketched in Fig. 4 is impossible when x % 1, for, with neglect of this term and the assumption that (8.10) holds in regions (ii) and (iii), yo satisfies

3q?; + u;;(qo - 2yth) = 300 3 (8.18)

instead of (8.12). However there is no solution of (8.18) satisfying the no-slip condition and (8.13). Instead all solutions must become singular at some finite value yo of y and, near y = yo,

-1- Professor Messiter has kindly informed thc author t h a t Neiland ( 197 1) had earlier arrived at the leading term of the asymptotic structure whcii Y % 1.

184 K . Stewurtson

This property of go effectively rules out region (ii) and makes the match with region ( i ) difficult. I t is not surprising, therefore, that the agreement between the numerical solution and the asymptotic expansion is unsatisfactory in region (ii).

IX. The Plateau

The asymptotic structure to the compressive free interaction, discussed in the previous section, necessitated a subdivision of the lower deck into three regions and thus the triple deck develops a quintuple structure when .Y is large. However, the added complication is only apparent, and in fact, the solution can be matched to a credible structure in the post-separation region or so-called plateau region where the pressure is virtually constant. Suppose, to fix matters, that the free interaction is provoked by a geometrical or physical feature at x* = x:, where x: > ,Y:, for exampie, by a concave corner or shock. Let LIS consider what form the flow in the plateau must take to match the compressive free interaction, assuming that the streamwise length scale is O(xX). We consider each subregion in turn, with x % 1.

A. LOWER DECK, SLJBLAYEK (iii)

In the triple deck sublayer ( i i i ) of the lower deck is a boundary layer of thickness O(x2 ”) and with 11 = O(,Y)(- I ’). In terms of the physical variables, its thickness is O(r;3(x* - X , ” ) ~ / ~ . Y , ? ~ ~ ’ ) and the velocity is O(r:2U: x,”l!’(x* - x : )~”~) . One, and the most obvious, implication for the plateau is that it contains a boundary layer near the wall, of thickness O(E~.Y; ) , in which fluid is being drawn with velocity O(c2U? ) toward s,?. The associated pressure variation is O(c4pf U*,2). This part of the plateau is denoted by IV in Fig. 5.

B. LOWER DECK, SUBLAYEK (ii)

In the triple deck sublayer ( i i ) of the lower deck is a region of inviscid flow of thickness O(x) with I I also O(.Y-”~) . In physical terms its thickness is

Midtistructured Boundury LUJVYS 185

.I X 5 ' X "

FIG. 5. The regions of the plateau (.Y,* < .Y* < Y:). Region V IS the continuation of region 111 when the provoking feature in\-* > .Y: is injection.

O(c2(.u* - x:)), and hence one might expect that the plateau contains a region of reversed flow governed by the inviscid equations lying immediately above the boundary layer IV. Its thickness is O(c2x,*), and the velocity and pressure gradients are the same as in IV. It is denoted by 111 in Fig. 5.

C. LOWER DECK, SUULAYER (i)

In the triple-deck sublayer (i) of the lower deck is a region of viscous flow of thickness O(Y"~), in which the vorticity climbs from zero on one side to unity on the other. In physical terms its thickness is O(~~x:~~ ' (x* - x:)' '), and hence one might expect that the plateau contains a shear region of thickness O(x,*c4) in which the direction of motion changes sign and U*

begins to increase rapidly. We note that the thickness of this region is obviously the same as that of the main deck, in triple-deck terms, when .x $ 1, and should therefore be considered together with it.

D. MAIN DECK

As explained above in Subsection C this region merges with the top of the lower deck when x 9 1, and indeed, the solution there is simply the Blasius flow with y displaced by A ( x ) . Hence it continues into the plateau as a thin shear layer of thickness O(c4x,*) with an initial profile consisting partly of the Blasius profile and partly of the profile u* = 0. It is denoted by I1 in Fig. 5. The exact method of evolution of this initial profile is universal and independent of the pressure gradient applied in 111. Further, it is so thin that it may be regarded as a single line 1*, so far as region I11 is concerned. The appropriate boundary condition on the reversed flow in I11 is that the stream function at I* is continuous with the value of the stream function at

186 K . Stewartson

J’* = - x, in the evolution of the Blasius boundary layer after the no-slip condition u* = c* = 0 at J,* = 0 is discontinuously changed to U* = 0 at y* = - cc. The precise shape of the boundary line I* is determined by the pressure in 111.

E. THE UPPER DECK

Since the pressure p tends to a limit as x ---f cc, the flow in the upper deck becomes uniform to order .Y-”-‘ as x -+ cc. In physical terms the mainstream changes direction by an angle O(c2), and the flow is thereafter uniform apart from the effects of pressure changes

O(r:4p;, u;2x:2’yx* - x\*)-’ 3 ) .

Thus, one might expect that in the plateau and above the evolving Blasius flow, the fluid motion is uniform apart from variations O(c4U.yX), while the increase of pressure above p: is constant apart from variations O(c4pZ ). This region is denoted by I in Fig. 5.

If the description of the plateau is correct, then the procedure for determining the flow field in it is a s follows. First solve the problem of the evolving Blasius profile, mentioned in Section IX,D, i.e., solve the standard boundary-layer equations with a zero pressure gradient for a compressible fluid in s* > 0, subject to the conditions

il* = U ; U0(~~*/c4.upY) when .Y* = 0 and J** > 0,

11* = 0 when .Y* = 0 and J’* < 0,

Lf* + u: as !‘* ---f x, .Y* 2 0, (9.1)J-

21” + 0 as J,* -+ - cc, s* 2 0,

where U , is the Blasius velocity profile, defined in (2.2) and (3 .2) . Suppose that according to the solution of this problem E ~ U ~ rB(s*) is the limit of I.* as y* + --x. This function has been computed by P. G. Daniels (Stewartson, 1974) and has the properties that uI3 = O(.Y*- ‘ I 3 ) as x* ---t 0, while cB = O(x*-”*) as x* + cc (Chapman, 1950). Second, solve the inviscid equations governing the flow in 111, namely,

subject to the boundary conditions

‘i Effectively Y * is a running variable in (9. I )

Multistructiired Boundary Layers 187

when y* = 0, .x: < x* < .Y$,

C* = c4u*, c ~ ( x * - x:)

when y* = aX(x* - x:),

L f * = i?U: U,(2'*)

when s* = .u;, 0 < J,* < r;(.uX - x:) = r:2~$d,.

(9.3)

In these equations j v * = ~,*(x* - x:) is the straight line to which 1* reduces on the scales of region 111, where a,* = i:2C"4i1"2(M2, - 1 ) - 1'4P0 is constant and determined by the pressure rise through the triple deck. Further- more r?U; U , is supposedly prescribed in some way from the particular feature provoking the plateau. The only internal requirement on U , arises from the balance of mass influx to region I11 across x* = x: with the mass entrainment into region 11, i.e.,

. ~~~* , i~+~ . so* 1 U1(y*) dy* = 1 c&* - .u?) rlx". (9.4) ' 0 . .r,*

Alternatively this condition may be regarded as fixing x: relative to s;. Given I/, and rB, a universal function, the pressure variation p* may be found uniquely. Wc shall show how this may be done in Section XI1 when we discuss the effect of strong blowing in x* > .xg, since it is only in that case that we are, at present, able to determine U, . We note here that if both 1 do 1 and 1 6, 1 are O(1), then (9.2) is the correct leading term in the expansion of the Navier-Stokes equations in powers of i: in region 11, the terms neglected being of relative order c4. Moreover, the boundary layer, region IV, under Il l , is of thickness O(c3.u,*) since the slip velocity is O(i: 'U:). Finally the behavior of the solution of I1 as x* + x$ + is dominated by the form of rB, which is O((x* - . x ~ ) - ' ~ ~ ) . Hence since region 111 converges linearly to a point, u* = 0(c2U*,(.x* - x:)-'!~) as x* + x:, in agreement with the requirements of sublayer (ii) of the lower deck of the triple deck. The pressure variation determined from (9.2) displaces the line I* but only through a distance O(C~X:) which is negligible to leading order.

I t is interesting to note that we have been able to construct a coherent self-consistent structure for a supersonic boundary layer which includes a substantial length of separated flow. Furthermore, the leading term is embraced within classical boundary-layer theory, except that the pressure gradient is determined from the displacement thickness. The only arbitrar- iness lies in the choice of x:, where separation occurs, and of U,(J~*), the reversed flow set up at the feature which provokes the whole phenomenon. Unfortunately at present we are only able to determine U I for one type of feature, namely blowing, but it is quite possible that it may also be comput- able for cranked plates and shockwave interactions. Should i t be thought

necessary to continue the expansion to the second term, of relative order c, then we must take note of the variation of pressure across the triple deck. Our tentative opinion is that the main effect will be to change the value of P o , while leaving the basic structure of the asymptotic form of the lower deck (Fig. 4) and the plateau (Fig. 5 ) unaltered. Hence, apart from this one quantity, which fixes I " , the second term for the plateau region can also be embraced within the classical boundary-layer theory. I

X. Comparison With Experiment

The fundamental approach adopted in this review is mathematical. Our aim is to set up an asymptotic structure to the solutions of the Navier- Stokes equations, valid in certain circumstances in the limit I: + 0. While we cannot claim that our results are established rigorously, we do claim that they are in the best tradition of mathematical conjecture, by which we mean that we do not believe that i t will be possible to prove any of them false, even though rigorous proof of their truth may never be obtained. The only valid criticisms of the results must be made within the terms of reference of asymptotic solutions of differential equations.

Nevertheless, it is hoped that an understanding of the asymptotic properties of the solutions of the Navier-Stokes equations will be helpful in understanding the behavior of real fluids in motion at high Reynolds numbers. If the results of an asymptotic theory, strictly valid only as a parameter I; -+ 0, are not in accord with accurate numerical results at finite values of i;, the usual explanation is that the numerical values of c chosen are too large and smaller values must be used in the numerical work. The correctness of this attitude can usually be demonstrated. However there are three special points to bear in mind when we consider the relation between asymptotic theories and the Row of real fluids. First, the governing equations may be insufficiently accurate to describe the flows; in the present case this can be disregarded. Second, if theory and experiment do not agree, i t is usually not possible to allow the relevant parameter i: to decrease indefinitely in the experiments for, at some stage, the flow becomes unstable and the theory ceases to have any hope of relevance. Third, the experimenters often have in mind a practical situation where the value of c is fixed by conditions which have nothing to do with the niceties of the mathematical argument.

Comparisons between theory and experiment should therefore be made in a spirit of humility and we should always be prepared for the possibility that, however exact the theory, i t is relevant only to situations which are of no interest to the engineer. In the present instance there are hardly any experiments at values of i: < 0.20, and in most 0.20 < i: < 0.35. The comparison is still useful, bccause it may give the mathematician confidence in his rational

Mirltistriictured Boundary Layrrs 189

but nonrigorous approach, it may reveal new features of the flow which he had not allowed for, it may help to delineate the range of c over which his asymptotic theory is a good approximation to the correct solution, i t may help to isolate parameters controlling the flow properties, and may even be of value as a quantitative predictor.

There are, of course, already a considerable number of predictive methods available for interactive problems, perhaps the most effective being based on the ideas of Lees and Reeves (1964). Comprehensive reviews on these methods have been written by Fitzhugh (1969), Murphy (1969), and Geor- geff (1972). Murphy comes to the conclusion that, although certain features of the phenomena are adequately described by these methods, for instance pressure variations resulting from weak shock impingements, there are a number of features for which the methods are inadequate, e.g., heat transfer. That there are deficiencies in integral methods, like the Lees-Reeves, is not surprising for they are primitive tools in boundary-layer theory and impute to the flow a number of properties which cannot be correct (Brown and Stewartson, 1969, p. 67). I t has been suggested that pressure variations across the boundary layer may be a cause of the discrepancies and indeed it is apparent from Section 111 that they will be important at the second stage of the asymptotic expansion. However the most pressing need would seem to be to improve the field-equation approach to the problem, initiated by Reyhner and Flugge-Lotz (1968), particularly with regard to the treatment of the reversed Row regime.

The asymptotic theory is not yet in a position to predict overall properties of these interactions, being restricted to the free-interaction component. In order to assess the relevance of the theory i t is convenient to continue the expansion to include the term O ( c 3 ) in the pressure: the necessary calculations have been performed by Brown and Williams (1974) and the results have kindly been made available for this review. One of the difficulties in making a comparison with experiment is that X;, the start of the interaction, is comparatively easy to observe, whereas the natural s* coordinate to use as origin is the position of separation x:. Theoretically it should not make much difference because the length of the free-interaction region is only O(e3x:) but the numerical factors are so large that there is a significant difference between 7: and Y:. Values of (.x$ ~ xZ~) / .x~ of about 0.3 are typical. The value of Xg is also rather vague in the theory and in this comparison it was decided to follow the theoretical definitions and convert, as far as possible, the experimental data. Again the length of the free-interaction region is not easy to define theoretically and so it was decided to compare the values of

(10.1)

190 K . SteM,artsoiz

which gives a good guide to this length, and may be measured on the experimental data with a ruler.

The numerical solutions were found by writing

u = U I + cu2 + O(e2), c = 1‘1 + t:cz + O(r:2),

p = p2 + cp, + O(?), (10.2)

in (4.5), where ~ i , , e l , and p2 satisfy (4.5) with boundary conditions obtained by setting I: = 0 in (4.6), (4.7) and (4.8). The equations for L I ~ , c2 , and 11, are

i u 2 i c 2

?x ?J + = 0,

(10.3)

with u2 + 0 as I + -K, u2 = r2 = 0 at = 0,

( 10.4)

as y + a, and

p3(x) = - R ( . ~ ) + 2 ~ ’ 9 , 5 / 4 AJM: - I ) ~ ’ ~ T : / T ; ) - l / 2 ~ ” (- 1. ( 10.5)

The solution of this linear system of equations includes a complementary function

u2 = CI(i.u,/ils), p 3 = M ( d P 2 / / A X ) (10.6)

for any constant CI. This is simply appropriate t o a shift in the origin of .Y. The values of p and d p / d x at separation are nevertheless uniquely determined as a series of powers of i;, of which we now have the first two terms. The results are

where

10.7)

10.8)

taking separation to occur a t the origin.

191

M r 2.0 2.0 2.3 2.3 2.4 2.4 3.5 3.6

7 R x l O - ' Y 28 9 I9 - I0 3 0 80 pI (0) 0.97 0.86 0.96 0.90 0.88 0.68 0.88 1.16

p l ( x ) 2.18 1.67 1.72 1.63 1.67 1.34 1.76 1.74 [/I; ((I)]- I 2.8 2.9 2.6 2.2 2.2 2.6 2.5 4.8

" Taken from data in Chapman c't r r l . (1958). " I n the table is defined in terms of the measured v;iIues of /I* and .Y* by means of

(4.4).

The standard experimental data at moderate supersonic speeds were obtained by Chapman er [ I / . (1958) at values of A4, up to about 3.6 and by Hakkinen ct a/. (1959) at M , = 2. Chapman's results were the first to give the variation of p*(x%) with M , and I: that corresponds to p ( 0 ) being constant, and were backed up by a qualitative argument. Unfortunately he did not include tables and in order to provide a detailed comparison with the present theory, the essential quantities have to be measured from his graphs, but a set of representative data is given i n Table 5. Similar data may be gleaned from Hakkinen's graphs and a set is given in Table 6.

Thus the leading term of (10.7) gives the experimental values of p*(.u*) quite well, and the agreement is even improved by using the second term as well. The plateau pressure is also close to the theoretical value of p 2 ( x - ) , (= l.8), but it is quite possible that the agreement will be less good when the next term is computed, or even if the present estimate for Po is improved. The theoretical estimate of [p ' (O)]- I is not in good agreement with [ p k ( O ) ] - ', however. The first term is substantially higher than the mean experimental

R x l o - ' 2.1 2.25 2.3 2.4 2.55 2.6 2.8 2.9 3.5

p1(0) 0.64 1.30 0.61 1.29 1.08 1.37 1.50 0.90 0.81

/ I ~ ( X ) ~ 1.35 1.68 1.66 1.61 1.80 1.74 ~- 1.58 1.55 [Pi (0)l 3.6 3.1 3.5 2.Y 3.8 3.8 3.4 4.0 3.5

' I Taken from data in Hakhincn ('1 ol. (1959). " In the table ,Atz = 2, atid /I, (.Y) i \ defined in tei-iiis oftlie mca\tired ~ a l u e b of /I* and Y * b j

means of (4.4).

192 K . Stewartsoii

value, and when the second term is added the value of [p ' (O)]- ' is decreased to 2.1 at c = 0.23, M, = 2, and is even smaller if hl, is increased further. These two terms, in fact, seem to straddle the experimental values, but we must accept that the value of 0, and hence of the width of the interaction region, is poorly approximated by the first two terms of the asymptotic expansion of 6. This confirms the conclusion reached by Stewartson and Williams (1969), although there the discrepancy was somewhat masked by the use of x2;, instead of xt, to define R , and consequently the value of p*(.x$) was overestimated.

Very recently, Carter (1972) has carried out some numerical integrations of the full Navier-Stokes equations for flows past a concave corner. The Mach numbers were 4 and 6.06, while the Reynolds numbers at the corners were about lo5. In the examples considered, the agreement he obtained between the theoretical calculations and the observed flow properties seem to be better than had been found hitherto. He confirms the general ideas of the triple-deck free-interaction theory qualitatively. Provided his definition of I: is based on xt rather than on ig, the point where the interaction first becomes noticeable, the quantitative agreement with p ( 0 ) is also good but, as with the experiments, his estimate of the length scale of the interaction differs from that of the leading term of the triple-deck expansion.

If M, 4 I , the formulas (10.7) and (10.8) may be written in terms of the hypersonic viscous parameter

and in particular when the wall is adiabatic, ;! = 1.4, and G = 1 , they reduce to

x = Cli2M; c4, (10.10)

p*(x;)/p*, = 1 + 0 . 8 2 8 ~ " ~ - 0 . 2 9 ~ ~ ' ~ + ...

pZ/x$p* ' (~$) = 1 . 6 7 ~ ' ~ ~ - 4.4%"' + . . . .

(10.1 I )

(10.12)

and

These formulas are only valid when 4 1 of course and the solution properties when % - 1 are still open. Some recent developments are described in the Addendum.

A number of experimenters have examined separated laminar boundary layers at high Mach number, including Sterrett and Emery (1960), Needham (1965), Miller rt al. (1964), and Watson rt al. (1969). It has not been possible to express the data they give in terms that can be directly compared with the present theory. For example, they naturally favor .UX as a reference length for e, and a change to .xP often cannot be made. Again the plateau pressure is recorded, rather than the separation pressure. Nevertheless the correlation of data made by Watson rt ul. is interesting from our point of view as a guide to future developments in the theory. In Fig. 6 we reproduce their correla-


X O

FIG. 6. Correlation of plateau pressures. = C' 'M.: t ; : , t:: being based on .?t. The uppcr curve is the theoretical estimate for /];and the lower is that for p:( .~:) . 0 . Watson ('1 r r l . (1969); x , Needham (1965); +, Sterrett and Emery (1Y60); . Hakkinen o r trl. (1959).

inn-

365

t FIG. 7. Correlation of the free-intcraction length. 0 , Watson et u / . (1969): x , Needhani

(1965); +. Chapman rt ( I / . (1958); , Hakkinen et rrl. (1959).

194 K . S twar tson

tion for the plateau pressure pp* as a function of lo together with the theoretical estimates ofp*(xT) and pp*, taking Po = 1.8. In Fig. 7 we reproduce their correlation of I , , the length of the free-interaction region and roughly equivalent to 8x: in (10.1), in terms of the boundary-layer thickness 6” and the Reynolds number R , at .UX, M, , and p,*/p: . Figure 6 supports the scaling implication of (10.1 I ) , but Fig. 7 implies that 0 is of the form t:’F(%). I t may be that the variation of c2 throughout the experiments is not significant, bearing in mind the spread of the data, or possibly that the flat plate was not maintained at adiabatic conditions, which would mean that (10.12) needs modification before a comparison with experiment can be made, or again that the free-interaction solution, even for adiabatic conditions, is not completely specified by x when j ! 2 1.

XI. Convex Corners in Supersonic Flow

Hitherto we have been concerned with the spontaneous evolution of a supersonic boundary layer by means of a triple deck, leading either to a pressure singularity in an expansive free-interaction or to separation and a plateau pressure in a compressive free-interaction. No direct notice was taken of any feature which might provoke such behavior, the assumption being that, in general, its main effect would be to control the (possibly virtual) position of .Y: or x:. A start on a more complete examination of the flow feature is now intended, with the aim of setting up a uniformly valid asymptotic expansion, about c = 0, of as much of the solution as is physically reasonable. We begin by considering the feature to be a convex corner so that the mainstream outside the boundary layer is expansive, and we can expect the pressure to fall as the fluid negotiates the corner. According to inviscid theory the fall in pressure is discontinuous. and this would provoke a violent reaction from the boundary layer. I t might be expected, therefore, that the boundary layer will anticipate the corner by evolving as an expansive free interaction with a (virtual) position of .Y: chosen to enable the pressure to make the adjustment downstream of the corner as required by inviscid theory. Since the pressure variation in the triple deck is O(f;’p:,) it can be expected that as the angle c(* of the corner increases from zero. the first significant new flow properties will arise when u* = O(r:’). It is convenient to define

cx* = r : 2 ) w 1 / 2 ~ l : 4 ( ~ ~ - 1 ) 1 / 4 ~ , ( 1 1 . 1 )

and take the origin O* of the triple deck to be the corner, distant r,* from the leading edge of the plate with (s, J,) axes parallel and perpendicular to

Multistrucfured Boundar.), Lajvrs 195

that portion of the plate. Then, if u = 0(1), the governing equations for the leading terms of the triple deck are the same as (6.1). The boundary conditions are now

i , - 2 ' + 0 as x+-a, u - y + A ( s ) as y+x ,

A'(x) = - p ( x ) ;

= r = 0 for ~1 = 0 if s < 0,

(11.2)

( 1 1.3)

while

ii = I; = 0 for J) + x s = 0 if s > 0. ( 1 1.4)

The last two conditions reflect the fact that the plate is cranked at 0". The easiest case to study is when x G 1, for the solution can then be written in the form

The equations for the linear terms are

and the boundary conditions at the wall are that t , ( l ) - - - c ( 1 ) - - 0 if . x < O , y = O

and (1 1.7) - x = 0, J1) = O if x > 0, J' = 0.

This problem has been studied by Lighthill (1953) and Stewartson (1970c, 1971). I t is found that

= -3 K \ 4e - if x < 0,

31'2 7 r 1 3 e - K l Y lit (11.8) I f x > 0,

27c .il, t8'3 + r4'3 + 1

while

where 1; is defined in (6.8).

196 K . Sttwwrtson

Thus ahead of the corner and within a distance O(C'.Y,*) of it, the pressure starts to fall until at the corner itself the pressure has fallen 75x) of the amount required by inviscid theory. The remaining 25 '%, of the fall occurs in x > 0. Simultaneously the skin friction rises ahead of the corner, returning to its Blasius value on the downstream side.

The linearized theory also applies equally well to concave corners where a < 0, the only difference being of sign: for example, the pressure rises upstream of the corner and the skin friction falls. Indeed, it is possible to make a rough estimate of the value a, of r , required to induce separation, by equating 1 + rz ' l ) to zero at x = 0. We find that?

-a, = 0.50. (1 1.10)

The nonlinear theory has been studied for convex corners only by Stew- artson (1970b), the numerical work being performed by P. G. Williams. The broad strategy of the numerical method is to use the expansive free- interaction solution of Section VII in x < 0, which is unique once the termination point xl. (> 0) is fixed. At .Y = 0, the solution changes, due to the change in geometry, and it is hoped that xr is chosen correctly so that p + --c( as .Y + a. Such a situation would conform with the requirements of the inviscid theory and with the exact solution

u = J' + EX, p = -a (1 1.11)

of (6.1), which satisfies the boundary conditions in x > 0. If, however, xf is chosen to be too large, then the fall of pressure in x < 0 is too small and a compressive free interaction is set up in .Y > 0, while if .'if is too small, the expansive free interaction continues, terminating in a singularity. I n either case the asymptotic form of the solution is quite different from the requirements of ( 1 1.11). Thus the numerical integration proceeds by trial and error until the correct value of xf is chosen.

The integration near x = O t requires some care. As can be seen from the linearized solution ii2pli?x2 K xp2'3 and C?zl?x 'x x - 1 / 3 as .Y + O + . A detailed study of the solution near x = 0 shows that it is double structured. There is a region near the wall where the solution may be expressed as a series of ascending powers of Y whose coefficients are functions of j ~ , ' . ~ ' ~ ~ , and a region further out where the solution may be expressed as a series of ascending powers of x whose coefficients are functions of y. The two series appear to have a region of common validity and are completely determinate in terms of the velocity profile at s = 0. Numerically a fine mesh is needed to deal with the singular region when y - x1I3 and x B 1, and so a variable

'i Numerical studies of t h e nonlinear equations by R. Jensen (privatc communication) suggest that i n fact 1.5 < - r , < 2.0.

Multi.structurc~l Boundary Layers 197

mesh size, after the manner of Cebeci and Keller (1971), is appropriate. The most refined version has been developed by Smith (1973a), who uses a uniform mesh in terms of j ~ / x ” ~ and .x113 near the wall and a uniform mesh in terms of J’ and x1 elsewhere. When the differential equation is written as a set of first-order equations the transition between the two meshes becomes straightforward.

Three cases were extensively studied by Williams in Stewartson (1970b) and his main results are set out in Table 7.

TABLE 7

PKINC‘IPAI. PROPFKI‘II;~ OF T H E

LOWER [)tC‘h: A 1 A C O N V E X COKNtK

2 0.16 1.69 26.83 - p ( O ) 0.12 1.37 25.91 ,\, 6.3 1.28 0.2x

I t is clear that as u -+ a, -p (O) /x -+ I , so that virtually the whole of the pressure fall takes place ahead of the corner. Using this property as a guide a consistent description of the flow field when x $ 1 can be made. For, when u 3 I the velocity at Y = 0 takes the form

from (6.1), p(0) = -2/h2, and 6(@ 1) is a number to be found but which we anticipate to be % (2/x)’I2. We now assume that the pressure variation in x > 0 is only O( 1) and may be neglected, so that the velocity distribution changes from (11.12) into the Blasius form

11 = (2/6)Fb[(y + “S)/h] (11.13)

within a distance O(6) of the origin, where FB satisfies Blasius’ equation, provided y + CIX = O(6). The change in profile is necessary because F g ( 0 ) # 0 due to the pressure gradient in x < 0, while F;;’(O) = 0. Asso- ciated with this change in profile, a normal velocity is induced which has the form

u , = -SFK(x/6) (1 1.14)

when (y + as ) /h is large; K is a function which has not yet been fully computed but we know that t113K( t ) is bounded as t -, 0 and t 1 / 2 K ( t ) is finite as t + m. In discussing the evolution of the velocity profile we have neglected the linear shear y in ( 1 1.12). This is not justified when J‘ = O(6) but the forms taken by u and v are then very simple, namely

I / = (2/6) + ( y + US) + SB(x/S), L’ = - (J + K.Y)B’(S/~) - 6 - K(s/iS) - NU,

u = (2/S)F0(y/S) + J’ (11.12)

( 1 1.15)

198 K . Stewartson

where B(r) is a function of r to be found. On substitution into (6.1) it immediately follows that

I

B’(t) = e 2 t 1 K(rI)e-2‘ l L/t, (11.16)

while p = --a - B’(x/d). Thus the fall in pressure, while the velocity profile converts itself into the Blasius form, is 0(1) and so 6 = (2/-a)’” as anticipated. The actual fall is given by B’(O), which has been estimated to be 0.67, in moderate agreement with the results in Table 7.

After the Blasius form has been reached, the velocity profile slowly changes back into the linear form CI = y + c(x, over a distance O(is-3), and the corresponding pressure changes are O(a2) and negligible.

There is a very simple interpretation which may be given to this solution when CI 9 1, namely, that the lower deck itself takes on a double structure of which the outer part is a simple shear y + A ( x ) , with a large velocity of slip at the wall [= A ( x ) when x < 0 and = A ( x ) - M X when x > 01. The inner part is a conventional boundary layer whose purpose is to reduce this velocity to zero. I t is driven by the pressure gradient p ‘ ( . ~ ) , which is related to the slip velocity by means of Bernoulli’s equation. Hence

’ 1

p ( x ) = -+A2(.\-) if x < 0 and

p(x) = - + ( A - -ax)’ if x > 0.

On adding the usual condition p = -A’(.x), we have

~ if x < O 2

P = - [(2/a)”2 - 4 2

and

p = -a if x > 0,

(11.17)

( 1 1.18)

as the only acceptable form for p. With this asymptotic form for p when x S I , it is possible to extend the

theory of convex corners beyond the triple-deck limitation that X* = O(c2), and consider angles which while small do not tend to zero as t; + 0, i.e., i ; ‘ < -a* < 1 for any t > 0. A partial description of the flow field in these circumstances was first proposed by Matveeva and Neiland (1967), and it may now be completed, using the results of the triple-deck theory. Introduce the scaling laws

Y * = x,* + x*- 1’z~;4~,*X, y* = x,*c* yM ( I 1.19)

199

and write

u*iU’“, = u , ( Y ~ ) + cx*”’A(X)Ub(YM) + O(cc*),

11*iu; =

(P* - P*, )/d Ut’ =

- x*A’(X)U,( Y M ) + . . ’ , ( 1 1.20)

.*p*(X) + ‘... This expansion is analogous to the main deck in Section III,A and is actually identical to it if a* = c2. The analog of the upper deck (Section III,B) leads to the formula

- p2 = - ( M : - l)-1’2A‘(x) (1 1.21)

connecting the unknown functions p 2 and A. At the bottom of this new main deck

u*/U’“, = (YM + x*’”A)Ub(O) + ”. (1 1.22) and, as anticipated, the lower deck is so thin ( - ~ : ~ a * - ’ ’ ~ x z ) that when the viscous forces come into play there, YM may be neglected in ( I 1.22). Con- sequently the pressure may be found from Bernoulli’s equation which takes the form

(MZ, - 1)-”2/1*, A’ = f [ & P [ U b ( 0 ) ] 2 , (1 1.23)

since p 2 + 0 as X + - x,, from our triple-deck study. Furthermore, from the known properties of p* when CI 4 1, we may assume that F 2 ( O ) =

- ( M t - 1)- ’ l2 . Hence by integration of ( 1 1.23), 1

if < 0. For large negative X this solution matches smoothly to the expansive free-interaction solution of Section VII as does the lower deck. Down- stream of the corner, the lower deck converts into a Blasius form within a distance O(a*”2e4~z ) , accompanied by a further pressure fall O(r:2pz ) and thereafter slowly reverts back to a uniform shear.

Experiments on convex corners are few. A discussion of the findings is given in Stewartson (1970b), who notes that the values of x* chosen in the experiments are too large for a proper comparison with the present theory. However Hama’s (1968) measurements of the pressure fall upstream of a right-angled corner have been analyzed by Olsson and Messiter (1969) who showed that they can be collapsed onto a single curve. Further, ifwe take the pressure ratio at the corner to correspond to a subsonic Mach number there, this curve is in good agreement with the predictions of the Matveeva- Neiland theory.

200 K . Strwurtsorz

The theory breaks down when x* = O( l), the prime cause of failure being that Z/i?.x* and t?/i?j>* become of the same order of magnitude sufficiently near the corner in the main deck. It is interesting to speculate on the likely flow structure when the convex corner has a finite angle. The pressure first begins to fall upstream of the corner in an expansive free interaction, when x,* - x* = O(e3x,*). On this scale the terminal point of the free interaction is the corner and, when c4x: < x,* - x* 4 e3x,*, the pressure is jointly controlled by the Ackeret formula in the mainstream and the effective slip velocity [’c A(x) ] at the wall, the lower deck playing a passive role. Once .x,* - x* = O(i:4x,*), the main deck begins to exhibit nonlinear behavior, the governing equations become elliptic, and a significant pressure gradient develops across it. Inviscid forces are now dominant and the slip velocity continues to rise. There are now two main possibilities. I f x* is not too large the total pressure fall around the corner is not sufficient to accelerate the fluid at the bottom of the main deck from rest to the sonic value, in which case one might expect that nearly all the pressure fall occurs in x* < x,*, and that the pressure fall downstream of the corner is O(i:’p?), just as when x* < 1. On the other hand if c(* is too large to satisfy this condition, it is conjectured that the slip velocity reaches, but cannot exceed, the sonic value at the corner, and instead a significant pressure fall occurs downstream, through a PrandtlLMeyer expansion fan, centered at the bottom of the main deck. Some studies of the structure of the main deck when the velocity ofslip reaches the sonic value have been reported by Adamson and Fishburn (1971) but their results have not yet been fitted into the present framework. Clearly the correctness of the description of the flow field when CY* = O( 1) is open to question, particularly the claim that supersonic velocities of slip cannot occur upstream of the corner, and it is hoped that more theoretical work will be done in the future.

XII. Injection into the Supersonic Boundary Layer

Of the features which can provoke a compressive free interaction the simplest, from a theoretical point of view, is normal injection, and it is also the only one which has been studied deeply enough to enable us to claim that a complete asymptotic expansion can, in principle, be set up. The problems of reattachment, flow near corners, and shock impingement are as yet imperfectly understood and impede progress towards a full understanding of the other features. I t is hoped that the insight gained from a study of normal injection will be of assistance to the study of cranked corners and shock- wave interactions for. although there is some experimental work available

Multistructured Boundury Laycrs 20 1

on fluid injection and the technique is of practical interest as a means of effecting a reduction in heat transfer, these other two cases are major goals of theoretical investigations of supersonic boundary layers.

We shall suppose that the injection begins at x* = rX and terminates at x* = ST, and that the normal velocity is V ; so that the appropriate boundary conditions on the plate y* = 0 are

u* = 0, u* = 0 when 0 < x* < x8 and when .Y: < .Y*,

while I/* = 0, c* v* ~ when xg < ,Y* < x?. (12.1)

The assumption that the tangential velocity is zero during the blowing is conventional, but it does not seem to have been established how correctly the condition reflects the practical situation.

For convenience we shall define weak b lowiq by the condition V ; = O(c4U;) and strony blowing by the condition V ; = O(i:’UU*,), intermediate rates of blowing being referred to as motlerate hlowiny. When V ; = O( U ; ) the blowing is said to be nzussiw; we shall hardly be concerned with this situation here, but observe that the transition from strong to massive blowing is not believed to involve any other regime and to be smooth. We shall define slot injecrion to occur when the length of the injection region xT - = O(r:-3.uT),and pliite injrctiorz to occur when the length is O(xy). There are some interface problems between these two kinds of injection, that have not yet been fully resolved.

Most early work on iiijection was concerned with weak plate-blowing, taking xX = 0. One characteristic of this blowing is that the interaction between the boundary layer and the mainstream may be neglected so long as separation does not occur; the theory then is a straightforward extension of the noninjection theory. A summary of the progress made is given by Gadd rt a/. (1963). The most seminal contribution was that of Pretsch (1944) who studied the similarity equations of the boundary layer, establishing that under moderate blowing it is blown off the wall and bcconies a vorticity layer separating two regions of inviscid flow; the plateau in Section IX is reminiscent of the structure he found.

In an adverse pressure gradient, injection hastens separation but in a favorable pressure gradient it merely thickens the boundary layer. Our interest in this review is with 6asically uniform mainstreams and the role of injection is then more subtle. Emmons and Leigh established (1953) that solutions to the Blasius equation are only possible if the injection is not too strong, while Catherall rt a/. (1965) found that in plate injection under a uniform stream with .YO* = 0, separation occurs at a value

.u,* = 0.7456(c4U*, /V;)’x,* (12.2)

202 K . Stewartsorz

of .Y*. Near separation the reduced skin friction

[(.Y,* - x*)/.x\*]' (log[(.x,* - x*)/x\*])2 ' T Z (12.3)

while the displacement thickness has a logarithmic singularity at .Y,*.

Presumably if xg > 0, the length of blow required to induce separation is less than that given by (12.2) for the same injection velocity. Little is known about the behavior of the flow after the classical boundary layer breaks down, and the interaction with the mainstream has to be considered. Kassoy (1970, 1971) and Klemp and Acrivos (1972) have considered aspects of the post-separation region where the boundary layer is apparently blown off the plate, and it is probable that their work can be included as part of a comprehensive theory of injection. The studies are for incompressible fluids with xT = a, so that the length of blowing is infinite, and similarity solutions, valid presumably in the limit x * + a, are mainly considered.

The range of V: permitted for weak blowing is limited. If V: < O(c4U;t: ) then the effect of the blowing is negligible, while if V: = O(c'UU*,) it leads to distortions of the streamlines in the mainstream greater than O( V:,/U: ). We can already see from (12.2) that, in strong blowing, separation occurs immediately. The main effort to understand strong blowing has been concentrated on supersonic flow, and the first major step forward was taken by Cole and Aroesty (1968), who set up an inviscid model for the region of blown fluid, determining the shape of the boundary by a formula similar to that given by Gadd rt al. (1963). The governing equations for this region are, in fact, the same as those for region 111 of the plateau (9.2), while the pressure gradient is determined by an appeal to linearized inviscid theory, as in Section III,B. Although their theory is incomplete in certain respects, it is, in fact, not difficult in principle to make the necessary additions using the notion of the triple deck. Smith and Stewartson (1973a,b) have examined the flow structure both for slot and plate injection and have shown how formal expansion procedures can be set up provided x g > 0; a review of the existing literature is also given to supplement the account presented here.

Cole and Aroesty (1968) recognized the crucial feature of injection, which is sufficiently strong to make a significant impact on the inviscid flow field above the boundary layer, namely, that the induced pressure gradient must, broadly speaking, be favorable in the blowing region to drive the blown fluid downstream. But this means there must be regions upstream and/or downstream of the blow where the pressure adjusts to its undisturbed value, or at least to its preassigned value. If blowing starts at the leading edge any adjustment upstream must take place in the leading-edge region but in a way still not understood; on the other hand, if .xX > 0, the antecedent pressure rise can only occur through a compressive free interaction. It is noted that

Mult istructurrd Bouizrlar!~ Lajier-s 203

the normal velocity in the lower deck of the appropriate triple deck is then O(c3U; ) and of the same order as the strong-injection velocity prescribed in the blown region. The pressure rise here is O(c2pE) , which then falls in the blown region as the injected fluid is driven downstream. Once blowing stops, the pressure must adjust once more to some final value which depends on the amount of fluid injected. This is at a rate O(p* V$( YT - xg)) and so one must expect a final pressure rise O( L‘: p: (.Y: - Y ~ ) / U * , D*), where D* is the width of the channel through which the supersonic stream is moving and where the plate is placed. Consequently, tli-e final pressure rise is at most O(c3p$) for strong injection and, since the triple deck implies a pressure variation O(i:’pz ),

where p is defined in (4.4). For slot injection this condition means that the integration of the governing equations must be continued beyond the termination of blowing, but for plate injection this condition (12.4) may justifiably be applied at x* = x:. The reason is that at the termination of blowing, the boundary layer is very thick and the detached shear layer is too far away from the wall for the change in boundary condition to have any significant effect on it, except on a long scale which hardly changes the pressure. Some authors, e.g., Lees and Fernandez (1970), have chosen different terminal conditions for reasons which seem to have as much to do with expediency as with the physics of the flow, while others, e.g., Taylor et ul. (1969), terminated the blow at a convex corner for which the above argument is inapplicable.

Let us consider the flow properties in strong slot and plate injection in some detail, restricting attention to uniform blowing velocities. Variable blowing rates can be treated by parallel arguments if necessary. It is not clear whether effectively uniform blowing rates can be achieved experimentally, but nevertheless the understanding of uniform blowing is essential as a step toward more realistic situations.

A. SLOT INJECTION

Here xT - YX = O(t3xT) and so the triple deck is relevant to the whole of the injection region. We choose x$ = .up and write

V ; = i.3/,3 4C”’8(M2, - l)”x(T;/TY,)1’2U$ V,, , ,3~-s/4c”x(M2, - 1 ) - 3 I y T / ) 3 2 * (12.5) .; - xp = $ 7’: x , X o ,

with a similar definition for X I . It is also assumed that the wall temperature

204 K . Stewartson

remains constant over the entire plate. The modifications necessary to the concept of the triple deck to deal with a discontinuity in wall temperature have not been worked out but there seems no difficulty, in principle, in developing the appropriate theory. Smooth variations in wall temperature present no problems in concept but lead to greater computational complex- ity. The flow field in the neighborhood of the injection region depends on the solution of the lower-deck equations of which the leading term is (6.1), subject to the boundary conditions (6.2), together with

u = 0, u = 0 if y = 0 when either x < X , or .Y > X , .

and ( 12.6)

u = 0 , P = V, if y = O when X , < x < X I .

The solution in .Y < X , is a compressive free interaction which, unless modified by the injection in x > X , , is assumed to separate at x = 0. Thus in any computational study the value of X , - X , is given but we have to find X , . I f X , < 0 separation does not occur before the start of blowing but if X , > 0 the blowing takes place in a separated boundary layer.

The solution for V , < 1 is easily obtained in a similar way to the corresponding results for convex corners (Section XI). We find (Smith and Stew- artson, 1973a) that the pressure rises in x < X , , while the. skin friction falls and

(12.7)

Ai being the Airy function and ii being defined in (6.8). The pressure then falls in X o < .Y < X , . rising again in .Y > X , to satisfy (12.4), and, as X I + x, p ( X , ) + - p ( X , ) . The skin friction reaches a minimum in X , < I < X , and steadily increases thereafter, its limiting value as .Y + rx: being unity. For V, 4 1 and X , --f cc we should expect an interface with the classical solution for weak blowing, and indeed the two solutions do formally match as x + a. and as .Y* + x$ +, respectively. Thus, as X I + x and x + cr, in that order, z - 1 = O ( X ' / ~ ) , which is just the behavior of z that occurs at the start of weak plate injection. Also p = O(x- 1:3), which means that beyond the triple deck the pressure rise is O(c4pt ) as is to be expected from the classical theory.

There is nevertheless a difficulty with classical theory that has not yet been resolved. The interaction between the boundary layer and the mainstream in weak injection may be neglected so long as separation does not occur, and the pressure rise is O(i:"p*,), which is of an acceptable order according to the arguments used in deriving (12.4). This happy state of affairs comes to an

Miiltistriictured Boirriilary Layc~rs 205

end when separation occurs, for the interaction is then very marked, leading to an abrupt pressure rise which has not yet been computed, but is likely to be O(c3pz ) . If the blow is now terminated shortly afterward, there will not have been a sufficient length of blowing into a separated layer to reduce the pressure rise to O(c4p*,), as needed far downstream, and the main boundary layer may be blown too far off the plate for the ending of the blow to have much effect on the pressure; in that event there cannot be a recovery to the ultimate value required from considerations of continuity. From these arguments, Smith and Stewartson (19734 were led to raise the possibility that as V: is increased in weak plate injection a transition occurs at the onset of separation. Suppose V: is the critical blowing velocity at which separation first occurs (when x* = x:) according to classical theory, then, when V: = V : + 0 it may happen that the separation point actually moves to the neighborhood of x* = x$ (the onset of blowing) after a compressive free interaction. If so this would support a view sometimes put forward that all supersonic separations occur after a free interaction, but it is emphasized that these remarks are merely suggestive and the wholc question is still open.

Returning to slot injection, at values of V, - 1, recourse must be made to numerical methods, and Smith and Stewartson (19734 have computed a number of cases. A difficulty, similar to that mentioned in Section XI at the convex corner, occurs at the onset of blowing and use was made of a two-tier mesh scheme devised by Smith (1973a). In all cases fully computed the pressure gradient is favorable throughout, the blow being adverse both upstream and downstream. Furthermore the computational procedure required that ~ ( x , ) > 0 so that no solutions were computed ofslot injection into an already separated boundary layer. In Fig. 8 we display the variation of p and z with .x for three blowing lengths and values of V' such that p ( X , ) = 0.5301 and z ( X , ) = 0.4107. In other, incomplete, calculations the adverse pressure gradient extended a short distance into the blowing region downstream of X , , and sometimes separation occurred with eventual reattachment. Smith and Stewartson (19734 were led to conjecture: ( i ) for any V , > 0 separation must occur if X , - X , is large enough, ( i i ) if p ( X , ) is fixed, the corresponding value of V , + 0 as X I + m, as do ( i i i ) z ( X , ) , and (iv) p ( X , ) . This last conjecture is not obvious from the numerical studies but was inferred as a consequence of (i i i) ; i t is also in line with (12.4), the terminal condition for plate injection. There is a clear need to extend the computations to include blowing situations in which z ( X , ) < 0 so that separation has occurred before blowing starts, but the appropriate numerical procedures have not yet been developed. As it is, the interface between weak plate injection, which leads to separation, and strong slot injection is not understood properly. The interface between strong slot and plate injection seems to be clearer because we can expect that strong plate injection must

206 K . Stowirtsori

X

I I I I I 0 10 20

c

Flc;. X. Variation of rcduccd prcssui-c and skin friction with Y for three blowing lengths and blowing ratcs adjusted so hat p(0) = 0.530: b ' w = 0.534 if X I = 5 . 1 ,, = 0.408 if X I = 10: V,l. = 0.448 if X I = 15.

occur in a fully separated boundary layer and that the flow structure upstream of the injection is of the kind described in Section IX. Of course, more studies of the slot-injection problem are needed when V, > 0 and X , B 1, but there is good reason to believe that such studies will not disturb the structure proposed. We now turn to consider the present state of the theory of strong plate injection.

B. PLATE INJECTION

Here we assume that the boundary layer has separated at Y* = Y:,

upstream of the onset of blowing, and that both sT - XX and YX - Y,* are O(xX). Furthermore, we shall suppose that the flow structure at x* = YO* - IS

M~iltistr~ictured Boundary Layers 207

the same as that described in Section IX, with suitable forms for U , (J.*) and 6, [see (9.3)]. The values of these quantities depend uniquely on I,': and xT - x:, and we shall indicate below how they may be determined. If they are both O( l), then with injection rates O(c 'U$) , the streamwise velocity component in region V of Fig. 5, where x* > x; and which is the continuation of region I11 into the blowing zone, is O ( c U z ) , while in region I11 it is 0(c2U:,). Again, the width of region V is O(c2x;) while the continuation of region I1 above it has width O(c4x;), so that, as in Section IX, region I1 may be replaced by a line of zero thickness, for the purpose of computing the flow in region V. Moreover, the entrainment velocity c4UXuR into region 11, which helps control the flow in region 111, may be neglected in region V as a first approximation. Hence, restricting attention to the leading terms only, the appropriate boundary conditions for region V, above the blown fluid, are

11* = 0 at x* = x; , 0 < y* < c2.u;so,

I)* = 0 at 4'* = r:2x;(s(x*), .Yd < s* < .uy,

I / * = 0, u* = v* at y* = 0, .Y; < x* < xT, (1 2.8)

where ~:~.x;(S(x*) is the thickness of region V, $* is the stream function, and S(xg*) = 6,. The injection velocity V: removes the necessity for a subboundary layer, analogous to region IV, below region V, which extends right down to the plate. Furthermore, the flow in this region may be regarded as incompressible and inviscid (Cole and Aroesty, 1968) but with strong injection and, since 0 < c2(' < 1, the governing equations are identical with (9.2). The pressure p* remains to be found but we know that

p* = p*, + p: U;*,2/2''2C'/4(M2 - 1 ) p ' / 4 c 2 ~ o

when x* = .Y: (12.9a)

and

p* = p*,, + p t U*,'c2(M:, - 1 ) p " 4 ~ , * d6/dx*

when x* > x; (12.9b)

from the linearized theory of supersonic inviscid motion. The final boundary condition is

I?*(xT) = P*, ; ( 12.10) by means of this condition we can fix the value of do, and hence of x,* through the formula

e 2 X ; f i , = E 2 1 2 1 / 2 c 1 / 4 ( ~ : - 1 ) - 1 / 4 ~ ~ ( ~ ; - .;*I. (12.11)

208 K . Stewartson

The required solution follows on noting that (9.2) has the formal integral

+p;11*2 = -P*(-Y*) +f($*)? (12.12)

where $* is the stream function and,f a function to be found; the integral is a form of Bernoulli's equation. The value of,f'is determined from the no-slip condition at the plate, and we finally obtain

* * t'2x;d x* - I/* 1 * I i 2 ( ) - & P W ) 1 M P * ( t ) - P*(x*)11'23 (12.13)

' x o *

a formula due, in principle, to Cole and Aroesty (1968). This integrodifferen- tial equation may be reduced to nondimensional form if we write

p* = p ; + p*, U;2/2w"4(M2, - l)-l'4P0F(z).

(S(.Y*) = ~ , A ( z ) , X * - x,* = x,*P,'S,(M: - l ) - " 4 i - I ' 2 C - 1 4 7 - >

(12.14) and set

In terms of these new variables, the equation for p ( z ) reduces to

where A(0) = 1, A'(0) = 1. (12.17)

Once the problem has been reduced to the solution of (12.16), the consistency of the structural form adopted becomes clear, as does the significance of the definition of strong blowing given at the beginning of this section. For, when V: = O(r:'U; ), (I) becomes independent oft' as does A. Consequently the assumptions that the width of region V is O(c2x;) and that the corresponding pressure change is O(e2p: ) are seen to be consistent. The appropriate streamwise component of velocity is also correctly assumed to be O ( E U ; ). The only remaining unknown quantity is x;* and if, from the solution of (12.16), we define the value of z at which p = 0 to be zl, so that

z1 is determined by

x; - x;* = (xT - x,*)/zl,

from (12.11) and (12.14).

(1 2.18)

(12.19)

Multistructured Boundmy Layers 209

A discussion of the properties of I, is given in Smith and Stewartson (1973b), from which the following table of representative values of z I has been taken.

(1) 0 0.2 0.4 0.6 0.8 I .o :, x. 49,000 45.x 8.87 4.088 2.532

(U 2 3 4 5 z , 0.813 0.416 0.336 0.259

so that under very strong blowing, V, + 1,

x; - xg = (.YT - x;)[vw,/(8P;)’!2]. (12.21)

Thus increasing the rate of blowing over a fixed length of the plate moves the separation point toward the leading edge, and it is clear that the theory must fail when

for then separation occurs at the leading edge. A further increase in V$ cannot alter xg - x,* much in physical terms, but as XP + 0, i; increases, and this is presumably how the steady flow would cope with the situation, for, an increase in i; permits a rise in the plateau pressure and allows a further increase in the width of region V. The present theory is now inapplicable, and one must also expect the downstream pressure condition (12.4) to fail at some stage.

Nevertheless it seems that a direct link can bc found with nza.s.sire bIowing problems in which V,* = O(U*,). We can expect that separation then occurs at the leading edge of the plate and a largely stagnant wedge of fluid is formed upstream of the blown region. The configuration might well be similar to that in the well-known aerodynamic spike (Birkhoff, 1960, frontispiece), in which case the blown fluid behaves toward the oncoming fluid like a finite body and the presence of part of the plate in front ensures that the oncoming fluid separates at the leading edge. Furthermore, since the

210 K . SteMartsoii

increase in pressure on the plateau is O ( F ’ ~ ; ) and i: increases without limit as s,* + O it is possible, in principle, for the increase to be as much as O(&) in massive blowing, which is in accord with the expectation that the resulting flow pattern is largely inviscid in character.

The experimental evidence about massive blowing is limited (Hartunian and Spencer, 1967; Bolt, 1968) and only available at values of R % 100. Some, but not all of the photographs in these papers support the view that separation occurs upstream of the onset of blowing. There is also a large body of theoretical work, chiefly semiempirical, on massive blowing which the interested reader can consult (e.g., Inger and Gaitatzes, 1971).

In the other limit, LU + 0, the behavior of @ is more complicated but can be put into the form

1 - p3 = 3dr’(3zp3’2) (12.23)

except near p = 0, wheref(t) z nt2 when t is small andJ‘(t) z log t when t is large. When F = O ( [ I ) ~ ’ ~ ) the behavior of I, may be expressed in the form

(zl - Z)/ZI z y(F/w2’3), ( 12.24)

where y(0) = 0 and y(t) z 1 - $ 7 t - ” 2 t ~ 3 j 2 e - ‘ ” ~ 3 as t + xm. (12.25)

Thus the pressure p falls from unity very slowly when w 4 I, taking an exponentially long distance to reach a small value. Nevertheless the final decay to zero, when j = 0((u2j3), is even slower and comprises the major part of the blowing region. One consequence of (12.25) is that as (I) + 0 the separation point moves up toward xg , in line with one’s expectation from the discussion on slot blowing. Indeed (12.21) can be rewritten in terms of triple-deck coordinates as

X O = (XI - xo)/zl, (12.26)

and it is conjectured that this is an asymptotic formula for slot iiijection, with the interpretation that if the length of blowing and the injection velocity are both given, (12.26) defines xo and hence the distance .yo( $ 1 ) downstream of separation at which blowing commences. For low rates of blowing x1 - xo must be large which gives some support to the conjecture made earlier that for any rate V, of blowing, separation must occur ahead of the blow if it takes place over a sufficiently long range of values of .Y.

In order to complete the formal theory, by determining a uniformly valid leading term in the asymptotic expansion of the solution in x* > 0, the prime remaining needs are to elucidate the nature of the transition regions near x* = so* and x* = xT. We report on their properties in detail elsewhere (Stewartson, 1974) but the main results are summarized here.

Multistructtired Boiindury L u j ~ ~ s 21 1

For the most significant part of the first transition region, near x* = .YO*, the appropriate scaling is E X ; in the x* direction and r;26,.xg* in the y* direction. We write

x* = x; + cx;,;c, y* = e26 0 x*- O Y ,

LI* = E 2 u; il, L'* = {;"; u, (12.27)

and when 2 = O( 1 ) and 0 < j < 1 the governing equations for the leading terms li and fi are still given by (9.2). The relevant boundary conditions are that

i . = o when ,;C < 0, i, = 0,

û = V,*/c3U; when 2 > 0, j* = 0, (12.28)

i j = o when = 1.

Furthermore, as .? + x',

ij + (v,*/c3u:)cos +xi',

and, as i + - m,

i? + 0, ( 12.29)

to leading order. The corresponding pressure changes are O(i;4p: ). and the match, with the plate-injection solution given above, when z 4 1 is possible because

ii z +xi( V,*/e3 U ; )sin +xi (12.30)

as ,? -+ oc, so that the associated value of u* = O(cU*,,), in line with the assumed behavior in x; < x* < sT. Part of the injected fluid is turned upstream and the length of blowing required for this purpose is Oexg*, where

O.Y;V,* = C ~ U * u&*) ds*, (12.31)

from (9.4). The form for U,(y*), defined in (9.3), then follows from an application of Bernoulli's equation analogous to (12.12). There is one important difference between the streamlines leaving the wall and moving upstream and those that move downstream. On the downstream side, even according to inviscid theory, the streamlines nearest the wall at some station .Y* > x;( 1 + O E ) have just emerged from it and hence the favorable pressure gradient ( d p * / d s * < 0) here has not had time to accelerate the fluid moving along them. Thus we can expect the no-slip condition to hold at y* = 0 (i.e., I!* = 0). On the upstream side, the streamlines nearest the wall when

. \,)*- \.*

!o

212 K . Stewartson

x* < xg emerged from the wall some distance away [in the interval xg < x* < x*( 1 + U E ) ] and, having been accelerated by a favorable pressure gradient [p* has a maximum at x* = xg(1 + cO)] the fluid moving along them now has a nonzero slip velocity. Consequently a subboundary layer must be formed below this region and is the genesis of region IV in Fig. 5.

The appropriate scaling for the most significant part of the second transition region, near x* = xT appears to be

x* = xT + cZ.uT.u,

u* = t;u: U I ( j ) + c”$1(,

I’* = E%*(XT)x;J:

v* = E3u: a, (12.32)

where cU:, iil is the x* component of velocity in region V as x* + xT -. The governing equations in this region are the small perturbation equations of incompressible inviscid motion, since the scales in the x* and y* directions are the same. The pressure variation across the region is O ( I : ~ ~ Y : ) and negligible, confirming the correctness of (12.1 1). An inviscid subboundary layer is needed below this region because of the slip velocities induced when 4; = 0, but it dies away as 2 -+ - m and a further subboundary layer is needed when .? > 0. Downstream of this second transition region the flow ultimately returns to the Blasius form if the plate is infinitely long, or to uniform flow if it is finite, but the pressure changes are then negligible. For further details the reader is referred to Stewartson (1974).

I t is of interest to contrast the properties of boundary layers with injection which we have just discussed to those with suction. Although the effect of suction has not been studied so extensively in the context of the triple deck, its main features are fairly clear. Ahead of the suction region, which we can suppose also extends from xg to xT, there is an expansive free interaction but it only extends a distance O(c3x;) upstream of xg. Furthermore the transi- tional suction velocities are now O(c4U:) in the sense that if 0 < - V,* < O(c4U2) the boundary layer is unaffected by the suction, whereas if - V: > O(c4U:) the change in c* across the boundary layer is negligible in comparison with - V;. Once this occurs the boundary layer is irrelevant to the main flow characteristics which are dominated by inviscid effects. The boundary-layer profile takes on the form of the asymptotic suction profile (Young, 1948) and plays a passive role. The chief outstanding boundary- layer problem concerns the metamorphosis of the expansive free interaction into the asymptotic profile in the neighborhood of x* = x g . Presumably the solution will follow lines parallel to those of Section XI.

If fluid is injected it is true that interesting flow properties can develop when V; = O ( E ~ U : ) , but the flow properties when the blow is strony and V$ = O(e3 U: ) are much more spectacular, involving an extensive region of separation and probably having an interface (in the mathematical sense) with t m s s i u r blowing. If fluid is sucked into the wall, it is the t77orlrrute rates i.e., (c4 U $ < V ; < c3 U*, ) that have the interface with massive suction.

213

XIII. The Trailing Edge of a Symmetrically Disposed Flat Plate

Although the theory of multistructured boundary layers can, in principle, be applied equally well to problems in supersonic and in subsonic flow, nearly all the explicit solutions found so far relate to supersonic flow. The reasons are partly that the spectacular nature of the free-interaction boundary layers, which have no counterpart in subsonic flow, have generated more interest among theoreticians and partly that the numerical work is simpler than in subsonic flow where the Hilbert integral (4.9) presents considerable difficulties. Thus little work has been done on iiijection into the subsonic boundary layer, from this standpoint, and on flow round corners, apart from a linearized study analogous to that at the beginning of Section XI (Stewartson, 1970c, 1971).

Most attention, for subsonic flow, has been concentrated on the neighborhood of the trailing edge, and although no complete numerical solutions are available, the structural properties have been extensively explored, and a sufficient comparison with accurate numerical solutions at finite Reynolds number is possible for confidence in the principal ideas to be established. The general problem of the trailing edge is of great importance in aerodyna- mics and, to give just one example, aeronautical engineers have been making strenuous efforts to improve flow characteristics there for many years. It is, however, only comparatively recently that a quantitative formal understanding of the nature of the viscous flow has been achieved and the notion of the triple deck plays a vital role in its elucidation. A measure of the importance of the trailing-edge region is given by the asymptotic expansion of the drag D on one side of a finite flat plate of length L* placed symmetrically in a uniform stream. Prandtl's boundary-layer theory gives the leading term in the form

CD = D/$p: UZ2L* = 1.328C':'/R'l2, (13.1)

with R now equal to U*L*/iie in the remainder of this review. The numerical factor 1.328 was given by Goldstein (1930). who also determined the principal flow properties of the continuation of the boundary layer into the wake beyond the plate. In order to improve this estimate for the drag it is necessary to take into account the interaction of the boundary layer with the inviscid flow outside, through the displacement thickness: for many years it was believed that this should be done by computing the over-all properties of the disturbed inviscid flow and using the induced pressure gradient on the flat plate to determine the perturbation in the boundary layer. A summary of these studies, which lead to a contribution O ( R - ' ) to D, is given by Van Dyke (1964). However this line of argument neglects the most important

214 K . Stewartso12

contribution, namely, that from the neighborhood of the trailing edge, which is O ( R p 7 ' 8 ) . The first step in the elucidation of this flow was made by Goldburg and Cheng (1961), who established that the flow properties described by Goldstein implied that the pressure becomes singular at the trailing edge. Later, Hakkinen and O'Neil (1969) examined the effect of a trailing edge on a uniform shear, using the full Navier-Stokes equations and found flow properties which could not be matched to Goldstein's solution. This showed the necessity for an intermediate region, and its properties were then examined independently and simultaneously by Messiter (1970) and Stewartson (1969), using the arguments of the triple deck.

In broad physical terms their results can be understood by noting that once the restraining influence of the plate is removed, the fluid, in the stream tubes which originally were near the plate, is now rapidly speeded up by the viscous action of the fluid in neighboring stream tubes. As a result these stream tubes tend to become more crowded together as do those in the inviscid Aow just outside the boundary layer. I t is then reasonable to expect that this property extends a little way upstream of the trailing edge from considerations of smoothness, so that there is an anticipatory fall of pressure just upstream of the trailing edge. In supersonic conditions this is effected by an expansive free interaction and for both types of flow the result is an enhancement of the drag. At the trailing edge the pressure reaches a minimum and must rise thereafter. The crowding of the streamlines continues, of course, so long as the pressure is less than its ambient value, the adverse gradient being needed to keep the streamlines smooth. On leaving the triple deck on the downstream side a pressure overshoot occurs and the pressure gradient becomes favorable before it finally returns to its ambient value.

We now describe in some detail the flow structure near the trailing edge, concentrating on the incompressible problem ( M x = 0). The subsonic problem is an easy generalization, and we shall comment on any special features of the supersonic problem as they arise. Goldstein's near-wake flow is found by solving the usual boundary-layer equations in .Y* > L* with a uniform mainstream velocity U;E , together with the boundary conditions

2;*= i, 7 u * / i 3 , * = 0 J at J'* = 0, ( 1 3.2) and an initial velocity profile ti* = UW(y*) = U z Uo(YM) , being the Blasius profile described in Section TI. The conditions at I'* = 0 reflect the smoothness and symmetry of the flow in the wake. Goldstein found that near x* = L* the solution is double-structured. When j,* = O((x* - L*)''3L*2 31:4) the .Y* component of the velocity u*, can be expanded in a power series in s* of the form

(13.3)

Multistructurerl Boundary Layers 215

where x = + ~ ~ ~ * 1 1 3 [ 2 ( ~ * - ~*)1-1/3, p = c 4 ~ * y M ,

and the prime denotes derivatives with respect to X. The function kO satisfies the differential equation

p; + 2kOk;; - p; = 0, (1 3.4) similar to (8.8), and the boundary conditions

kO(0) = &(O) = 0, F ~ ( w ) = 182, (13.5) while the remaining k,, satisfy linear equations, boundary conditions at j j = 0 identical with (13.5) and a condition as j j + 03 to er?able (13.3) to be matched with the initial Blasius profile at x* = L*. On the other hand, when YM = O( I), u* must be expanded in the form

(13.6)

where again U*, Uo(YM) is the velocity profile at x* = L* (see 3.2). The remaining U,, can all be determined explicitly by insisting that (13.3) and (13.6) have a common region of validity when 2 % 1 and Y, 4 1. Goldstein actually worked out the first eight terms of (13.6), but for our purposes it is sufficient to point out that

UI(YM) = 2.045 the numerical factor being given by

( 1 3.7)

(13.8)

The necessity for an intermediate region separating the Blasius solution when x* < L* and the Goldstein wake when x* > L* can now be made clear by consideration of the behavior of u*. For, as x* + L* -, o * / ~ : ~ U * , remains finite but as x* -+ L* +,

V*/&"U*, = - 2.045 UO( YM)L*213/[2(X* - L * ) y (13.9) and has a strong singularity. The streamlines in the boundary layer and just outside it are deflected sharply toward J'* = 0, in consequence of the speed- ing up of the fluid there. Also when y* = 0,

u*/U*, = 1.611/22/3[(x* - L*)/L*]'I3 ( 1 3.10) as x* + L*+.

If now we apply the standard arguments of the triple deck we see that Goldstein's inner region (2 - 1 ) corresponds to the lower deck and his outer region ( YM - 1) to the main deck. The upper deck is additional but rendered necessary by the singularity in r* in the double limit Y,, -+ ~ , ' x * -+ L*+.

2 16 K . Stewartson

Furthermore, the properties of the main and upper decks follow immediately from those of the lower deck, the governing equations for which reduce to (6.1) after the transformations (4.4) are made. The appropriate boundary conditions are that

u - y - + O as x - + - m , u - y - + A ( x ) as y + m, (13.11) u = v = 0 at y = 0 if x < O ,

u = O~LI/(?J> = 0 at J’ = 0 if .Y > 0. (13.12)

P ( X ) = (1/n) I [A’(x,) dxt/(x - .*)I (13.13)

for incompressible flow, and indeed for all subsonic flows. In parentheses we note that the appropriate condition determining p in supersonic flow ( M , > 1) is

p ( x ) = - A ’ ( . Y ) . ( 13.14) These conditions effect a smooth junction with the Blasius flow upstream of the trailing edge, and to join with the Goldstein wake we need

p + 0, 11 -+ 4(: .~)”~4(1) , A -+ 0.8920.~’ (13.15) as x -+ a, where k, = /2-2/3k0, x = /2”3j. Provided we insist that p -+ 0 as x -+ x, so that the reduced pressure is o( 1) on leaving the triple deck on the downstream side, the remaining conditions there are superfluous.

No complete numerical solution of (6.1) has yet been obtained,? although Messiter (1970) has given an approximate solution. We can obtain some of its properties, which indicate that the full solution will not be easy to get. When x is large and negative

while (13.16)

(Goldburg and Cheng, 1961; Stewartson, 1969) so that the decay of the triple deck into the Blasius profile is only algebraic, in contrast to the exponential decay in the free-interaction problems of supersonic flow. At the other extreme, when .Y is large and positive,

In addition . I

. ~ I’

p(x ) % [- 1.784/33 ’ ( - . Y ) ~ ’ ~ ] + . . . . A % 0.326/(-.~)+ . . . ,

1 + [0.3106/(-.~)4/3] + -,

A = 0 . 8 9 2 ~ ” ~ + O(.Y-”~) , ( 1 3.17) while U(.Y. 0) has a form equivalent to (13.10).

t This has now been done by Jobc and Burggraf (1974) who estimate that i , bclow is approximatcly I .34.

Multistructurerl Boundury Layers 217

At finite negative values of x, the pressure gradient is favorable and the skin-friction parameter T increases monotonically to a finite value A,, (> 1) as x + 0-. Strictly speaking, of course, T should be infinite at the trailing edge, just as it is at the leading edge, but the singularity develops only within a distance O(c6L*) of the trailing edge. As a result t is finite in the double limit I: + 0, x + 0- in that order. It is expected that A , p, and their first derivatives with respect to x are finite as x + 0-. On the other hand, the reduced pressure develops a singularity as Y + O + . The reason for this can be seen by considering the effect of a finite pressure gradient on the solution in the lower deck. Thus for small x the solution is virtually identical with that obtained by Goldstein (13.3) and leads to a form for A which includes a singular term identical with (13.17) except for a power of A l . As a result, p - x-2’3 as x + 0 + , from (13.13), which is a contradiction. Hence we must introduce a singular pressure gradient

dpldx = cox - 1‘3 (13.18)

as x + 0 + , which implies that near x = 0 + ,

u “N +($4”3Kb(X), (13.19)

where now

K: + 2K0 Kg - K Z = 27.24’3~0,

with boundary conditions Ko(0) = KG(0) = 0, K g ( a ) = 18A1: the last condition ensures a match with the velocity profile U , ( y ) at x = 0-, when J’ < 1. Given co, the solution is unique: let us suppose that as + x.

Kb - 1 8 A 1 ~ -+ d o , (1 3.20)

where d o is a constant. Then, matching this solution to an outer solution, when p - 1, we find

and hence

x- ‘13[,4(.\-) - A ( ( ) ) ] + +don;’ as x + O + . (13.21)

A contradiction is now obtained with (13.18) on using (13.13) unless do = 0, and this is the condition needed to complete the solution for K O . The properties of K O have been worked out by Hakkinen and Rott (1965), and we deduce that

u(x, 0) = 0.8991R:’3x”3 + . . . , (13.22)

p(x) p(0) + 0.6133A,:’3x2’3 + . . .

218 K. Stewartson

as x + O + . It is inferred that the corresponding form for p as x + 0- is

p = p ( 0 - ) + xp'(0 - ) + O(x2 log( - x)). (13.23)

Thus the pressure gradient is adverse when x is small and positive but nevertheless the centerline velocity u(x, 0) rapidly rises from zero at x = 0. For larger values of x, p reaches positive values and decreases again to zero a sx -co .

TABLE 9

~N(.OMPRESSlHI.l! Fl.ow PAST A FLAI. PLATE AT FINITE REYNOLDS

NUMBERS.

I 1 3.79 0.94 10 0.750 0.773 1.00 20 0.688 0.504 1.08 40 0.630 0.323 1.08

100 0.562 0.187 1.15 200 0.516 0.123 1.14

From this discussion we can infer the form of the second term in the expansion of the drag coefficient of a finite flat plate in descending powers of R. We have

1.328C"2 2A- 1/4C718(T: /Tz)3 /2 CD = R1/2 + I M ; - 1 ) 3 / 8 ~ 7 / 8 0 , + '..,

where . o .

0 , = 1 (7 - 1)dx. (13.24)

The most reliable numerical computations for incompressible flow past a flat plate at finite Reynolds number appear to be those of Dennis and Chang (1969) and in Table 9 we give their values of CD and the equivalent values of U,, which would be a constant if (13.24) were exact. The agreement is encouraging? but the values of E are rather too large to expect the higher powers of e in (13.24) to be negligible.

There have also been a number of detailed studies of the solution in the

- a

Jobe and Burggraf (1974) report that 0 , = 1.021. As a result the agreement between (13.24) and the numerical values of Dennis and Chang (1969) is astonishingly good. Thus at R = 1, the discrepancy is only 6 "(,.

Multistructured Bourzdary Lajiers 219

neighborhood of the trailing edge (e.g., Plotkin and Flugge-Lotz, 1968; Sch- neider and Denny, 1971). Although they are open to criticism (Messiter and Stewartson, 1972; Schneider and Denny, 1972), they do indicate some support for the triple-deck structure to the flow described here.

When the mainstream is supersonic the triple-deck structure is almost the same as for the subsonic problem, the only difference being that (13.14) is appropriate instead of (13.13). The details are however easier to work out. For, in x < 0, we have an expansive free-interaction boundary layer of the kind discussed in Section VII with xf to be found. At x = 0+ , the singular pressure gradient is the same as in (13.22), and now remains adverse for all positive .Y. The value of xf has to be chosen so that p + O + as x + m. The necessary computations are similar to those described in Sections XI and XII, and the special difficulties near x = O + are overcome by the use of a double-mesh procedure (Smith, 1973a). The calculations have been performed by Daniels (1974) who finds that

0, = 0.768 ( M , > 1) . (13.25)

The insertion of a triple deck between the Blasius and the Goldstein regimes does not render the resulting solution completely smooth. In the main deck the chief remaining singularity is due to the reduced pressure gradient which is finite as x + 0-, but - x- ' I 3 as x + O + . This may be smoothed out in turn, by the introduction of yet another layer, whose width and length are both O(e4L*) and in which the velocity variations needed to account for the properties of p are O(r :X /3Uz) . The detailed properties of this interior region in the main deck have not yet been studied in detail and necessitate solving a linear second-order partial differential equation of the kind envisaged by Lighthill (1950), but it seems that the additional slip velocity induced at the bottom of the main deck (Y , + 0) gives rise to an extra contribution to the drag coefficient of order R - 'I6. For, since the length of the layer is O(c4L*), the subboundary layer needed to reduce the slip velocity to zero is only O ( E ' ~ / ~ L * ) in thickness. Similar difficulties occur in the studies of flow over convex corners discussed in Section XI and with suction and injection discussed in Section XII.

A more fundamental failure of the triple deck occurs in the limit x2 + y 2 + 0 for all terms of the Navier-Stokes equations, which could safely be ignored in studying the rest of the flow field, now simultaneously become important. From the lower-deck solution

u = I l ,y if x < 0, u = + ( $ x ) ' / ~ K ~ ( x ) if x > 0, ( 1 3.26)

in the limit x2 + y 2 = 0; (13.19) and the forms for p are also different. The central region necessary to achieve the transition between the forms of the

220 K . Stewartson

solution when x < 0 and x > 0, is somewhat analogous to that for the leading edge, but is rather thicker owing to the more slowly moving fluid just outside. The scaling laws are

x* - L* = pL*2, y* == pL*F,

I)* = c8u: L*Y(2, P), (13.27)

so that

u* = c”$(dii/?F), u* = -cW:(?q/dz). (13.28)

The reduced stream function 9 satisfies the equation

v4ii = d ( i P , V 2 q ) / d ( z , F), (13.29)

with boundary conditions

9 = H / ? F = o at

q = (Q9/aF2 = o at

9+.& F2 as 2 + -m,

P = 0,2 < 0,

F = 0,2 > 0, (13.30)

= 2 - 1 / 3 ~ 2 / 3 ~ , , ( ~ ) , = ~ / 3 ( 2 2 ) 1 3, as 2 -+

and

’? = +A,?l F2 as ? + nc, for fixed 2. (13.31)

Some studies of the properties of 9 have been made by Hakkinen and O’Neil (1969) when 2’ + F2 is large. They expanded 9 in descending powers of distance from the origin, but omitted two sets of eigenfunctions, one arising from the inviscid part and one from the boundary-layer part into which the equation then splits. These were subsequently added by Capell (1972), who also showed that the expansion may then be related to the exact solution of an approximate form studied by Stewartson (1968). The approximate form was effected by replacing 9 on the right-hand side of (13.29) with fAi, F2. The equation is thereby reduced to a linearized form and may be converted into an integral equation of the Wiener-Hopf type. Readers of that paper (Stewartson, 1968) have, regrettably, to contend with the author’s curious error of totally neglecting the triple deck which surrounds the neighborhood of the plate being considered, and imagining that it has interfaces with the Blasius and Goldstein regions. A similar error appears to have been made by Talke and Berger (1970) in their study of the flow near the trailing edge, using the method of series truncation. They infer that the appropriate length scale in the x* direction is O(c6L*), which is the same as that for the central region under discussion here.

The approximate solution of (13.29), when properly interpreted, suggests

Mult istructured Boundary Luyers 22 1

that a match can be effected between it and the triple-deck solution sur- rounding it; the role of the interior region of the main deck, however, still needs consideration in relation to this central region. According to (13.29), the skin-friction parameter 5 [defined in (4.10) and (4.1 l)] has the form .F(r?) in the central region, so that the additional contribution to the drag coefficient CD from this region is O(Rp5/4). As r? -+ - m, .F + ,il and as 2 + 0-, Stewartson's studies indicate that

3 z L O ~ ( A A ~ ) ~ ' ~ / A ( - r?)li2, ( I 3.32)

which agrees with the view taken by Carrier and Lin (1948) of the nature of the singularity at a sharp edge. The approximations made probably only affect the numerical constant in (13.32). Dennis has privately informed the author that the numerical studies he has carried out with G.-Z. Chang (1969) indicate that z has a singularity resembling (13.32) as 2 --$ 0-, whose magnitude is almost independent of R when R is large.?

The pressure variation in the central region is O(c4p2,), of higher order than in the triple deck, and remains finite on the plate as r? + 0-. In the wake, the x* component of velocity

I /* z 1.09(;13L1)3/4U?: cp'[(x* - L*)/L*]1'2 (13.33)

when

(s* - L*)/L* 4 2,

and, from (13.22), u* z 0.2997(iLi. l)2~3U~ [(s* - L*)/L*]1'3 ( 1 3.34)

when

c6 4 (s* - L*)/L* 4 c ~ .

Stewartson's approximate solution differs from (13.34) in the numerical coefficient: his value is 1.0651/21/3 = 0.64. For completeness we note that

u* 2 1.61 1 A 2 ' 3 U ~ [ ( s * - L*)/L*]'

when

c3 < (s* - L*)/L* 4 1 . (13.35)

A somewhat related problem is the flow near the edge of a circular disc of radius a*, rotating about its axis with angular velocity O* in a quiescent fluid. With R = R * U * ~ / V ; and s* and y* denoting distances perpendicular to the axis and the disk, respectively, the boundary layer in s* < a* was

'r Further using Jobe and Burggrafs value 1.34 for In (13.32) to have the limiting value of 1.074 iis R - x .

he estimates the numerical coellicient

222 K . Stewurtson

calculated by von Karman (1921) and Cochran (1934) and, from our point of view, has the noteworthy feature that the equivalent mainstream velocity U z = 0. If 0 < x* - u* 6 a*, the boundary layer has the characteristics of the Goldstein wake, except that the displacement thickness changes smoothly, because U z = 0 (Leslie 1972). Consequently, (S. H. Smith 1973) the triple deck is absent and the discontinuity ofslope between the streamlines in the Goldstein and the von Karman solutions is smoothed out by an interior inviscid region of length O(c4u*), similar to that discussed on p. 219 above. A subboundary layer is induced, of thickness O ( ~ * E ' ~ / ~ ) and a central region is needed in the immediate neighborhood of x* = 0 and y* = 0 to complete the description of the flow field. The solution in this region is governed by Eq. (13.29)-(13.31).

XIV. Trailing-Edge Flows for Bodies with Finite Thickness

An obvious extension of the theory described in the previous section is to include bodies of finite thickness-to-length ratios r * . It is necessarily more complicated, if only from geometrical considerations, but the possibility of separation is at present the chief obstacle to progress. Indeed if the body is smooth and E* is held fixed and positive as t: -+ 0, the flow must separate before the rear-stagnation point is reached. Even if the flow is supersonic it is not yet clear whether the separation is of the free-interaction kind, which is regular and permits an extension of the theory to include reversed flows in the boundary layer (Section VIII), or is of the Goldstein (1948) type, which implies a singularity at separation that remains anomalous (see Section XVI). In subsonic flow there is no analogy to the free-interaction boundary layers so that the separation problem is intractable at present. If r* is allowed to fall to zero with c, so that the condition a* < E~ is satisfied, separation can apparently be avoided, but a consistent theory is not yet available. There have, of course, been many approximate theories of flow past thin bodies, including discussions of separation, but none has been carried to the stage when even the beginning of a rational theory, of the kind we have in mind in this review, can be laid down, and so we shall not consider them, notwithstanding their practical utility.

The main attack, from a rational standpoint, has been restricted to bodies with sharp trailing edges. Let the trailing edge be the origin O* of coordinates and O*x* be in the direction of the undisturbed stream. Then the body shape near the trailing edge may be defined by y*/L* = a*( -x*/L*)" where MZ is a positive number. If m < 1 the edge is not sharp and if m > 2 the pressure gradient near O* is favorable so that separation does not occur.

Multistructured Bouidury Luj1er.s 223

The flow structure for a wedge, with m = 1, is representative of sharp trailing edges, and we shall discuss it in some detail, considering subsonic flows first. Since a* 4 1, the pressure gradient over most of the body is small, and the slip velocity U ; is almost uniform, but near the trailing edge

u; zz u:a,(-x*/L*)"*, (14.1)

where a , is a positive number, and so the pressure rapidly rises. Thus the Blasius profile in the boundary layer is modified, even without taking the triple deck of the trailing edge into account, by setting up an inner boundary layer, governed essentially by (6.1). Its thickness 6* - E L * and its streamwise extent - M * ~ ' ~ L * . The boundary conditions are the same as in (6.2) and (6.3) except that the condition A' (x ) = -p(.x) is omitted and instead the pressure gradient is determined from (14.1). The reader is referred to Riley and Stewartson (1969) for a discussion of the matching of this subboundary layer with the Blasius solution as x + - m and as y + m : the technical details are complicated but there are no real difficulties. The prescribed adverse pressure gradient inexorably leads to catastrophic separation before O* is reached unless a* is sufficiently small so that the favorable pressure gradient induced by the mechanism described in the previous section comes into play. A numerical solution leading to an accurate determination of the separation point x has not yet been carried out, and approximate methods like Stratford's (1954) are not sufficiently accurate. The best estimate uses the method of series truncation and gives

-x, = 4.5 if -x dpldx = 1 . (14.2)

The critical scaling for an interaction between the two kinds of pressure

a* = ~ 1 / 2 c 1 / 4 I M ; - 1 1 1 / 4 ~ 2 ~ , (14.3)

where CY is fixed as E + 0. Then if we adopt the scaling laws of Section XI, the governing equations are again the same as (6.1) and the boundary conditions are now

u - ( y - ax) + 0 as x --t -a , y > rx, (14.4)

u = u = 0 when y = ax, x < 0, (14.5)

2' = ?u/?y = 0 when y = 0, x > 0, (14.6)

gradient is a* = 0 ( e 2 ) , and so we write

while

(14.7)

224 K . Strwartson

for subsonic flow. As formulated this problem is very similar to that for convex corners, the main differences being that M , < I , a > 0, the x axis no longer coincides with the plate upstream of the corner, and the plate ends at 0". It also embraces the Riley-Stewartson theory when a $ 1. When a < I , on the other hand, it reduces to the trailing-edge problem of Section XIII, with a dominantly favorable pressure gradient in x < 0. Thus a - 1 is a watershed between smooth flow off the trailing edge and catastrophic separation. , An interesting series of three questions can now be formulated, but not answered. First, is it possible to find a number a, such that if a > a, separation occurs and if a < a, separation does not occur? Second, granted the existence of a,, are the separations always catastrophic if a > a,? If they are, we do not have a mechanism for computing the solution at all because it cannot be found downstream of separation, and the upstream solution depends on the over-all properties of A , which determine the pressure through (14.7). It may be necessary to creep up on from below by computing the solution for smaller values of a and examining carefully what hap- pens when a. - a is small. Third, if the separation is regular in a similar way to the compressive free interaction in Section VIII, we should be able to compute the solutions when a > a, , and it would be interesting to discover how they behave as a + a. For, according to Section XVI, the catastrophic separation that occurs when a 9 1 cannot be removed by a triple deck. How then does the regularity break down ?1-

For supersonic flow, the situation is more straightforward, since the inviscid flow is uniform right up to the trailing edge (where a shock wave occurs). Hence the triple deck at the trailing edge joins up with the Blasius profile. The governing equations are the same as in the subsonic problem except that (14.7) is replaced by

p(x) = -A ' (x) - a. (14.8) Consequently, in x < 0 the solution is simply a free interaction, and there is presumably a number a1 such that it is expansive if a < CI, and compressive if a > a,. We note that if separation occurs there is a limit to the pressure rise upstream of O* owing to this mechanism, namely

(14.9) As the limit is approached, the lower deck becomes very thick, so that a pressure rise in x* > 0 of any significance is unlikely. It is inferred that as r o + Po( + l.8), x, -+ -a, and indeed that the separation point retreats a finite distance upstream of 0*, until a point is reached where another mechanism can be brought into play to ensure that p* = p z far downstream of the body.

p * ( o ) - p*,, I p*, - 1 )- 1 / 4 c 2 ~ , .

t Sychev's (1972) argumcnts are relevant here. See Addendum.

Multistrtictured Boundary Layers 225

XV. Viscous Correction to Lift

The theory of the triple deck can also be applied to study the viscous corrections to the lifting forces on aerodynamic shapes at high Reynolds numbers and to show that the Kutta condition, which determines the lift for inviscid flows, can be embedded in a formal asymptotic expansion of the flow field in powers of E . We restrict attention to thin two-dimensional wings, which, to a sufficiently close approximation, may be supposed to occupy the part - L* < x* < 0 of the x* axis. In addition, the undisturbed fluid velocity upstream of the wing has components ( U z , cc*Uz) along the .x* and y* directions where cc* is a small positive constant. In order that we may treat the wing as a line of zero thickness near O*, the thickness of the wing must be < O(c*L*) there, and in order that we may be able to exclude the possibility of leading-edge stall, the thickness of the wing must be > O(cc*L*) near x* = - L*. These conditions are not incompatible and are satisfied, for example, by a simple Joukowski aerofoil. It is noted in parentheses that the problem of leading-edge stall is very important and not fully understood. The techniques of the triple deck may possibly be helpful in elucidating this phenomenon too.

With the wing approximated by a flat plate and the restriction that the fluid is incompressible, the inviscid solution to the flow field shows that on the wing (y* = 0, -L* < x* < 0),

u* = 0, u* = U Z - c c * ~ ~ { ( x * + B ) / [ - ~ * ( L * + ~*)]"~}sgn y*, (15.1)

while on the wake centerline (y* = 0, x* > 0)

u* = u* 3 0 9 u* = U* &* (x* + B)/[x*(L* + x*)]'''. (15.2)

In deriving these results the Kutta condition has not been applied so that B is an arbitrary constant, but we adopt the view that it does give the correct limit flow as c: + 0, and shall in fact assume that B = O(e3L*), justifying it a posteriori.

Thus upstream of the trailing edge the boundary layer develops in a similar way to that described in the previous section, i.e., it remains close to the Blasius profile over the majority of the wing but then changes rapidly in the neighborhood of 0". The main difference is that on the lower side (y* = 0-) the pressure gradient is favorable, so there is no tendency to separate. Again subboundary layers develop and are the geneses of the lower decks of the trailing edge. The reader is referred to Brown and Stewartson (1970) for details of these properties and their relation with the main parts

226 K . Stewartson

of the boundary layers, and we pass on to the consideration of the triple decks which are, in principle, independent of these details.

After making the usual scale transformations, writing CI* = ,1/2/2”’8a, (15.3)

and using the subscripts plus (+ ) and minus ( - ) to denote flow properties in J’ > 0 and J’ < 0, respectively, we find that the governing equations of the triple deck are the same as (6.1). The boundary conditions are

u i --+ l y l as x-+ -M,

u* - I J’ I ? 3%( -+)! I y 1”2/(

u* = u+ = 0 if y = 0, .I

pi = + ( l / X ) * * [ A&’ I - ?

- i ) ! + A & ) sgn y , (15.4)

x < 0; (15.5)

dx‘/(x - x’), (15.6)

p* (x) + .( -x)1’2 -+ 0 as x-+ -a, (15.7)

p+(x ) = p-(.x) when x > 0, (15.8)

while ( t i + , v,) and (u- , u - ) are smooth continuations of each other in the half-plane x > 0. The double star before the integral sign in (15.6) signifies that the finite part is to be taken. This formulation of the lower-deck problem may be regarded as including the subboundary layers induced by the prescribed pressure gradient associated with the slip velocity (15.1) and which we mentioned earlier. Either we neglect (15.6) and assume (15.7) holds at finite values of x, or we take c( S I to obtain these subboundary layers. The reader is reminded that the constant B is assumed to be O(c3) and that its value will emerge from the solution of the triple deck. In x < 0 the pressures in y > 0 and y < 0 are different but when x > 0 they must be equal. The pressure variation across the wake in x > 0 can be computed from (3.8) and is 0(c3p*,).

A complete solution is not yet available but we can determine a few of the flow properties. First, when x is large and negative the lower decks are dominated by the pressure form (15.7) and in particular the one on the top side of the wing probably separates at a finite value of x if a is large enough, the best estimate of x, available being

- X, + (0.326)- 6 ~ 6 . (15.9)

Brown and Stewartson (1970) have suggested that if separation does occur, then it provokes the phenomenon of trailing-edge stall. However, if is the critical angle of incidence in the sense that separation always occurs if CI > ol, and does not occur if a < a,, they estimated ol, = 0.4 which leads to rather low stalling angles (= 2”) in realizable situations.


The form of the pressure when x is large and negative is

p*(x ) = f ( - x ) ” 2 k ab,/(-x)”2 + . ” ) (1 5.10)

where

B = e32-5!4 bl (15.11)

and b , is determined from over-all properties of the triple deck. An approximate solution of the equations gives

b , = 0.79, (15.12)

which means that the lift coefficient on the wing is given by

C L z 27m*(I - 1 . 5 8 ~ ~ / 2 - ” ~ ) . (15.13)

Provided separation has not taken place? the velocity profiles at x = 0 have different (positive) slopes A, and A- at x = 0, which must be smoothed out immediately downstream of the trailing edge and consequently generate an adverse pressure gradient in x > 0, in a similar way to the symmetrical problem. When s % 1, it appears that

-24(x) A + ( x ) + A - ( x ) Z -$Lxx~’~ - 4G!b1X”2 (15.14)

and A +(x) - A - (x) 2.83($~)’ /~ . (15.15)

It follows that after the two lower decks unite at x = 0, they form a wake which remains within a distance 0(x’l3) of the curve y = $(x), which may therefore be taken as the centerline of the wake. Furthermore the form of$, when x + 1, is compatible with the conditions of the triple deck and with the equations governing the larger scale properties of the flow outside.

Some numerical studies of the problem are needed before we can safely pursue further the details of the solution. It is worth pointing out however that the critical range of values of the incidence is practically inviscid. When a* 6 0 ( R - ” l 6 ) the effect of incidence is negligible and when a* + O ( R - ‘ / I 6 ) it provokes separation which might well be catastrophic.

From time to time it is suggested that the wake curvature just outside the triple deck is at least as important for the determination of the viscous correction to the lift. This is not so, and the appropriate correction may be shown to be of smaller order than that due to the triple deck in the following way. From (15.2) the centerline of the wake is given by y* = $*(x*) where $*’(x*) = a*[x*/(x* + L*)]’” (15.16)

tSychev (1972) has discussed a possible structurc of the solution of (15.4)- (15.7) when separation docs take place. Further details of his ideas may be found in the Addendum.

228 K . Stewartson

and 4*(0) = 0. Hence the pressure jump across the wake is m

p*, U*,'4*"(x*) J (1 - u*2/U*,2) dy* -7

= p*, U * , ' ~ * ' ' ( X * ) E ~ A ~ L*, (15.17) where c4A3 L* is the sum of the displacement and momentum thicknesses of the wake and varies from 3 . 3 7 ~ ~ ( 2 C ) " ~ ( T,*/T*,)L* at x* = 0 to ~ E ~ A C " ~ L * at x* = m.

In order to calculate the effect of this pressure variation on the inviscid slip velocity past the plate to leading order, it is sufficient to assume that the corrections (u;, 0;) to the velocity components satisfy the following conditions at y* = 0:

uz =0 , if x* < -L*, v 2 = O if -L* < x* < 0, UT = -fa*4*"(x*)~~A~ U*, sgn y* if x* > 0. (15.18)

The inviscid motion satisfies the harmonic equation and u$ is an odd function of y*. Hence the distribution of uz on the plate may easily be written as a Hilbert integral. In general uz = O(c4U:), but it is singular as x* + - L* + and as x* + 0 -. The leading-edge singularity is similar to that in (15. I ) and may be absorbed in it, but near the trailing edge

(1 5.19)

Hence as we approach the upstream end of the triple deck

u ; / ~ * , = 0 ( ~ * ~ 5 / 2 log = 0(>;3 log (1 5.20)

when a* = O(c'l2). This slip velocity appears in the upper deck (see Section- III,B) where the dominating velocity perturbations are O(c2 U z ) , and con-

sequently, we may expect the contribution to C, from (15.19) to be O ( E ~ log E ) and weaker than the effect we have been considering earlier. We remark that over the main part of the wing the induced pressure makes a contribution O ( E " ~ ) to C L , as does the boundary layer.

Compressibility effects may be taken into account without formal difficulty. When the flow is subsonic the basic equations of the lower deck are the same, the only difference being that now a is defined by

cI* = C 1 / 2 ) w W 8 c 1 / 1 6 (1 - M',)"'"T:/T;)3'4a (15.21)

instead of (15.3). In supersonic flow the inviscid slip velocity does not have a singularity at the trailing edge, being, in fact, uniform over the wing and the wake centerline is straight. In consequence the critical order of magnitude for a* is g 2 instead of c1 l2 and if we define a as in (14.3) the boundary

Multistructured Boundury Layers 229

conditions are the same as (15.4)-(15.8) except that now

p i + f a as x + -a, (1 5.22) p + f CI = f dA+/dx. (15.23)

It is expected that an expansive free interaction occurs upstream of the trailing edge on the lower side of the plate; on the upper side the free interaction changes from expansive to compressive as a* increases. From the origin O* on, the two pressures must equalize, to leading order, and, as x + m, pk -+ 0, A , z -ax. As with the symmetric problem in Section XIV we anticipate no difficulty if separation occurs on the upper side of the wing and that as a -+ Po ( = 1.8) this separation point retreats to a finite distance from the trailing edge to enable another mechanism to be brought into play to ensure that p* + p z downstream of the wing.

XVI. Catastrophic Separation

As a final application of the notion of the triple deck we may consider the neighborhood of a point of catastrophic separation where, on classical theory, the boundary-layer solution comes to an end. Taking the separation point to be the origin O* and the imposed pressure gradient to be finite, Goldstein (1948) showed that the solution becomes double structured as x* + 0-. Writing y = Y,/( - X)lI4, where X and Y, are defined in (3. l) , he showed that the solution in x* < 0 can be expressed as a series of ascending powers of ( - X ) whose coefficients are functions of y when y - 1 and are functions of YM when Y, - 1. In particular

T = (-X)’’’, u* w c4U*, UO(YM)/(-X)”’ as X + 0-. (16.1)

Furthermore the solution cannot be continued into X > 0. It is natural to ask whether this singular behavior can be removed by resorting to the full Navier-Stokes equations in the neighborhood of O*, allowing the boundary- layer equations to hold on either side. One might have in mind a locally large pressure gradient dependent on the boundary-layer properties, which reacts with the boundary layer and permits reversed flow to be set up without a singularity occurring. The appropriate mechanism seems to be of the triple-deck kind and Stewartson (1970a) has considered this possibility in detail. Some modification to the scaling laws (4.4) is clearly necessary, because A @ 1 at the onset of the interaction, but the principal ideas are clear enough. He found that the length scale is now c’L* in the x* direction, the upper deck has width E’L*, the main deck c4L*, and the lower deck has width

230 K . Stewartson

E~”L*. The mathematical argument is simplified because the lower deck is governed by a linear differential equation, which can be fully solved for any prescribed pressure gradient, and he finally reduced the problem of determining the flow properties near the origin O* to that of solving the integro- differential equation

f ” ( x l ) d x ,

‘ x (xl - x)1’2 p ( x ) + x = 1

if the mainstream is subsonic, and

. x f ” ( x , ) dx,

- a (xl - x)1’2 f 2 ( X ) + x = - 1

(16.2)

(16.3)

if the main stream is supersonic, where x is a scaled distance along the wall downstream from O* and .f is a scalar multiple of the skin friction 5 . Thus superficially these equations for ,f appear to be consistent with the requirement (16.1) from the Goldstein theory when x is large and negative.

However it may be established that (16.2) cannot have a solution at all because a satisfactory behavior cannot be found forfwhen x 9 1, and the solution of (16.3), apart from not being unique, must terminate at a finite value of x.

The only certain conclusion that can be drawn from this discussion is that it is impossible for a catastrophic separation to be embedded in an asymptotic expansion of the solution of the Navier-Stokes equations which includes a single-structured boundary layer both upstream and downstream of separation. It may be that catastrophic separation is impossible: i.e., the prescribed pressure gradient must be such that separation is always approached in a regular? fashion with dz/d.u finite at x = 0. It may be that, if the motion is generated from an impulsive start, the flow near O* never becomes steady at high Reynolds number even if instability is inhibited elsewhere. It may be that the flow does become steady but that it exhibits pathological features downstream of separation when c < 1, as suggested by Brown and Stewartson (1969).

For a supersonic mainstream it is now well established that regular separation can occur through a compressive free interaction, and there is a view, which has considerable backing, that such separations are the rule. For a subsonic mainstream, instances of regular separation in problems of physical interest are rare, and it is not even known whether they can occur through a triple deck. In fact, the properties of separating flows in multistructured subsonic boundary layers is a major long-term problem of the

t Sychev (1972) has argued cogently that this is indeed the case. Some account of his views is presented in the Addendum.

Multistructured Boundary Layers 23 1

subject, and it is hoped that this review may provide a stimulus to further work in this area. A necessary first step is to devize a suitable numerical scheme for integrating the lower-deck equations.

XVII. Addendum

Since this review was written a number of further applications of multi- scaling techniques have been reported. Sychev (1972) has considered the analog to compressive free interactions that might arise when the mainstream is incompressible. He begins by adopting the view that the limit solution, as t‘ --f 0, of flow past a bluff body is given by the Kirchhoff free- streamline theory in which the curvature of the free streamline is finite at the point of detachment from the body. This means (Woods, 1955) that the pressure gradient is favorable upstream of, and vanishes at, separation. Sychev asserts that when 0 < c 4 1, there is a weak singularity in the pressure gradient at separation on the body-scale in that

as x* + x,*- where ko is a numerical constant to be found. On the other hand,

as x* + x,* +. The passage through separation and the smoothing out of the singularity in (A.l) is effected by means of a triple-deck closely similar to that described in Section XV. Indeed the fundamental equation for the lower deck in this region is the same as that for p + (x) in (15.4)-( 15.7), except that the condition x < 0 is omitted from (15.5). Thus Sychev’s theory may be thought of as the limit of the studies in Section XV when CI is large enough for z to vanish at a large negative value of x.

dP*yx* = 0 ( & 7 ~ : uz2/x,*)

Since A ( x ) + 0 as x -+ - co, it follows from (15.6) that as x + + rn

p + 0, A ( x ) + -+CIx3’2 = A , , ( x ) , (‘4.2) provided of course that the system of equations (15.4)-(15.7) as modified has asolution. [Brown and Stewartson (1970) inclined to the view that it does not.] Sychev ignores this question and uses (A.2) to determine a flow structure for the lower deck as x --f co, similar to that in Section VIII. If y > - A , ( x ) , u --f y + A,(x); if < = 0(1), where <x1l3 = y + A , ( x ) , $ is given by (8.7) and (8.8); if x11/12 < y < - A ,

(A.3) and if y - ~ ~ ~ 1 ’ ’ there is a sub-boundary layer having a “backward facing” self-similar flow of the Falkner-Skan type with = - 10. Messiter and

u z - ( ~ C , / ~ C ( ) X - s’6, p z - ( 9 C $ % ~ ~ ) x - 5/3:

232 K. Stewartson

Enlow (1973) have also examined the effect of the pressure gradient (A. 1 ) and in addition have considered other, possibly viable, forms of dp*/dx*.

Further downstream, i.e., beyond the triple deck, Sychev’s structure is consistent with a plateau region similar to that in Section IX. Here the mainstream would be bounded from below by a free streamline, the continuation of the boundary layer from upstream of separation being a free shear layer and governed by (9.2). Between this shear layer and the wall there is a region of slow, inviscid, reversed flow in which u* - c712U$ and a subboundary layer of thickness O ( C ~ ~ ~ . X ? ) . This structure is consistent with the requirements of Kirchhoff free-streamline theory and Sychev concludes that this theory is relevant to all laminar separated flows at high Reynolds number. He adds “the scheme of flow, assuming that the separation of the stream on the smooth surface occurs as a result of an adverse pressure gradient distributed over a finite segment of the body surface,. . . apparently does not have any place in actuality.”

There is certainly something to be said for this point of view. Consider, for example, the flow past a circular cylinder. The numerical studies of Takami and Keller (1969) and Dennis and Chang (1970) and the experiments of Acrivos, Snowden, Grove and Peterson (1965) are at least partially consistent with it. Further, the rapid rise of pressure known to occur just before separation in high Reynolds number flow (e.g., Woods, 1955) might be evidence of a triple-deck phenomenon. It is a strong argument to weigh against the views of Brown and Stewartson (1969) on the nature of high Reynolds number flows. On the other hand, Schubauer’s experiments (1935) on a thin ellipse at R - 47,000 revealed an extended region of adverse pressure gradient before separation which could not be joined directly to a triple-deck of the kind envisaged by Sychev.

The Sychev theory of separation opens up exciting possibilities for boundary-layer theory and for the study of high Reynolds number flows in general. It is clearly important, however, that the existence of the solution of the governing equations be settled. The value of x , or k , , in the theory appears to be indeterminate and Sychev appears to think that it cannot be determined from the triple-deck theory alone. If that were so, the situation would be different from that in a compressive free interaction, but it may be that a solution can be found for only one value of c(. Since viable numerical methods are available (Jobe and Burggraf, 1974; Veldman and Van de Vooren, 1974) for incompressible triple decks now, these questions may not remain long undecided.

A start has been made on the hypersonic analog of a compressive free interaction. In the limit 1 + m (10.10), the initial departure from the standard boundary-layer solution is not exponentially weak as when (6.5) holds and occurs at the leading edge, instead of at an arbitrary station of the wall.


Its structure may be deduced from the theory of hypersonic boundary layers (Stewartson, 1964b) in which the pressure on the plate is written in the form

where P(y ) is a function of y only equal to 0.555 when j! = 1.4, and H ( t ) is a power series in t'" with H ( 0 ) = 1 in the classical theory. It was realized by Neiland (1970) that H contains an eigenfunction which may be found by writing

H = 1 + at" + ... and deducing the value of n which enables the compatibility conditions between the viscous and inviscid parts of the boundary layer to be satisfied when a # 0. O n using the tangent-wedge approximation to describe the inviscid flow he obtained a unique value for n of 25.3 when y = 1.4. This result was confirmed by Werle et al. (1973) and extended to include flows with heat transfer. Subsequently Brown and Stewartson (1974), using the exact equations for the inviscid part of the boundary layer and taking the wall to be thermally insulating, obtained values of n of 26.6 when y = 1.4 and 14.5 when y = 513.

In addition to the numerical work on the trailing-edge triple deck (Section XIII) by Jobe and Burggraf (1974) already briefly mentioned in the main body of the review, an independent study has also been carried out by Veldman and Van de Vooren (1974). They find that d, = 2.66 (p. 149) and 2, = 1.345 (p. 217).

Smith (1973b) has considered the effect of a small hump, of height O(r:5x:) and length O(Axg) in the wall, on the laminar boundary layer. Earlier Hunt (1971) had considered such humps when c6 < A < c4 as a step toward the elucidation of the phenomenon of trip-wire transition. One limitation to Hunt's theory is the assumption that the main stream has no first-order effect on the pressure and, when he allowed for this, Smith was led to the triple-deck theory which then enabled him to discuss all values of A 5 1.

Three other papers may be noted. Guiraud (1973) has extended the theory of trailing-edge flows in Section XI11 to include three-dimensional effects. Daniels (1974) has considered the numerical solution of the differential equations of Section XV when the main stream is supersonic and (15.22), (15.23) hold. His work is restricted to flows in which separation does not occur on the plate, so that the conjecture made at the end of Section XV is still unresolved. Finally Brown and Daniels (1974) have examined the theory of oscillating aerofoils and discussed the modification required to the Kutta condition.

234 K . Stewavtson

ACKNOWLEDGMENT

The author is grateful to Dr. S. N. Brown and P. G. Daniels for discussions on the subject of this review and for constructive criticism of the manuscript. I n addition he acknowledges a special debt to Dr. S. N. Brown who undertook the task of proof reading during a time when he was ill .

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VAN DE VOOREN, A. I . , and DIJKSTKA, D. (1970). The Navier Stokes solution for laminar flow

Aero/Taut. Tech. Note D-618.

Cambridge Phil. Soc. 46, 182- 198.

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Math. 1 1 , 399-410.

1-37.

Oxford Univ. Press, London and New York.

London, A 306. 275--290.

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Roy. SOC. Lotdot?, A 319, 289-305.

Mtrth. 23. 137-152.

A p p l . Math. 24, 387-389.

The transition regions. J . FInid Mech. 62, 289-304.

A 312, 181-206.

9% 108.

Cooncil Rep. M r m 3002, London, England.

GUZU, 3, 47--59.

sible Buid past a circular cylinder. Phys. F h i t l s Suppl. 11, 51-56.

161-189.

large surface mass in.jection. A.I .A.A. J . 7, 1268- 1273.

Phil. Trans. Roy. Soc. London, Ser. A 253, 55-100.

past a semi-infinite flat plate. J . Eng. Math. 4, 9-27.

Multistructured Boui?dary Layers 239

VAN DYKE, M. D. (1964). ‘‘ Perturbation Methods in Fluid Mechanics,” Chapter VII. Academic

VAN DYKE, M. D. (1969). Higher order boundary layer theory. Annu. Reu. Fluid Much. 1,

VFt.rmAs. A. E. P.. and VAN DF VOOREN, A. I. (1974). In preparation. VON KARMAN, TH. (1921). Uber laminare und turbulente reibung. Z . Amqew. Murh. Mech. 1.

233-252. WATSON, E. C., MLJKPHY, J. D., and ROSE, W. C. (1969). Investigation of laminar and turbulent

boundary layers interacting with externally generated shock waves. Nut . Adc. Corntn. Aeronuut. Tech. Note D5512.

WERI-E, M. J., DWOYER, D. L.. and HANKEY, W. L. (1973). Initial conditions for the hypersonic-shock/boundary-layer interaction problem. A.1.A.A. d. 11, 525-530.

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265-293.


Response Curves for Plane Poiseuille Flow

DANIEL D. JOSEPH? Department of Aerospuce Engineering and Mechunics

University (f Minnesotu, Minneupolis, Minnesotu

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. The Solution of the Basic Equations for Laminar Poiseuille Flow . . . .

111. Global Stability of Laminar Poiseuille Flow . . . . . . . . . . . . . IV. The Fluctuation Motion and the Mcan Motion . . . . . . . . . . .

VI. Laminar and Turbulent Comparison Theorems V. Steady Causes and Stationary Effects . . . . . . . . . . . . . . . .

. . . . . . . . . . . VI1. Turbulent Plane Poiseuillc Flow-An IJppcr Bound for the Response Curve

VIII. The Response Function near the Point of Bifurcation . . . . . . . . . IX. Some Properties of the Bifurcating Solution . . . . . . . . . . . . . X. Inferences and Conjectures . . . . . . . . . . . . . . . . . . . .

Appendix: A Formal Bifurcation Theory for Nearly Parallel Flows . . . References . . . . . . . . . . . . . . . . .

24 1 245 246 249 250 25 1 253 256 26 I 263 268 216

I. Introduction

A response function for a fluid motion can be defined as a scalar function that measures the response of the flow to the external forces which induce the motion. For example, in problems of thermal convection, the response function can be taken as the heat transported and the external forces can be regarded as the applied temperature difference. The dimensionless response function relates the Nusselt and Rayleigh numbers. In the flow between rotating cylinders, the response function relates the torque and angular

?This work was supported in part by a grant from the National Science Foundation. I had the good for twe to be able to complete the work at the University of Sussex through a grant from the British Science Research Council and thc hospitality of the Partial Dificrential Equa- tions group. I am indebted to Dr. T. S. Chen for many constructive suggestions and for computing the values given in Table 1.

24 I

242 Daniel D. Joseph

velocity. In the example considered below--flow through a plane channel- the external force is the pressure gradient and the response function is the mass flux. The dimensionless response function relates friction factors and Reynolds numbers.

The response function is generally obtained by evaluating a response functional on a suitably defined set of solutions. We study statistically stationary solutions of the Navier-Stokes equations for flow through a plane channel. The solutions are defined in Section V ; their chief property is that the horizontal average of such solutions is time independent. This is trivially true of laminar Poiseuille flow; we show in Section IX that it is also true of the time-periodic motion which bifurcates from laminar Poiseuille flow, and we shall assume that other solutions observed as turbulence have the property of statistical stationarity. This assumption gives a sense in which fluctuating flow in a steady environment can have steady average properties.

The purpose of this study is best served by drawing a distinction between laminar Poiseuille flow and all the other statistically stationary flows, including the time-periodic bifurcating flow. The subscript I will be used to designate laminar Poiseuille flow. The analysis is conveniently framed in terms of the friction-factor discrepancy f - f ; and the Reynolds number R (or in terms of the friction factor and a Reynolds number or mass-flux discrepancy). Response curves define relations betweenfand R. In Figs. 1 and 2 we have given various response curves for flow in a channel. The experimental results of Walker ef al. (1957) are shown as circles in the diagram. The reader should note that for a given channel, the experimental points appear to fall on a single curve. The essential ideas to be explored here are all represented in Figs. 1 and 2. We aim at an understanding of these figures.

To understand the response diagram, it is necessary to develop the concept of stable turbulence. Stable turbulence may not be a stable solution; rather, we envision a stable set of solutions; actually each solution in the stable set need not be very stable but each solution exchanges its stability only with other members of the stable set, and other solutions, outside the stable set, are never realized. The concept of stable turbulence is at the heart of the conjectures of Landau (1944) and Hopf (1948) about the “transition to turbulence through repeated branching of solutions.”

Landau and Hopf regard repeated branching as a process involving continuous bifurcation of manifolds of solutions with N frequencies into manifolds with N + 1 frequencies. Here the attractive property of the stable solution is replaced with the attractive property of the manifold. For example, when the data are steady and Reynolds number is small, all solutions are attracted to the steady basic flow. For higher Reynolds numbers, the steady flow is unstable and stability is supposed now to be claimed by an attracting

Response Curves fbr Plane Poiseuille Flow 243

FK. I . Response curves for Poiseuille flow. The circles represent the measured response (from Walker et ul., 1957). Dashed and solid lines represent unstable and stable solutions, respectively. The reader should verify, using Table I , that

- 1.0430 = tl In,f/d In i? I R , < d In,f/d In R ~ R ( , ~ , o l < ~I

for 1.021 = ct* < x < 1.0964, where a* is the minimizing wave number defined by

R , = min R(0. 3) = R(0, Y * )

I t follows that the slopes of successive bifurcating solutions for 2 > 1.021 cross each other. This suggests that the lower branch of the envelope

R(c2) = min R(cZ, x) = a([;', z(c'))

of two-dimensional bifurcating solutions is taken on for wave numbers c((I: ' ) > Y * .

manifold of time-periodic motions differing from one another in phase alone. Arbitrary solutions of the initial-value problem will be attracted to one or another of the members of the attracting set according to their initial values. At still higher Reynolds numbers, the manifold of periodic solutions loses its stability to a larger manifold of quasiperiodic solutions of independently arbitrary phase. Now, arbitrary solutions of the initial-value problem are attracted to the manifold with two frequencies, and so on.

Important details of the Landau-Hopf conjecture are in need of revision (Joseph, 1973, 1974) but the essential idea-the motion of stable sets of solutions-may yet provide a basis for a correct and fertile mathematical definition of stable turbulence.


In practice, stable turbulence appears to have the property of consistent reproducibility on the average. By this we mean that in a given channel there appears to be a curve, which we have called a response curve, which defines a functional relation between the Reynolds number and the friction factor.

FIG. 2. Response curves for Poiseuille flow. This figure is the same as the previous one except that the estimate (10.7) of the upper bound (7.1) is shown. The shaded region contains many unstable two-dimensional bifurcating solutions.

The existence of such a curve, widely accepted as natural even in elementary books, is actually a remarkable event since the curve is defined over a set of fluctuating turbulent flows each of which differs from its neighbors. In this sense, the response curve may be regarded as giving the steady average response of a fluctuating system subjected to steady external forces.

Response Curves .for Plane Poisetrille Flow 245

11. The Solution of the Basic Equations for Laminar Poiseuille Flow

Consider the flow driven through a channel by a constant pressure gradient p > 0. I t is best to visualize this flow as occurring in the annulus between concentric cylinders when the gap is small. The flow proceeds from left to right in the 2 direction increasing where 2 is the axial coordinate. The coordinate i is perpendicular to the bounding planes and j is the other coordinate. The equations which govern the motion of the fluid in the channel are

(2.la)

(2.1 b)

?V ~

?t^ + V . Vfi = -Vk + e ip + vAV,

div V = 0 in

$ =[2, j ,21-m < x = , j < c o , - + d l i l ) c l ]

V( i , i,, *+ d, r ) = 0.

k ( i , i,, 2, r ) - P i ,

and (2.lc)

Here p > 0 is a constant whose value is determined by the applied pressure drop and the total pressure at a point in the fluid is

where k(2, j , i, r ) is uniformly bounded in Y .

dimensional variables We shall work with a dimensionless statement of the problem (2.1). The

[i, j , i, ;, Pi, P,, Pi , k, P ] are obtained from dimensionless variables

[x, y, z, t , K , v, , v, , .n, P]

d, d, d, d2 v v v 1’2

by multiplying the dimensionless variables by the scale factors

The domain occupied by the fluid, described in dimensionless variables, is

f r = [x, y , z I - cc < x, 4’ < co, -$ _< z I +I. We shall need to designate the horizontal average

246

and the over-all average

Daniel D. Joseph

. 112

( f ) = 1 .frlz. .' ~ 112

It will be convenient to introduce a Reynolds number based on the average of the velocity over a cross section

(2.2)

The average velocity is proportional to an average mass flux. In a loose terminology we shall call ( V , ) a mass flux.

In dimensionless variables the equations which govern the motion of the fluid are

(2.3a)

div V = 0, (2.3b)

(2 .3~)

(?V/&) + V VV = -Vn: + e,P + AV,

V(x, y , _+f, t ) = 0. The simplest solution of (2.3) is laminar Poiseuille flow

v = (VX, v, 3 K) = [U, (z ) , 0,0], where U,( z ) is the function

U , ( z ) = (U,)$(l - 4z2) 3 <U,>UO(Z),

which solves (2.3) when

P = P,[U,] = - (d2Ux/dz2) = 12(U,).

111. Global Stability of Laminar Poiseuille Flow

When R is small, laminar Poiseuille flow is stable and unique. To prove this, consider the difference between any solution of (2.3) and the laminar solution (2.4). This difference we call a disturbance of laminar Poiseuille flow

u = V - e ,U, (z ) = (u, c', w).

The disturbance satisfies the evolution equation

(?U/?f) 4-

and u and n: are functions :

H = [u, n:

U x e x - Vu + u * VU,e , + u * Vu = -Vn: + Au, (3.la)

elements of the space H of kinematically admissible

Response Ciwes , for PIanc Poisetrille Flow 247

where AP(x, y) designates almost periodic functions of s and J-.? Initial conditions u = u,, for (3.la) are elements of H .

The proof of global stability of Ux(y) starts from the energy identity

3(11/11t)( 1 u 1') + (u * vu, - u) = -( 1 vu (2). (3.2) The energy identity follows by integration by parts of the equation ( u * (3.la)) = 0, using the properties of H . The first term of (3.2) is proportional to the energy, the second term is an energy-production integral, and the third term is an energy-dissipation integral. The production integral may be written as

(u - vu, * u) = - 12( U , ) ( Z W U ) = -24R[U,](z,vu). (3.3)

We shall define the values

(3.4)

and

A = min[2( 1 vu I"/( I u I')]. (3 .5) I1

Energy Stahilitj, Theorem. Suppose that

24R[ U x ] < 2 = 24R, (3.6) then

( 1 u)') I ( I u,, )')exp(-;l[l - (24R/I)]tJ. (3.7) The field which solves problem (3.4) gives the form of the disturbance whose energy increases initially at the smallest value of

R > 2/24, (3.8) Reinark. 2 > 96. Hence, global monotonic stability holds when R < 4. Numerical computations (Busse, 1969) show that 2 = 24Rc 2 793.6 corresponding to R, E 33.3. Proof. The inequality (2.7) follows directly from (3.2) written as

d( I u (')/dt = -2( 1 VU / ' ) ( I - 24R(zw~r)/( I VU 1')) I -2( I Vu 12)(1 - 24Rb).

An almost periodic function is a uniformly boundcd function whose value at an .Y, !'point (for fixed t and z j is repeated, nearly, at some distant point of the Y , 1, plane. The graph of an almost periodic function may appear chaotically irregular. The overbar average of an A P function always exists and, in fact, forms the natural scalar product for thc space of almost periodic functions; an A P ( s , y j function f ' ( . ~ , J,) vanishes uniformly if and only i f , f 2 = 0.


If (3.6) holds, then d( 1 u I2)/dt I -( I u I2)A(1 - 24R/;?)

and the inequality (3.7) follows by integration. Suppose that initially the disturbance is in the form given by the solution of the maximum problem (3.4). Then if (3.8) holds,

d( I u (’)/dt = -2( 1 VU 1 2 ) ( 1 - 24R/X) 2 0. This proves the theorem.

flow which is possible when R < 1/24 z 33.3. The result just given shows that the laminar flow (2.4) is the only steady

To prove that 1 > 96 we note that when -3 I z I 0:

with an identical inequality for w. Then 0

u2 I ( z + 4) [

I 4 I z 1 ( z + $) 1’

I vi 12 dz‘ - - 1 / 2 ~

with an identical inequality for I IV 1’. Then zwu I I z 1 (W21TZ)”* I $ 1 z I ( w 2 + 112)

0

I v u 12 dz‘ . - 1 / 2

(3.9)

and

0

i (1/96) 1 I Vu l 2 dz‘. (3.10a) . ~ 1/2

A similar argument relative to the wall at z = $ gives . l / Z

z’wu dz‘ I (1/96) 1 I Vu 1’ dz‘ ‘ 0

(3.10b)

Adding (3.10a) and (3.10b) we find that

(zwu) I (1/96)( I Vu 1 2 ) . Better a pviori estimates than 1 > 96 can be derived using the Fourier series for AP functions and the constraint div u = 0.

In the proof of the energy stability theorem, we compared flows in a family characterized by one and the same constant part of the pressure gradient P = P, . This pressure gradient corresponds to the unbounded (as x --f ? m )

Response Curves ,for Plune Poiseuille Flow 249

or non-almost-periodic component of the pressure. There is no loss of generality in dividing the different flows in this way; we get all the possible flows by allowing P to range over positive values. We could just as well have considered families of flows having different pressure gradients and the same mass flux (V,) = (U, ) . Though a term ( P - Pl )e , would then appear in (3.1a), it would again disappear in the energy identity (3.2) because (u) = 0.

The energy criterion guarantees monotonic and global stability (monotonic stability to all disturbances when < 33.3). This criterion gives sufficient conditions for stability but is silent about instability. In fact, experiments (see Fig. 1) suggest that all disturbances eventually decay when R < R , 'v 650. There is certainly a best limit R , for global stability. Theor- etical methods for obtaining the value & are at present unknown.

IV. The Fluctuation Motion and the Mean Motion

The motion of the fluid in the channel may be decomposed in several ways. In Section 111 the motion V was divided into laminar flow U,(y) plus a disturbance u(x, y, z, t) . Now we want to resolve this motion into mean and fluctuating parts

[V, 7c] = [v + u, 71 + p ] , (4.1 ) where the fluctuations u and p have a zero mean U = p = 0.

equation is that An elementary consequence of the horizontal average of the continuity

v, = 0. (4.2) To obtain the equations for the mean and fluctuating motion, insert (4.1)

into (2.3). Using (4.2) we find that a v a u + ~ + v * [VU + u v + UU] at a t

d 2 V ?Z ?Z2

Î-

(1 7c = -Vp - - e , + Pe,+ + Au.

The average of (4.3) is

(4.3)

(?V/?t) + V . Uu = -e,(&/;iz) + e,P + ( i2V /Sz2 ) ,

((:U/?f) + V . (VU + uv + uu - uU) = -Vp + Au,

(4.4)

(4.5)

and the difference [(4.4) - (4.3)] is


where V * u = 0 in f and, a t z = *$,

It is possible to consider an initial-value problem for the mean and fluctuating motions. Since the initial velocity field (V,, v,, il) is arbitrary, we may assign ( V , , Vp, 0) and (u , c, w ) arbitrarily, provided only that the fluctuations have a zero horizontal average.

Associated with the decomposition into a fluctuation and mean motion are two energy identities, one for the mean motion and one for the fluctuation motion. These identities follow from ( V . (4.4)) = 0 and (u * (4.5)) = 0, respectively:

(4.7)

The total energy is

The energy source for turbulent Poiseuille flow is P( V , ) work of the pressure gradient on the mean flow.

V. Steady Causes and Stationary Effects

We face the task of describing in some useful sense all the solutions of the Navier-Stokes equations that can arise when the steady external conditions are those giving rise to Poiseuille flow. Some progress with this hard problem can be made if we admit the basic assumption that steady external conditions can have a stationary effect even when the motion is fluctuating. This assumption is supported by the consistent reproducibility of certain average values in turbulent flow.

The stationary effects of steady external conditions need not imply unique steady solutions. Only when R < RG d o all flows tend to a unique steady

Responso Ciirzw for Plane Poiscwillt~ Flow 25 1

flow. When R > R , , there are at least two solutions possible: the unstable laminar flow and any one of the motions that replace laminar flow. The limiting flows that actually occur when R > RG are those that are in some sense stable. The stable solutions with R > R, need not be unique; indeed we envision stable sets of solutions. Though such solutions would lack uniqueness in the ordinary sense, it is consistent with observations to postulate the existence of stable sets of solutions sharing common properties in the average.

The basic property we shall assume here is (1) that all horizontal averages are time independent. This assumption says that a consequence of steady exterior conditions (boundary conditions and pressure drop) is that horizontal averages are steady. The fluctuation fields themselves can be very unsteady. We also shall assume (2) that velocity components have a zero mean value unless a nonzero mean value is forced externally. Property (2) implies that v, = 0. Following Howard (1963) we call fields which share properties ( 1 ) and (2) statistically stationary.?

VI. Laminar and Turbulent Comparison Theorems

Assuming properties (1 ) and (2) of statistical stationarity, equations (4.4) may be written as

d / L / Z [ H ’ I l - ( d V , / d z ) - Pz] = 0 (6.1) t / /dZ[l lT] = 0, (6.2)

@ l Z [ W 2 + 711 = 0, (6.3) and Eq. (4.8) becomes

( W L l ( d V Y / d Z ) ) = -( I vu 12). (6.4) Equation (6.2) shows that Kr is a constant whose value is zero at the boundary and elsewhere.

A basic and important consequence of statistical stationarity is that (6.1) has a first integral

w11 - ( ,cu) = Pz + (LlVJCk). (6.5)

i The observation that steady exterior conditions should be expected to lead to stationary turbulcnce needs qualification. Stationary turbulence evidently cannot exist in Hagen- Poiseuille flow and plane Coiictte llow when the Iluctuations arc infinitcsimal (the lincari7.cd stability theory shows that t r l l infinitesimal disturbances decay). The analysis given here applies to stationary turbulcnce when it exists. I t should also be noted t h a t motions which are here called statistically stationary nccd not be turhulent. Steady laminar inotiohs fit our definitions and arc to be included in the class of statistically stationary turhulcnce.

252 Duniel D . Joseph

Using (2.5) we may write this integral as

WLl - ( w u ) = (P[ V,] - P,[U,])z + (d/dz)( v, - U.r).

P(zwu) = ( 1 v u ( 2 ) + ([w - ( \ V U ) ] 2 ) .

( Z W M ) = (P[V,] - P,[U,])/12 + ( U , - V,) > 0.

(6.6)

(6.7)

(6.8)

Combining (6.4) and (6.5) we find that?

Forming (z(6.6)) = 0, we find, using (6.7) that

Equation (6.8) relates the pressure-gradient discrepancy P[ Vx] - Pi[ U,] and the mass-flux discrepancy ( U , - V , ) and forms the basis for the following laminar-turbulent comparison theorems.

hilass-Flux-Discrepunc~, Theorem. Statistically stationary turbulent Pois- euille flow has a smaller mass flux (( U~y) > ( V , ) ) than the laminar Pois- euille flow with the same constant component of the pressure gradient (P[P,.] = P,[U,]). This theorem was first proved by Thomas (1942); the proof given here is essentially due to Busse (1969, 1970).

Pre.s.sure-Grudiozt-Disc.repa/?cy Theorer?7. Statistically stationary turbulent Poiseuille flow has a larger pressure gradient P[ V, ] > P I [ V , ] than the laminar flow with same mass flux (( U,) = ( V , ) ) .

For the case of equal pressure gradients one may prove the following theorem.

Equal-Sheur-Stress the ore^^. Suppose that P[ v,] = Pi[ U,] and V , ( z ) is an even function of z, then

d V , / d Z = (LIU, /dZ) I:=+ ,,2 .

In Fig. 3 we have sketched the comparison for the equal-pressure-gradient case.

f

F"i. 3. Mass-flux discrepancy theorcm. I f the prcssurc gradicnt is fixed, thc laminar llow U , and turbulent mean flow V , have the same slope at the wall. The mass efflux of the laminar flow exceeds the turbulent flow by a n amount ( C j r - V , ) = (xw) > 0.

+Reynolds (1895) used Eq. (6.7) to find critical vnlucs of 1'. He notes tlic presence of the quartic integrals makes (6.7) " . . . very complex and difficult of intcrpretalion except in so far as showing that the resistance varies as a power of velocity higher than the first."

Resporise Ciirces for Plarzr Poiseuille Flow 253

VII. Turbulent Plane Poiseuille Flow-An Upper Bound for the Response Curve

Busse (1969, 1970), following earlier work of Howard (1963) on turbulent convection, and Howard (1972) have considered the variational problem for the response curve which is implied by (6.7). In a convenient formulation one introduces the mass-flux discrepancy

p = ( U , - V , ) = (zwu) > 0

as a parameter and seeks the minimum value

F ( p ) = min Y[u; ,u] H

of the functional

(7.1)

(7.2)

over the space I7 of kinematically admissible fluctuation fields; H is the subspace of H with zero mean values U = 0.

The values P(p) give a lower bound for the pressure gradients possible in statistically stationary turbulent Poiseuille flow. The reader is referred to Busse (1970) and to Howard (1972) for further details. We note when , ~ i = 0, the variational problem for p - (0) coincides with the problem (3.4) except that the competitors for the maximum of (3.4) do not need to have U = 0. The zero mean condition is satisfied by the winner of the competition (3.4) and therefore

P(0) = 1 2 793.6.

t G / t l p = ([\w - (wlf)]~)/(zwLf)2

(7.3) Using a method given by Howard (1963) one can show that

and @/dp is a decreasing function of p. A complete mathematical solution of (7.1) is not yet known [see Howard

(1972) for the most recent discussion]. The solution seems to generate repeated bifurcations as p is increased in a manner reminiscent of the Landau-Hopf conjectures discussed in the Introduction (see Joseph, 1974).

Fortunately it is possible to construct u priori estimates of the solutions of

254 Darziel D. Joseph

(7.1) which give explicit upper bounds for the response function. One such estimate is constructed below.?

The response function P ( p ) , ,LL = ( U , - V , ) , satisfies the inequalities

P ( p ) 2 P ( p ) 2 &)

where p* = [ P ( O ) + 48]’/576.

The bound (7.4) is reinterpreted in terms of the friction factor and Reynolds number in Section X and is shown graphically in Fig. 2.

To prove (7.4) we must first establish the estimate

([wu - (WU) - I ~ z ( z w u ) ] ~ ) 576 (ZM’11)2 > D + 4 8 ’

where D = ( 1 VU 1 2 ) / ( ~ ~ ~ ~ { ) .

(7.5)

Assuming (7.5), for the moment, we note that from (7.1) and (7.2)

P ( p ) - 12p 2 D + 576p/(D + 48) 2 min[D + 576,~i/(D + 48)]. (7.6)

This minimum is attained when D + 48 = ( 5 7 6 , ~ ) ” ~ = 2 4 , ~ ~ ” ~ and is equal to 2(576p)”’ - 48 = 4 8 d ~ - 48. However, by (7.1) D 2 P(0) and therefore D + 48 cannot equal 2 4 / ~ ’ / ~ if 2 4 ~ ~ ’ ~ ~ < P(0) + 48; that is, if p < [P(0) + 4812/576 = p*. Therefore when p < p*, D + 576,~~/(D + 48) is an increasing function D which must be minimum when D has its smallest positive value D = P(0).

n

Hence we may continue (7.6) as

- jP(0) + 576,~/[1‘(0) + 481 -

\48& - 48 completing the proof of (7.4).

It remains to prove the estimate (7.5). We first note that the relation

( 1 w - (MW) - 122(zWLr)) = 0

l V l l - (wu) - 12i(Zl\.U) = 0.

, f ( C ) = 11’11 - (W.) - 12(< - +)(..U).

(7.7) implies that there is a value z = 2, -+ I 2- I i, such that

It is convenient to use the coordinate i: = z + $ and to define

.bThe est imate (7.4) is d u e t o V. G u p t a atid D. Joseph

Response Curws ,for P l m r Poisruille Flow 255

At t = i + 5 we havef 'f t) = 0. Define

.i - Doc = 1 1 VU la, d i ;

' 0

clearly

Do; = (Do,/Do1)Do1 = MDo,

D.1 = D01[1 - (Do:/~o,)I = ( 1 - " )Do , ,

and

where s( = Do;/Dol . Following the derivation of (3.5) we may find

and with [I = ( w u ) / ( z w u ) and D = D, , / ( zMw) ,

At < = 0 we have 1 f ( 0 ) I = y(0) = 16 - (ZWU). At i = 1 = 16 - [j I! (12 + + r D ) we have g ( T ) = 0. Since 1 f ( i ) I 2 g(i) when 0 < < < < [ f ( t ) = 01 we must have

1 > r. (7.10)

When ( < r, (7.8) holds and I f (0 I 2 I Y(<) 1 . Using (7.9) and (7.10) we find that


Analysis relative to the wall at z = f (introduce i = z - i) leads to an inequality of the form (7.11) with 16 - f l I replaced with 16 + /) I and ct replaced by 1 - c(.

We have

([wu - (wu) - 12z(zwu)]2) (zwt1)2

(7.12)

where 0 I c1 I 1 , -a < ,/I < cc. The minimum of the right-hand side of (7.12) is found at f i = 0 and a = f. Hence

2 5 76 - -

([X - (wu) - 1 2 z ( z w ~ ) ] ~ ) 2 63 (zLvu)2 3 1 2 + $ D D + 4 8 ’

This completes the proof of (7.5). A slightly better estimate than (7.4) which is valid for large values of the

mass-flux discrepancy p (but not small values) has been given by Busse (1969). Busse considers the variational problem (7.1) in a weakened class of fluctuation fields u which need not be solenoidal. He solves the Euler equations for this problem in the limit p -+ and finds that

P ( p ) > 12p + 96Ji/$ + O ( P - * ’ ~ )

instead of (7.4). The present estimate (7.4) holds for all ,u 2 0. Yet better estimates can be obtained (Busse, 1970) when additional assumptions are made about the form of the minimizing solution of the variational problem (7.1).

VIII. The Response Function near the Point of Bifurcation

Now we turn away from the energy estimates. We shall consider the linear theory of instability and the time-periodic solution which arises from this instability in the neighborhood of the point of bifurcation. We want first to show how to enrich the physical content of the perturbation theory by a proper choice of the amplitude parameter. This is accomplished by defining

Response Curves ,fix Plane Poiseuille Flow 257

the amplitude as a friction-factor discrepancy. To obtain an expression through which the friction-factor discrepancy may be related to the bifurcating solution, it is necessary to compare two different resolutions of the same motion I/:

V = V,(z)e, + u(x, y, z, t ) = U,(z)e, + u’(x, y , z, t ) , (8.1) where V, ( z ) is the mean motion, u is the fluctuation velocity, U,(z ) is laminar flow with mass flux ( LIx), and u‘ is the disturbance of U,( z ) (formerly called u). Equations (8.1) imply that

w = w’, v = d, and V , + u = U , + u‘. (8.2) The relation

V, ( z ) = U,( z ) + U’(z)

follows directly from the horizontal average of (8.1). We note that the equality (8.1) leading to (8.3) is possible only if the bifurcating flow is statistically stationary (the stationary property is verified in the remarks closing Section IX).

We shall make use of the following relation: ~-

m = u’wI. (8.4) To prove (8.4) we use (8.3) and (8.2) to write

~~~~~

r w = K4’ = ( U , - I/, + U’)W’ = ( U , - V,)W’ + Ti‘ = fir.

Using (8.4) we may rewrite (6.8) as

(ZvV’Ll’) = &(P[V,] - P , [ U , ] ) + ( U x - V , ) . (8 .5)

This basic relation will be used to relate the time periodic bifurcating solutions to the response diagrams measured in the experiments.

In Section IX we shall construct statistically stationary periodic flows which bifurcate from laminar Poiseuille flow at the critical point of the linear theory of stability. We will restrict our analysis to two-dimensional motions. This restriction is actually forced by Squire’s theorem; this theorem shows that the solution which bifurcates at the smallest value of R is actually two dimensional. The bifurcating solution is unstable and would not be easily observed in experiments. The relation of the two-dimensional solutions to experiments will be discussed in Section X.

For any two-dimensional motion we may introduce a stream function q. The resolution of the motion into Poiseuille flow with mass flux ( U , ) plus a disturbance

3 = ‘T’ + ‘I”, U , = ? q / ? z , (u’, - w ’ ) = [(d’€“/?z), (?‘€”/?.x)] (8.6)

258 Duriirl D. J o s e p h

leads to the following problem for Y ’ :

2 AY’ + ( U,)( I/&) i’

?t (’X + J(Y‘ , A”’) - A 2 Y ’ = 0,

dz2 r3x

(8.7a)

where

?Y1 i ?Z ?.K ?s ?Z

J ( Y ’ , AY’) =

The total mass flux ( v,) of any two-dimensional statistically stationary solution of (8.7) may be written as

where ( U , ) is the mass llux for a suitably chosen laminar flow with pressure gradient P,[U,]. The bifurcating flow and laminar flow have the same mass flux (U , ) = (V,) if and only if Y’ I==

It is convenient, and completely general, to restrict one’s attention to the special case Y’ I I ,z = 0 in the construction of the bifurcating solution. To completely specify the bifurcation problem, it will be necessary to fix the spatial periodicity. Then the time-periodic solution which bifurcates from laminar flow is determined uniquely to within an arbitrary phase. This unique solution may be computed relative to a laminar flow for which ( U , ) = ( V , ) . To show this we will now reduce problem (8.7) to the study of bifurcation of laminar flow with ( U , ) = ( V.x) .

The stream function for the bifurcating solution may always be written as

= 0.

+ = e(,) + Y’, Y’ = aqz) + Y”, (8.9)

where O(z) is a function of z alone which can be chosen so that

@( --+) = do( - $ ) / d Z = d @ ( + ) / d Z = 0, a($) = (U’), (8.10)

and ‘f”‘(x, z, t ) is a flow satisfying (8.7a)-(8.7c) with zero mass flux

( ( ? Y ” / ? z ) = Y” I _ = ] ,2 = 0. (8.1 I )

R P S ~ ~ I I X Curves ,for Plane Poiseuille Flow 259

@ ( z ) is a Poiseuille flow. The mass flux for the Poiseuille flow q ( z ) + @(z) is ( V , ) . Moreover,

and

(z1v”u”) = &(P[ V,] - P/[ V , ] ) . (8.13)

Proqf: Substitute (8.9) into (8.7) to find that d4@/dz4 = 0 and, using (8.10)

Q, = $(.’)(z - $2) + +(U’). (8.14)

I t follows that

(?/?z)(P + Q,) = U , ( z ) + tlQ,/dz

= $( u, + .’)( 1 - 422)

= +( VX)(1 - 4z2) = ( V , ) V , ( z ) .

The pressure gradient for this laminar flow is

P,[ us + 11’3 = - ( d 3 q + @)/dz3 = 12( V , ) = P,[ V , ] .

This proves (8.12). Noting next that (w’lf’(z)) = 0 for any function of z alone, and using (8.9) and (8.12), we may rewrite (8.5) as

A ( P [ V , I - P/[U,I) + ( U . Y - V , ) = &(P[ V,] - P,[ V , ] ) = (zw‘u’)

= (zw’”u” + (dD/dz ) ] ) = (zw”u“),

proving (8.13).

from (8.7) using (8.9) and a suitably scaled stream function Y : The bifurcation problem for the double-primed variables may be obtained

Y” = 2( V,)Y = 21:RY, (8.15)

where I?[ V,] = I?[ V,] and

C Z ( V,)2( Z\YU) = A(P[ V,] - P,[ V , ] ) . (8.16)


Now, by choosing a normalizing condition in the form

( Z W U ) = 1/48

we find that

c2 = .f - i ; (8.17)

is the friction-factor discrepancy where, in dimensional variables,

(8.18)

and D, = 21 is the hydraulic diameter (the ratio of four times the area of the cross section of the annulus to the wetted perimeter and P = pl3/1?).

With these definitions established we may now state the bifurcation problem in terms of the friction-factor discrepancy:

d 2 U , dY 1 + c J ( Y , A'€') - -A*'€' = 0, [CO CIS ' + U,(Z);~) AY - d Z 2 dx 2 R

(8.19a)

'€' = d'€''/?z = 0 I = = + 1 1 2 , (8.19b)

Y is 2n periodic in EX and s = wt, (8 .19~)

( Z W U ) = 1/48. (8.19d)

Equations (8.19) are a complete statement of the mathematical problem for the bifurcating solution. With a given we seek solutions

[Y (x , z ; E , cc), R(cZ, a), crl(&z, .)I (8.20)

of (8.19). Joseph and Sattinger (1972) have shown that the bifurcating solution is necessarily time periodic when the solution of the spectral problem associated with (8.19) is unique and time periodic. The solution (8.20) of (8.19a,b,c) may be developed in a series of powers of c. The normalization (8.19d) leading to the definition (8.17) was given by Joseph and Chen (1973).

The spectral problem for (8.19) may be reduced to the familiar Orr- Sommerfeld problem; uniqueness of solutions to the Orr-Sommerfeld problem means uniqueness to within a multiplicative constant fixed by normalization; time periodicity refers to the fact that at criticality

Rrsponse Curces for Plurir Poiseuille Flow 26 1

o = icu,, = ico(0, a) # 0. Yih (1973) has recently shown O I ~ / X lies between the maximum and minimum values of U(z). t

IX. Some Properties of the Bifurcating Solution

It follows from the theory of Joseph and Sattinger that the solution (8.20) of (8.19) may be developed as a power series in c:

where 2 = 1/R. This solution is unique to within a n arbitrary change in the

The earliest studies of the spectral problem of linearized theory Icd to contradictory results with some investigators claiming stability and othcrs instability. Lin (1945) following the asymptotic theory developed by Heisenberg (1924) and Tollmein (1929) calculated a critical Reynolds number. Lin’s results were chccked and confirmed by the numerical computation of L. H. Thomas (1952). Nowadays critical values and cigenfuuctions for the spectral problcni are computcd by digital computation.

The earliest nonlinear studies of the stability go back to Meksyn and Stuart (1951). They used an energy approximation in which the shape of thc bifurcating solution is taken a s given by the critical eigenfunction ofthc spectral problem; the unknown amplitude of the disturbance is then determined by the over-all energy balance. Stuart (1960) and Watson (1960) gave a formal algorithm for dcterinining finitc amplitude solutions for disturbances of Poiseuille flow and other parallel flows. Their construction presumes an amplitude equation which at lowest significant order reduces to Landau’s (1944) equation. Landau had conjectured the form of this amplitude equation in thc context of his more general conjectures about transition to turbulence through repeated supercritical branching. The perturbation method of Stuart and Watson was extended and clarified by Reynolds and Potter (1967): they computed the first bifurcation results for plane Poiscuille flow: more computational results were then given by Mclntire and Lin (1972).

Closely related to Stuart’s (1960) formal method o f amplitude expansions is the forrnal theory of Eckhaus: this method was applied to the problem of bifurcation of plane Poiscuille Ilow. A relatively thorough review of these formal theories and more refercnccs can be found in the review paper of Stuart (1971).

Rigorous theories of bifurcation of time periodic flow from steady flow, assuming simplicity of the principal eigenvalue of the spectral problem at criticality, have been by Yudovich (1971) Iooss (1972) and Joseph and Sattinger (1972). Thc work of Joseph and Sattinger extends the Hopf (1942) bifurcation theorem for systems of ordinary differential equations to systems of partial diffcrcntial equations. Hopf‘s results are complete and rigorous. Hopf felt that his results might apply to partial differential equations, and he refers to the Taylor problem and other famous problems of hydrodynamic stability. The work of Yudovich extends the theory of Lyaponouv -Schmidt to the case of bifurcation from a simple, complex-valued cigenvalue. His theory has becn applied to the problem of bifurcating Poiscuille flow by Andreichikov and Yudovich (1972).

262 Daniel D. Josrph

origin of the time. The odd-order derivatives all vanish:

( 0 2 / , + I = 3.2,+ 1 = 0.

The first nonlinear corrections are given by the values w2 and A 2 .

TABLE 1

NONI.INIIAK STAnIl.I-rY CI IAKACTERISTI(’S Ot PI .ANt POlSELJll.l.1~ FLOW

0.650 14950 0.700 10903 0.750 8307 0.800 6588 0.850 5427 0.900 4643 0.925 4360 0.950 4138 I .000 3877 1.021* 3848 I .050 3927 1.075 4209 1.090 4683 1.095 5075 1.0964 5298 1.0964 6238 1.095 6597 1.090 7478 1.075 9538 I .050 12907 1.020 17573 1.000 21264

0.2483 0.2734 0.2975 0.3204 0.341 7 0.36 12 0.3701 0.3782 0.39 I8 0.396 1 0.3996 0.3988 0.3935 0.3886 0.3858 0.3745 0.3705 0.36 I5 0.3438 0.322 I 0.3008 0.288 I

192.37 158.45 133.03 113.35 97.59 84.44 78.50 72.80 61.37 56.12 47.43 36.32 23.67 14.60 9.70

- 10.01 - 17.33 - 35.00 - 75.58 - 140.66 -228.75 -296.80

-0.3798 -0.371 1 -0.3426 -0.2860 -0.1883 -0.0280

0.0870 0.2356 0.7014 I .0220 1.7650 3.3149 6.9112

13.32 I4 21.7853

-27.6987 - 17.3983 - 10.2626 - 6.4708 -4.9225 -4.2075 - 3.9373

311.89 264.38 241.39 237.45 25 1 .56 287. I7 316.11 355.89 492.57 595.05 852.01

1442.00 292 1.63 5660.52 9322.37

- 7885.29 - 12344.70

-4840.71 -3305.22 -2755.16 -2536.25 - 2450.08

- I’ R = (Vy) d’2r (based on average velocity and half-height) * Minimum critical point.

The values of I” , and o2 for Plane Poiseuille flow were computed by Chen and Joseph (1973); they use a different definition of c. The values of A2 and w2 using the friction-factor discrepancy to define c [Eq. (8.17)] were computed by Joseph and Chen (1973) for flow through annular ducts. In Table 1 we have listed values of the parameters for plane Poiseuille flow when 1;’ = j - f; . The working equations for these computations will not be given here. The solutions to the perturbation problem at zeroth order (c = 0) depends on x and t only in the combination cxx + (of = 0. This property is retained in the higher-order solutions. I t follows from this that the bifurcating solutions are statistically stationary.

A most important result of bifurcation studies is the determination of the

Respoiise CLirues for Plane Poiseiiille Flow 263

direction of bifurcation. Bifurcating solutions which exist when the point (A, u ) is in a region where laminar Poiseuille flow is stable according to the linearized theory of stability are called subcritical. Supercritical solutions exist for points (A, u ) for which laminar flow is stable.

The Floquet stability analysis (Joseph and Sattinger, 1972) of the time- periodic flow which bifurcates from laminar Poiseuille flow shows that subcritical bifurcating solutions are unstable when [2(c2, a), c] is near the point of bifurcation (A,, 0). Supercritical solutions are stable to small two- dimensional disturbances of the same basic periodicity as the bifurcating solution.? This sense of stability is too weak to be physically meaningful.

Referring to Table 1 we note that solutions for which i 2 Z 1 > 0 are subcritica1.J In particular the solution which bifurcates first (the one marked with an asterisk) is subcritical and unstable. This unstable subcritical bifurcating solution is shown in Fig. 1 as a dotted line on the left emanating from the point of bifurcation. By choosing other values ofcc, and in other ways, we can show that an unstable solution bifurcates subcritically from every point on the laminar flow line (the 45" in Fig. 1 ) when R > R , = R(0, 9,).

X. Inferences and Conjectures

Figure I is a bifurcation diagram in the plane ( R , f ) of the response curve. The experimental points, shown as circles, are the values observed by Walker et ul. (1957) in experiments on turbulent flow in annular ducts. The ratio of the inner to outer radius is 111.01, and we shall accept that this configuration is sufficiently representative of flow in a plane channel. In this regard we note that values computed by Chen (Table 1 ) for plane Poiseuille flow are only very slightly different than the values for the annular duct of Fig. 1 (see Joseph and Chen, 1973). The coordinates Re, andf2 used by Walker et ul. are related to R andf by

R = ReJ4, .f'= 4f2 .

The response curve for laminar flow,f, = 24/R appears as a straight line in a log-log plot:

d lnj;/d In R = - 1.

The 45" laminar line is a lower bound for the friction factors in all possible motions with steady average values.

t Hopf (1942) was the first to prove that subcritical bifurcating solutions are unstable and that supercritical bifurcating solutions are stable. His construction for the Floquet exponents is not clear; a clear construction has been given by Joseph aiid Sattiiigcr (1972).

t < is the real part of m(2.) for solutions of the linearized cquatioiis proportional to c - " ' ~ ' ' .

2, = c / ? ( i ) / c / ; . when 7. = io = 2(0, u ) = R-'(O, 3) .

+^- .


In Fig. 2 we have also graphed an upper bound for friction factors in statistically stationary turbulent flow. To convert the bound (7.4) into ( R , f ) coordinates we note that (7.4) holds for flows with equal pressure gradients:

P[V,] = P,[U,] = 12( U x ) .

f = P/R2 = 12( U,)/R2

,Ll = (UJ - ( V , ) =fR2/12 - 2R 2 0.

Then, using (8.18) and ( V , ) = 2 R we note that

and

These two relations are used to change variables ( p , P ) + (R, , f ) in (7.4); we find that

(10.1) (24/R) + (12/R2)(R, + 2)(R - I?,) for for R 2 7 ,

R, I R I ;' f < f = ' - l 3 + (36/R) + (12/R2)

where y = 2(R,; + I), R,: = 33.3.

only if Summarizing; statistically stationary flow through a channel can exist

,f; = 24/R < , f ( R ) < , T ( R ) . (10.2)

The slope of the response function for the time-periodic bifurcating solu-

d Inf/d In R = - 1.0430. (10.3)

This slope appears as the dashed line in Fig. 1 (the bifurcation is subcritical and is therefore unstable).

tion at the minimum critical Reynolds number is (see Table 1)

The slope (10.3) is computed as follows: One notes that

= f - . f i = (2 - i0)/iL2 + 0([;4).

Then, using (10.2) we find that

Inf = 1n.h + In{ 1 + I/24/12(a)[ 1 - R(e2, u)/R(O, u ) ] } + O(c4).

Differentiation with respect to In R(c2, a), when u and R(0, M ) are fixed gives

d ln,f/d In R = - I - [24/12(a)]-'. (10.4)

Bifurcation always takes place above the 45" laminar line. When the slope is negative, the solution bifurcates to the left; when the slope is positive, the solution bifurcates to the right.

It is clear from (10.4) and Table 1 that even when three-dimensional bifurcating solutions are disallowed, there are a continua of solutions with R(M, q ) 2 RL which bifurcate in every direction above the 45" laminar line.

Response Cur iw for Plane Poiseuille Flow 265

Many of these solutions may be closer to the laminar line when c # 0 than the solution which bifurcates first at a = x , . Further study of bifurcating solutions in two dimensions requires the computation of the envelope A(E’, a(c2)) of bifurcation curves a ) depending on the parameter x. The envelope condition

?I”/& = 0 ( 1 0.5)

is to be an identity in c2. Points on the envelope give the smallest values R(c2, a(c2) ) = iL- (c2, u(c’)) for which two-dimensional bifurcating solutions can be found. At second order in powers of c2 we have

~ ( a ~ , a(E2)) = ~ ( o , .,) + 2 , (42 + [ I ~ , ( ~ , ) + ~ ~ , ~ ( 4 ~ ~ 1 ( : 4 + o(P), ( 10.6)

where the derivatives

are all evaluated at I: = 0, a = a*. These derivatives can all be computed by analytic perturbation theory; in particular, the value of r 2 arises from differentiation of the identity (10.5) with respect to I:, at c = 0:

where

We shall now summarize what can be inferred and conjectured from analysis and experiments about the response curve for flow through chan- nels. The major points are most graphically explained as an interpretation of results shown in Figs. 1 and 2.

First, when R < R,: z 33.3 (or,f>,fi rr 24/R,) all disturbances of plane Poiseuille flow decay monotonically from the initial instant; if R,: < R < R,;, then all disturbances decay eventually but the decay need not be monotonic. Plane Poiseuille flow can be unstable when R > R,; . Experi- ments suggest that R, z 650. For R, < R < R,* laminar plane Poiseuille flow is stable to small disturbances; the experiments indicate that the stable disturbances must be very small. If natural disturbances are suppressed, however, one can achieve laminar flow with R, < l? < R, . For R > R , , laminar flow is unstable. A t R = R L , a time-periodic solution bifurcates from laminar flow. The bifurcating solution is subcritical and unstable. Un- stable bifurcating solutions cannot attract disturbances and solutions which escape the domain of attraction of ianiinar Poiseuille flow snap through the


bifurcating solutions and are attracted to a stable set of solutions with much larger values of the friction-factor discrepancy.?

The experiments suggest that when R > R G , there are a stable set of solutions, called stable turbulence, which appear to share a common response curve. The surprising and noteworthy observation is that at a given R > R , it is possible to reproduce the same value, as far as “sameness” can be ascertained from experiments, of the friction-factor discrepancy. The sur- prise stems from the fact that the solutions which are observed are turbulent and all different; despite this, each of these infinitely many turbulent solutions leads to an apparently common value of the friction factor.

Assuming that the stable turbulence is statistically stationary, the response function is bounded from above by the estimate (10.5) which arises from estimates of functionals defined in the variational theory of turbulence. The response of statistically stationary solutions is bounded from below by the response of laminar flow (the 45‘ line in Figs. 1 and 2). There are surely very many statistically stationary solutions in the region between the 45‘ line and the upper bound. The bifurcating solution shown in Fig. 1 is but one example; there are also at least a continua of solutions depending on E with R(0, z) > RL = R(0, E * ) which bifurcate subcritically. Many of these solutions are demonstrably unstable and all of them may be unstable.

The significance of the solution bifurcating from Poiseuille flow at the lowest critical value RL is that it is the first solution to bifurcate. The heavy dashed line shown in Fig. 1 gives the slope of the response curve of the two-dimensional bifurcating solution at the point of bifurcation. The computation of the actual curve requires, at least, the computation ofmore terms in the power series for 3,(c2, E) .

The physical significance of the bifurcation analysis is limited in the case of subcritical bifurcations by the fact that subcritical bifurcating solutions are unstable when t i2 is small and therefore cannot be achieved in permanent form. In addition, there are many who hold that the bifurcation analysis is of little physical significance since the theory leads to the conclusion that two- dimensional disturbances bifurcate first, and two-dimensional disturbances are not observed in real turbulent flows. This criticism ignores completely the fact that this same criticism could be applied to all of the unstable three-dimensional solutions which exist in the shaded region of Fig. 2. In

t I t may be possible to observc the time-periodic bifurcating solution as ;I transient of the snap-through instability. Small disturbances of laminar Row with a fixed value of R(R,; < R < R , ) which arc marginally attracted from Poiseuille Row may take on the properties of the two-dimensional bifurcating solution; given R , the solution could be expectcd to oscillate with a frequency m(t?(R)) . The measured friction factor for this flow would be given by ./ ( R ) =

j , ( R ) +. ? ( R ) = 24,IR + i : ’ (R) . This transient periodic solution might exist for a t h e before being destroyed by instabilities.

Response Ciirves ,for P l m e Poiseuille Flow 267

fact, most kinds of solutions are not observed in turbulent flows; the description of what is observed fills tens of thousands of printed pages. The really striking feature of regularity in the observations is the response curve for stable turbulence. Solutions which may exist in the shaded region of Fig. 2 are not observed in experiments. They are unstable as in the case of time- periodic bifurcating flow or they are weakly stable with only a small domain of attraction as in laminar Poisseuille flow. At a given R only those solutions which have the value (or small range of values) of the friction factor on the response curve for stable turbulence seem to persist. These are the circles in Figs. 1 and 2.

I want to raise the possibility that the response curve for stable turbulence (the circles) can be achieved on subsets of a stable set of solutions of permanent form. I do not know what limits can be placed on the definition of solutions of “permanent form.” Examples of the kind of permanence I have in mind are periodic or almost periodic functions of time. The usual objection that physically realized turbulence does not have this or that analytical property specified by permanence is beside the point being made here. It is almost certain that observed turbulence is never realized on solutions of permanent form. The domains of attraction of the stable permanent solutions are probably too small to capture all the disturbances which occur in real flows. Given this, we should expect that at each instant the observed solutions are transients which tend now toward one and then another solution in the stable set of permanent form. Though the permanent solutions are not fully realizable, all the realized transients are attracted by permanent solutions which lie in the stable set.

The exciting possibility is that the statistics of observed turbulence could be computed on elements of the stable set ofsolutions of permanent form. If true, this would enormously simplify aspects of the analytical problem associated with turbulent flow. For example, we could investigate the possibility that the response curve for stable turbulence can be computed on the envelope of two-dimensional, time-periodic bifurcating solutions. Numerical studies (for example, see Zahn et al. 1973) suggest that the envelopef’(R) is a double-valued function of R. The lower branch lies close to the dashed lines representing the bifurcating solution in Figs. 1 and 2; the lower branch is almost certainly unstable. The upper branch is more problematic; experience with other stability problems suggests that the upper branch of the envelope is stable, at least to more disturbances than the lower branch. The interesting possibility is that the upper branch of the envelope is stable to small disturbances and coincides with the response curve for stable turbulence (the circles). The computation of higher terms in the perturbation series, and of (10.6) in particular, is one way, though not a decisive one, to further study the possibility.


Appendix: A Formal Bifurcation Theory for Nearly Parallel Flows

In this appendix I am going to present a formal theory of bifurcation for nearly parallel flows. This theory is based on a triple-perturbation series; it uses an extension of the Poincare-Lindstedt method (described in Section IX) to treat the nonlinear effects and the method of multiple scales to treat the effects of slow spatial variation of the main stream.

Multiple scale theories for treating the linearized stability theory of nearly parallel flows have been given independently by Bouthier (1972) and by Ling and Reynolds (1973). These theories make the quasiparallel approximation which leads to the conventional Orr-Sommerfeld theory at the zeroth order.

The earliest mathematical study of the effects of nonparallelism (Lanchon and Eckhaus, 1964) already indicated that though the quasiparallel approximation is valid in the case of the Blasius boundary layer, this same approximation could not be expected to correctly give the linear stability limit for flows like those in jets.? This observation appears to be sound and its impli- cations for perturbations are great. I f a flow is not well represented at the zeroth order, it cannot be approximated by perturbations. I t is for this reason that the perturbation method of Ling and Reynolds fails in the case of the jet when the wave number of the disturbance is small.

The problem of the correct zeroth order is basic in developing a perturbation theory which will apply equally to flows in boundary layers and jets. 1 believe that the solution of this problem lies along the lines laid out by the work of Haaland (1972). Haaland has noted that the difference between flows of the boundary-layer type and flows of the jet type can be characterized by the behavior of the velocity component P normal to the main stream at distances y + cci far away from the axis y = 0 of the main flow. The boundary layer grows by the diffusion of vorticity and does not require inflow from infinity. On the other hand, the conservation of the axial momentum of the jet

I

M = 2p (’ o2 rl P

together with the slowing of the jet with distance r? downstream requires the entrainment of new fluid. The spreading of the jet implies a nonzero inflow ( V # 0 ) at infinity.

In his study of the linear theory of stability of nonparallel flows, Haaland modifies the Orr-Sommerfeld theory to include some of the effects of inflow. The effect of the retention of these terms is to confine the vorticity ofdistur-

‘ 0

t Tatsumi and Kakutani (1958) note that the parallel llow approximation may not apply to jets.

Response Curves ,for- Plune Poiseuille Flow 269

bances to the regions of the main flow where viscosity is important and to prevent the spillover of vorticity into regions where the flow is essentially irrotational. These inflow terms make a big difference in the critical Rey- nolds numbers especially when the wave numbers are small (see Fig. 4).

The formal perturbation theory developed below allows for a certain flexibility in the choice of a zeroth order. For definiteness, however, we shall

1.2 -

* 1.0-

5 L w

0.8 - n

c

$ 0.6 - P

0.4 -

0.2 -

R

FIG. 4. Neutral curves for the Shear laycr and the Bickley jet (after Haalaiid. 1972). (a) Conventional Orr-Sommerfeld equation: (b) modified Orr-Sommerfeld equation.

consider the stability of Bickley’s jet and use Haaland’s linear theory as the zeroth order. The perturbation scheme then corrects this zeroth order for effects of nonlinear terms and of linear terms which are neglected at the zeroth order.

We begin with a mathematical description of the Bickley jet. Throughout the Appendix we denote differentiation by the “comma followed by subscript ” convention; e g ,

A,Av = ?2A,f?)b (1,~.

The Bickley jet satisfies the boundary-layer equations

ij0.x + P0.p = vu,i,-, u.2 + v j = 0; (A.1) the axial momentum M of the jet is conserved. In the local theory we fix our attention on a point f o which is an arbitrary distance downstream of the origin of the jet. We introduce a scale length L o , a scale velocity oo, and a

2 70 Daniel D. Joseph

Reynolds number R :

Lo = ( 4 8 ~ 2 ; \12/M)1’3,

oo = (3M2/32p2 jO v)1’3,

and R = O o Z o / ~ ~ .

Then we may write the similarity solution of (A.l) in dimensionless variables:

U(y, x) = fl’’ sechjj ,

I/ = /zW(y, x) = 2/2f(2fj sech2f4, - tanhj j ) , (‘4.2) where

,f= (1 + 6 ~ ) - ~ ’ ~ , x = AX, 2 = 1/R,

x = (2 - 20)/zo, y = F/Lo. A two-dimensional disturbance

- - u = Y,”, z j = -Y.,

of the similarity solution (A.2) satisfies

Z., + Ub., - q , , A U + /2(Wt,,. - Y,rAW)

+ S,,Z., + ‘P.,Z., - AAt = 0, (A.3) where

= A’P = Y,xx + Y.,., is the disturbance vorticity and

9 - + 0 ( , 4 ,

Equation (A.4) implies that the nonlinear terms in (A.3) become increasingly less important at large distances from the jet.

The conventional quasiparallel assumption requires that A U be replaced by U,,”,. and AW = 0. This assumption is usually justified by noting that on the neutral stability curve 1- is generally small; this is not the case near the nose of the neutral curves shown in Fig. 4. The quasiparallel assumption is not uniformly valid for jets; all of the coefficients which depend on the main flow in (A.3) do not vanish at large values of I y I ;

lim [ U , AU, LW, LAW] -+ [O,O, -2A, 01. ( A 3 ) y + * x

Haaland has shown that if the W terms are set to zero at the outset, as in the

Response Curves fiw Plarie Poiseuille Flow 27 1

conventional quasiparallel theory, the vorticity of disturbances with small wave numbers will decay much less rapidly than the vorticity of the main flow. On the other hand, retention of the inflow terms results in confining the disturbance vorticity to the jet. A rough and not fully correct argument demonstrating this point follows from comparing the asymptotic solutions of

[.t - 214'f,y - ).A[ = 0 ( A 4

Equation (A.6) arises from (A.3) by applying ( A S ) . Equation (A.7) also arises from (A.3) when the additional assumption W = 0 is made. Continu- ing the rough argument with yet another approximationJ'= 1 we find that the decaying solutions of (A.6) are in the form

e , p = c1 e ~ ( a x + i o t ) -YY

where c1 is a constant and

y = 1 + ( I + u2 + iw/L) ' " ,

" 9 - - u + iw/L .

whereas for (A.7)

For small values of CI, solutions of (A.7) decay much less rapidly than the vorticity of the main stream. In fact a)/)" + 0 as SI + 0 on neutral solutions of the Orr-Sommerfeld equation. In the limit SI + 0 the disturbance vorticity associated with (A.7), but not (A.6), exists deep into the region ofirrotationa- lity outside the jet. In contrast, the inflow W transports disturbance vorticity to the jet interior and the disturbance vorticity does not escape into irrotational regions of the main flow as in (A.6). A more refined argument, given by Haaland (1972), leads to the same result; neglecting the convection of disturbance vorticity leads to sharply different results i n the neutral curves for jets and shear layers when the wave numbers are small (see Fig. 4).

In preparing for the perturbation theory we call attention to the two spatial scales x and x = Ax, where x is slowly varying. We next introduce a frequency w and amplitude c:

5 = (or, 9 = c'~'((t, x, y ; x, n ) (A.8)

and note that


where

and V2Y = Y,xx + Y.??

P Y = 2Y,xx + iY ,xx .

( A + ).B + AP)Y + e N ( Y , Y ) + eAM"Y, Y) = 0,

Using (A.8) and (A.9) we may rewrite (A.3) as

where

/lY = OV2Y', + uv2Y.x - u,yyY,x;

BY = -wyyY .g + WV2Y,, - V4Y;

clv = W9%JT + U 9 ? Y x + UVZY,, - Y'.xu.,J + A U 9 W x - Au,xxY,.y - i 2 Y . J J x x

(A.lO)

- my wxx + AWPY,, - 2 A V 2 9 j Y - i2V4Y';

N(Y, Y ) = Y,yv2Y,x - Y . p P y ;

MyY, Y ) = Y,y9%Jx + Y,,V2yx + LY,y9AY,x

- Y.x 9l:'.Y., - Y.xv2Y,J - B",, .

The next step in our theory is to introduce the notion of a local solution qf' permanent form. This is a solution of (A.lO) valid near = 0 (local) which vanishes at infinity and is 2n periodic in EX and z (permanent). Local solutions of permanent form are different from the localized transient solutions which were studied by Stewartson and Stuart (1971) in their effort to explain turbulent bursts. We seek mathematical expressions which will describe the Tollmien-Schlichting waves which are frequently observed as a permanent feature in certain boundary-layer flows. Such waves first appear with zero amplitude at a critical distance down from the leading edge; their amplitude, wave length, and period may all change with distance downstream. At each station there is a characteristic amplitude, spatial period, and frequency which may be permanently maintained. In our construction we fix the wave number and allow the frequency to vary. Then at each station downstream, there is a family of nonlinear solutions depending on the amplitude and wave number. The values of c( which minimize R = l / l on permanent solutions of fixed amplitude, of course, depend on the position downstream so that our method of computation does not preclude (nor assume) spatially varying wave numbers or constant frequencies.

The construction of a local solution of permanent form proceeds by a method offalse prohlerns. This method is introduced so that we may pivot the perturbation series around a problem which can be computed by separating

Response Curves jbr P lum Poiseiiille Flow 273

variables but does not neglect inflow (the modified Orr-Sommerfeld problem studied by Haaland, 1972). This requirement then leaves extra terms which are proportional to 3,. 3. is small (at worst, in the shear layer 3, 4) but decidedly not zero. We just replace /1 with p in the extra terms; then we perturb with 111. The solution of the false problem coincides (formally) with the true solution when i = p .

We shall start the construction of local solutions of permanent form with a complete statement, followed by explanation, of the false problem:

( A + LB + pP)Y + c N ( Y , Y ) + CpMqY, Y ) = 0,

Y + O as j ' + f x ,

Y is 27c periodic in CIX and T ,

1 = -[V 9J, z:]. (A. 1 1)

The operators C'' and MI' are obtained from C A and M A by replacing 3, with p and 9' with 9". The terms involving p are the extra terms. The square bracket is used to designate the scalar product

where

and the overbar designates complex conjugate. The last condition of (A.1 I ) is a normalizing condition which defines r : ; explanation of this condition requires a little preparation.

The pivot problem for the perturbation may be obtained from ( A . l l ) by putting

[t;, ,LL, Z, A (0, u, W ] = [O, 0, 0, &oo , ( ~ j o o o , U o , Wo], (A.12)

where

U o = U(Y> O), Wo = W(4: O),

and the first equation of ( A . l l ) becomes

~ o o o y o o o = (A", , + ~ L o o o B o " " ) y o o o = 0. (A.13)

The pivot problem (A.13) is autonomous in the periodic variables 2.x and T

and accommodates separable solutions of the Orr-Sommerfeld type

Yooo = z, + z,, z, = t''(Z'+T)Q) (Y). (A.14)

214 Daniel D. Joseplz

We call (A. 13) the Orr-Sommerfeld problem with inflow. Solutions (A.14) of (A. 13) and the side conditions of (A. 11) exist when the parameters wooo(cc) and ,looo(a) = 1/R have certain values which were computed by Haaland (see Fig. 4).

The adjoint Lgoo to Loo, is defined by the requirement that

[a, Loooh1 = [ G o o a , hl for all functions LZ and h which vanish at infinity and are 271 periodic in SIX

and z. We find that

L&J,a = -(uooov2u,, - V2(Ua.,) + Uo.y).u,,

+ Aooo{ -V4u - V2(aWO), , + ( U W ~ , ~ ~ ) , ~ ] ~

The adjoint pivot problem is then defined as

G o o % o o = 0, where Ygoo satisfies the same decay and periodicity conditions as Yooo. One may verify that

ygo0 = zy + 27, = e i ( a . y + r ) @ ( y ) .

We may now interpret the last of the conditions ( A . l l ) as equivalent to requiring that E be the projection of the vorticity of bifurcating solution into the eigen subspace of the pivot problem; that is,

i: = -[AY, Z:] = -e[AY’, ZT].

We should like to have the solution of the false problem in a series

This series would correct the solution of the Orr-Sommerfeld problem with inflow for the extra terms and for the effects of the nonlinear terms. Given the series for A(c, ,u, 0) we should take the solution of the false problem with

/ 1 = +;, P , x) as best approximation of the true solution.

It is not possible however to carry out the solution given by (A.15) because the perturbation computation for the extra terms requires differentiation with respect to the parameter 1 [see (c) below]. To circumvent this difficulty we develop the basic flow ( U , W ) , as well as the solution, in powers

Response Curues for Plane Poisruille Flow 275

of x:

(A.16)

Substitution of (A.16) and (A.17) into the false problem ( A . l l ) leads to a sequence of perturbation problems in the form

L O o o ~ n / p = Gnlp 3 (A.18)

where Gnlp is of lower order and Ynlp vanishes as J’ + _+ E and is 2n periodic in M X and z with

[ V 2 Y , , , > ZT] = 0

if 1 + n + p > 0. If there is a solution of (A. 18) it must necessarily satisfy the condition that

[G,lpI = [G,,, 1 ZT1 = [ L 3 0 0 y n l P 9 ZT1

= [Y,,, 1 L;,,zT] = 0. (A.19)

Equation (A.19) is usually complex valued and is equivalent to two real- valued conditions. These may be satisfied by an appropriate choice of the constants and Anlp which appear in the functions G n l p .

Let us consider some typical problems which arise in the perturbation. (a) Problems with (n, I , p ) = ( n , O , 0) give nonlinear corrections of the Orr- Sommerfeld problem with inflow without accounting for the extra terms. These problems are analogous to those considered in Section IX. The first of these problems is stated below:

~ 0 0 0 ~ 1 0 0 + ~ 1 0 0 ~ 0 0 0 ~ 0 0 0 + ( ~ ~ l o o V 2 ~ 0 0 0 . r + w o o 0 ~ y O 0 0 ) = 0.

21 o o [ ~ o o o y o 0 0 1 + ( 0 1 00[V2~000.TI = 0.

As in Poiseuille flow, [N(YOo0, YOo0)] = 0 and (A.19) gives

Hence, i,,,, = (oleo = 0. (b) Problems with (11 , I , p ) = (0, 0, p ) correct the Orr-Sommerfeld problem with inflow for variations with 1. This perturbation can be avoided by redefining variables; x may be swept into similarity variables so that one computation of the Orr-Sommerfeld problem with inflow for all x > 0 requires one computation and no perturbation. However the expansion in x is required when n + 1 + 0. (c) Problems with (n, I , p ) =

276 Duiiiel D . J o s e p h

(0, I , 0) correct the Orr-Sommerfeld problem with inflow for the extra terms. I t is not possible to compute corrections for the extra terms without prior computation of terms which involve x derivatives. To illustrate this we shall show that the first correction for the extra terms requires prior computation of YOOl. When (n, I , p ) = (0, 1, 0) we have

~ O O O ~ O t , + ~ - 0 , 0 ~ O 0 0 ~ o " 0 + ( ~ 0 1 0 V 2 ~ 0 0 0 . r + G 0 0 ~ O 0 0 = 0,

(A.20)

where

and

Hence to form the Eq. (A.20) for the first correction for the extra terms, we need first to find the functions YOo0,, = 'Pool. A similar prior computation of the derivatives with respect to the slow spatial parameter x is required at higher orders. The perturbation problems may be solved sequentially provided that an appropriate order in the computation is observed.

The final result of computations carried out for solutions which are 2n periodic in cc-yand z is the series (A.17). A X-dependent wave number X ( E , p, x) is automatically generated by maximizing )L over x > 0:

i(c, p, x, a({:, p, x)) = max I.({:, p, x, 2). Z > O

In the end we compute the values of the series for the first positive root of the equation

p = 4 8 , P, 31, a([:, p> x)).

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euille channel flow. J . Fluid Mrch. T o be published.

The Theory of Polar Fluids?

S . C . COWIN

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 A . A Guide to This Article . . . . . . . . . . . . . . . . . . . . . 281 B . Historical Note . . . . . . . . . . . . . . . . . . . . . . . 282 C . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 D . A Synopsis of Polar-Fluid Theory . . . . . . . . . . . . . . . . 286

11 . Foundations of the Theory . . . . . . . . . . . . . . . . . . . . . A . Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 290 B . Dynamics and Thermodynamics . . . . . . . . . . . . . . . . . 295 C . Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . 301 D . Formulation of Boundary-Value Problems . . . . . . . . . . . . . 306 E . Typical Boundary-Value Problems . . . . . . . . . . . . . . . . 321

111 . Related Theories . . . . . . . . . . . . . . . . . . . . . . . . . 335 A . Fluids with Couple Stresses . . . . . . . . . . . . . . . . . . . 338 B . The Cosserat Continuum with a Deformable Director Triad . . . . . 339 C . Fluids with Deformable Microstructure . . . . . . . . . . . . . . 341

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Note Added in Proof . . . . . . . . . . . . . . . . . . . . . . . . 347

D . Dipolar Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 342

I . Introduction

The theory of polar fluids and related theories are models for fluids whose microstructure is mechanically significant . The applications of these theories have been. for example. to suspensions. to blood flow. and to mean turbulent flow . Even very small microstructure in a fluid can be mechanically significant if the characteristic geometric dimension of the problem considered is of the same order of magnitude as the size of the microstructure .

?Dedicated to the Memory of Walter Jaunzemis

219

280 S. C. Cowin

Thus these theories are also being applied to thin lubricating films and to problems involving surface films. Real fluids for which the viscous Newton- ian model is adequate when the characteristic geometric dimension of the problem considered is large may be more adequately modeled as a structured fluid when the characteristic geometric dimension of the problem considered is small. Another area of application of the theory of polar fluids is to real fluids carrying charged particles and subjected to an external electromagnetic field that causes the particles to rotate relative to their neighbors. This application was the one originally proposed for the theory and arises naturally when one considers the particle or statistical model from which the theory was originally developed.

The theory of polar fluids was originally developed from a statistical mechanics model that assumed noncentral forces of interaction between particles. If the interparticle forces are not central forces in a particle- particle interaction there is an interparticle couple as well as an interparticle force. Under the action of this couple the fluid particle will have a tendency to rotate relative to its neighbors. The essential idea of a polar fluid is obtained by introducing a kinematic variable to model the rotation of the particle relative to its neighbors and a skew-symmetric stress tensor to model the forces that balance the action of the couple. While this is a slight simplification, it is the conceptual origin of the theory of polar fluids. Kin- ematically, the theory differs from other theories of fluid behavior in that a particle angular velocity is defined independently of its velocity field. Me- chanically, polar-fluid theory differs from other fluid theories in that angular momentum effects such as couple stresses and asymmetry of the usually symmetric stress tensor are considered.

One of the principal predictions of the theory is an increased effective shear viscosity close to solid surfaces and in narrow geometries such as bearings or capillary tubes. There is experimental evidence that indicates some real fluids behave in this manner. A summary of related experimental literature has been given by Henniker (1949). Henniker notes the evidence for abnormally high viscosity in the neighborhood of a solid surface is extensive. He mentions an experiment in which a tenfold increase in viscosity was found within 5000 A of a solid boundary and one performed by several people in which a tenfold increase in the viscosity of water between glass plates 2500 8, apart was observed.

This article is a presentation of the th;ory of polar fluids. The emphasis is on the foundations of the theory. Applications of the theory are discussed, but not pursued. Of the burgeoning literature concerned with applications only the key references are given. Also, only a few typical solutions to boundary-value problems are presented here. Again, the literature of this topic is rapidly expanding and the proper journals to consult can be

The Tlieory of Polar Flirids 28 1

determined from the list of references. The relationship of polar-fluid theory to theories of a similar nature is described in the last section of the article. These similar theories are only sketched, their foundations can be developed in the same manner as are the foundations of polar-fluid theory.

A. A GUIDE TO THIS ARTICLE

This section is introductory and contains this guide, a historical note, a section on notation, and a synopsis of polar-fluid theory. The synopsis of polar-fluid theory is a brief introduction to the theory.

Section I1 is the largest section and it presents the foundation of the theory of polar fluids. It is divided into five subsections. The first (A) is concerned with the kinematics of polar-fluid motions, the second (B) with dynamics and thermodynamics, the third (C) with the formulation of constitutive relations for polar fluids, the fourth (D) with the formulation of boundary- value problems, and solutions to typical boundary-value problems are presented in Section II,E. The foundations of the theory are described in terms of contemporary studies on the foundations of continuum mechanics and thermodynamics. The ideas of Noll (1955, 1958) concerning material objectivity, of Green and Rivlin (1964a,b,c) and Noll (1963) concerning consistency in the basic system of balance equations, of Coleman and Noll (1963) and Coleman and Mizel (1964) on how to use the entropy inequality are discussed and employed. After the formulation of the boundary-value problems, the theorems of Serrin (1959) concerned with the universal stability of viscous fluid motions are extended to polar-fluid motions. Section II,E contains solutions to boundary-value problems which illustrate the features of the theory. Generally these problems are analogous to problems solved for Newtonian viscous fluids using the Navier-Stokes equations. The solutions to these problems show that the effective viscosity of a polar fluid is always greater than or equal to the Newtonian shear viscosity and that the effective viscosity depends strongly on a ratio of the characteristic geometric length of the flow situation (e.g., a pipe diameter) to a material property ofdimension length.

In Section 111 related theories of fluid behavior are presented. The theory of fluids with deformable microstructure and the theory of micropolar fluids with stretch are generalizations of polar-fluid theory. The theory of fluids with couple stress is a specialization of polar-fluid theory. The theory of dipolar fluids is related to polar-fluid theory by the slightly indirect route illustrated in Fig. 12. The purpose of the location of this material at this

282 S. C. Cowin

point in the development is to simplify the presentation. It is thought that the reader new to the subject can best approach it by understanding the simpler theory completely, then viewing the more sophisticated theories from this better vantage point. These related theories are not developed here to the extent of the polar-fluid theory, but such developments are possible.

B. HISTORICAL NOTE

I t is a kinematical property that distinguishes the theory of polar fluids from the theories of other fluids. In a polar fluid the rotational motion of the fluid particles is not completely determined by the vorticity field. Instead, an axial vector field characterizing the rotational motion relative to the vorticity is constructed and the rotational motion of the fluid particles is considered to be proportional to the sum of these two independent fields. The notion of a polar fluid appears to have been introduced first by Born (1920) who associated the resistance to the relative rotational motion with the skew-symmetric part of the stress tensor. About 30 years later and subsequent to a statistical mechanics study that served as motivation, Grad (1952) introduced the linear constitutive equations for polar fluids. These constitutive equations relate the skew-symmetric part of the usual stress tensor to the relative angular velocity described above, as well as relating couple stresses to the gradient of the total angular velocity. The Newtonian law of viscosity is preserved in that the symmetric part of the usual stress tensor bears its usual relationships to the rate of deformation tensor.

The same constitutive equations have been advanced from different viewpoints by Aero et al. (1963), Cowin (1962), Condiff and Dahler (1964), and Eringen (1966). Aero and coauthors (1963) simply introduce the velocity and particle angular velocity fields and postulate a dissipation function that leads to Grad’s constitutive equations. Eringen (1966) obtained the same constitutive equations by specializing his more general theory of microfluids (Eringen, 1964). Dahler’s (1965) development is from statistical mechanical considerations. The development of Cowin (1962), which is followed here, was based on the Cosserat continuum. As there are a number of viewpoints from which the theory can be developed, so there are a number of names used to describe the theory. Condiff and Dahler (1964) appear to be the first to call this the theory of polar fluids. Eringen (1966) called a polar fluid a micropolar fluid because he obtained it by specializing his theory of microfluids. The terms “asymmetric hydrodynamics ” and ‘‘ fluids with rigid substructures ” have also been used.

The features of the Cosserat continuum are best understood by comparing it with the ordinary continuum used in classical continuum mechanics. In ordinary continua the only geometrical property of a particle is its position.

T h e Theory of Polar Fluids 283

A complete geometric description of ordinary continua is obtained by simply specifying the point where each particle is located. The particles of a Cosserat continuum have the geometric properties of position and orientation. They are infinitesimal replicas of a rigid body and possess the six geometrical degrees of freedom of a rigid body. The orientation of the Cos- serat particle is accomplished by outfitting it with a triad of vectors, called directors. The director triad is considered to be rigid, but is allowed to rotate independently of the translational motion of the particle. This continuum model was presented and developed by the Cosserat brothers (1907, 1908, 1909), but except for an exposition by Sudria (1935), it lay dormant until the papers of Guenther (1958) and Ericksen and Truesdell (1958). Since 1958 the Cosserat model has been applied to such diverse fields as the theory of rods and shells, the continuum theory of dislocations, anisotropic fluids, and generalizations of the polar fluid. As was noted above, the development presented here of the polar fluid is based on a Cosserat continuum.

The solutions to a number of boundary-value problems involving polar fluids have been presented. The reason for this productivity is the fact that the governing differential equations are, like the Navier-Stokes equations, linear if convected acceleration terms are not present. Plane Couette flow has been considered by Pennington (1966), Ariman and Cakmak (1967, 1968), Cowin (19684, Rajagopalan (1968) Hudimoto and Tokuoka (1969), Kirwan and Newman (1969), Pennington and Cowin (1969), and others. Plane Poiseuille flow was also considered by most of the authors mentioned above. Hagen-Poiseuille flow, or pipe flow, has been considered by Aero et al. ( 1 963), Eringen (1966), Pennington ( 1 966), Pennington and Cowin (1 970), and others. The problems of flow between two concentric cylinders in relative rotation has been studied by Condiff and Dahler (1964), Pennington (1966), Stokes (1966), Ariman Pt rrl. (1967). Rajagopalan (1968), and Pen- nington and Cowin (1970). Creeping flow about a sphere has been analyzed by Aero et a/. (1963). Rao et al. (1969) developed a solution for the slow steady rotation of a sphere in a polar fluid, and Bhatnagar and Rajagopalan (1968) studied secondary flows induced by the rotation of a sphere or coaxial cones. Cowin (1972b) gave the solution for flow in a spinning cylinder under an axial pressure gradient. Rao and Rao (1971) presented an analysis of the oscillations of a sphere in a polar fluid. Kline and Allen (1969a,b, 1970) and Kirwan and Newman (1972) have considered some cases of unsteady flow.

Generalizations of the polar fluid are easy to construct using the director triad introduced by the Cosserat brothers (1907, 1908, 1909). While the triad is required to remain rigid for polar fluids, the theory of fluids with deformable microstructure deals with fluid behavior associated with a deformable director triad. This theory is discussed by Eringen (1964), Allen et al. (1967), Allen and Kline (1968), Kline and Allen (1968a), and represents a useful

284 S. C. Cowiiz

specialization of the more general considerations of Green and Rivlin (1964b). The more specialized theory of Eringen (l969), called micropolar fluids with stretch, is obtained by constraining the motion of the deformable triad so that all three directors remain perpendicular during the deformation and experience identical rates of length change. Both of these theories reduce to polar-fluid theory when the appropriate kinematical constraints on the director triad are introduced. The theory of dipolar fluids presented by Bleustein and Green (1967) is a special case of the theory of fluids with deformable substructure obtained by constraining the motion of the director triad to coincide with the local motion of the region. Both polar and dipolar fluids reduce to the theory of fluids with couple stress introduced by Stokes ( 1966).

I n passing, it should be noted that the theory of anisotropic fluids created by Ericksen (1960a,b,c) is not considered here. Ericksen employs only one director, which then makes each fluid particle, in a certain sense, transversely isotropic. We consider here only theories obtainable from the three-director deformable triad. This applies also to the work of Allen and DeSilva (1966) and Leslie (1968, 1969).

The application of these theories, particularly the theory of polar fluids, to the explanation of a number of physical phenomena has been suggested. Cowin (1962), Kline and Allen (1968a, 1969b), Kaloni and DeSilva (1969a,b,c, 1970), Erdogan (1970a,b), Erdogan and Kadioglu (1971), and others have considered the application of these theories to the modeling of suspensions. In particular Kline et ul. (1968), Kline and Allen (1969c), Valanis and Sun (1969), Ariman (1971), and Cowin (1972a) have considered the applicability of these theories to blood. The application of polar-fluid theory to the lubrication problem was suggested by Cowin (1968a), and has been worked out in the case of a plane slider bearing by Green (1969) and Allen and Kline (197 1). The suggestion that polar-fluid theory might serve as a model for turbulent flow was made by Nikolaevskii and Afanasiev (1969) and Liu (1970) executed by Liu (1970) and applied by Peddieson (1972).

C. NOTATION

Cartesian tensors will be employed throughout this article. Both the indi- cia1 notation and, when the meaning is not obscure, the direct notation for Cartesian tensors will be used. The direct notation consists of representing the vector whose Cartesian components are I', in bold-face type as v and the second-rank tensor whose Cartesian components are T j by T. The transpose of a tensor is denoted by TT, the trace of a tensor by Tr T. An orthogonal tensor has the property QQ" = Q'Q = I , where 1 is the unit tensor whose Cartesian components are given by the Kronecker delta h j j .


In general, vectors will be denoted by Latin minuscules, second-rank tensors by Latin majuscules, and third-rank tensors by Greek majuscules. The following rule is laid down for associating axial tensors with absolute three- dimensional tensors of rank two and three. Given a second-rank tensor q, and a third-rank tensor A j k m , the axial vector denoted by pi and the axial second-rank tensor denoted by Aim are computed by the formulas

(1.1 a,b) In these equations eijk is the permutation symbol whose value is + 1 if ijk is an even permutation of 123, - 1 if (jk is an odd permutation of 123, and 0 for all other values of ijk. The identity

F . = - - 'e . 2 iJk T j k , Aim = -3e.. 1.1k A . j k m '

eijke,,,,, = hi,dj, - hinhjm ( ' .2)

7;mnI = - e i m n T i 9 A I m n j j = - e i m n A i j . (1.3)

may be used to invert (l.l), thus A -

where the brackets indicate the skew-symmetric part of the tensor with respect to the enclosed indices

T m n j = +(Ti, - T m ) , ' I m n l j = + ( A m n j - A t i m j ) . (1.4) Since any tensor has a unique additive decomposition into symmetric and skew-symmetric parts with respect to any two indices, i t follows that any second-rank tensor G,, and any third-rank tensor Ymnj whose symmetric parts vanish

G ( m n ) = 3 ( G m n + G n m ) = 0, y , m n ) j = 3 ( v m n j + y n m j ) = 0, (1.5)

may be completely represented by the associated axial vector Gi and the associated axial tensor Yim. All the axial tensors encountered in polar-fluid theory have this property, but not those encountered in the generalizations of polar fluids.

The notation x will be used to represent the ordered set of Cartesian coordinates xl , x2, ?c3 of a point. The gradient operator V will denote differentiation with respect to the coordinates x and will be treated as a vector with components d / 2 x j :

V ( ) = (r?/r3xj)ej, (1.6)

(1.7)

(1.8)

where ej is the Cartesian base vector. The vector cross product is written V x v = e . . ~1 . e . = - e . . u . e. 1Jk k . ~ I I j k j . k I .

The vector identity

v x v x v = vv - v - v2v will be frequently used.

286 S. C. Cowin

All cylindrical and spherical components of tensors employed here will be physical components.

Sets of functions of independent variables will be denoted by script majuscules, .A', 9,. . . etc. The elements of a set will be specified by enclosing them in brackets, ./fl = {. . .i. If the elements of a set are defined by a restriction or constraint, the constraint on the set will be specified after the elements of the set separated from them by a vertical line. N = j . . . 1 . ..}.

To facilitate comparison with the material presented here, Table 1 listing the notation of other authors is given.

D. A SYNOPSIS OF POLAR-FLUID THEORY

The basic equations for the mechanical behavior of polar fluids will now be presented.

The Eulerian description of the instantaneous motion of a polar fluid employs two independent vector fields. The first is the usual velocity field v(x, t ) , where x is a spatial position and t is time. The second is an axial vector field G(x, t ) which represents the angular velocity of the polar-fluid particle at the place x at time t . For an ordinary continuum the only independent field is the velocity field; the angular velocity field is equal to one-half of the curl of the velocity field. We interpret the usual angular velocity field as a regional angular velocity and denote it by W,

\iv = +v x v. (1.9) The angular velocity of a particle relative to the angular velocity of the region in which the particle is embedded is denoted by H(x, t ) and defined

, . - by H G - W = G - 4V x V. (1.10)

H is called the relative angular velocity. The spatial gradient of the total angular velocity G will be denoted by 3,

- y.. I J - = G . 1 . J . . (1.11)

The rate of strain or rate of deformation tensor D, which is the symmetric part of the velocity gradients, is given by

D . . 1.1 = - 0 . . ( 1 . J ) ' (1.12)

where the parentheses indicate the symmetric part of u ; , , ~ . The forms of the conservation laws of mass, linear and angular momen-

tum appropriate for a polar continuum will be considered now. The conser-

TABLE I

NOTATIONS FOR POLAR FLUIDS

Name Symbol in the present Eringen Grad Aero et ti/.

text (1966) (1952) (1963) ~

Stress tensor Thermodynamic pressure Couple-stress tensor Body force per unit mass Body couple per unit mass

Velocity vector C h ' k '1 '1

Rate of deformation tensor 4, d t j ' ( t . j ) ' ( i k )

Regional angular velocity cli;j 0)LJ '[Ll c"i.Jl

Particle angular velocity ' k ' k ' e 2 k i J w . . 1 , Qh

Q.j

Relative angular velocity fir Particle angular velocity gradient j ' z . j tei"lri%l",J

Traditional viscosity coefficient /. /. ~ iZ + (+);.I i. Traditional shear viscosity coefficient L' p y + t K , i.' P Rotational viscosity +tir i Viscosities of the gradient of c( a,. 2. ' 211

I'h ~ w k t ' k ~ j ( ' ~ , , - "[,,j)) i ' t j k ' j k

T -

particle angular velocity lj +i,. + @, i U + T

$7,. - 9, i fl ~ T

288 S. C. Cowiii

vation of mass has the same form as in an ordinary continuum, and is expressed by the continuity equation

i) + pV - v = 0, (1.13) where p is the density and the superimposed dot indicates the material or substantive time derivative. Except for the fact that the stress tensor T need not be a symmetric tensor, the conservation of linear momentum also has the same form as in an ordinary continuum

V T + pb = pv or Tj , j + phi = phi, (1.14)

where b denotes the body force per unit mass. In an ordinary continuum the conservation of angular momentum re-

duces to the requirement that the stress tensor be symmetric. For a polar fluid it is necessary to consider angular momentum in a much less trivial fashion. From a physical viewpoint this is necessary because local force couples are considered and must be balanced. From a mathematical viewpoint it is necessary because an additional unknown axial vector, namely the particle rotation G, has been introduced and an axial vector equation from which this unknown can be determined must be introduced. Since a couple can be represented as an axial vector, the same arguments that lead from the concept of a stress vector to the stress tensor can be used to develop the couple-stress (axial) tensor A from the couple-stress vector. The balance of angular momentum is

or V * .A + 2T + pC = p k 2 G

(1.15) Aii,i + 2Ti + pci = p k 2 & , ,

where T is the axial vector associated with the skew-symmetric part of the usual stress tensor, C is a body couple field, and k is the radius of gyration of the polar-fluid particle.

The mechanical constitutive equations for the compressible polar fluid are the following relations between the stress tensors T and A, the kinematical fields v and G, and the thermodynamic pressure p :

(1.16)

(1.17)

The coefficients i, 11, T , ct, p, and y can depend upon temperature and density. Except for the last term on the right-hand side, (1.16) is the Newtonian law of viscosity; 1 and p are the usual viscosity coefficients. As the last term is the only term on the right-hand side of (1.16) that is not symmetric, the axial vector equation associated with (1.16) is

T = -2zH. (1.18)

T + p l = A 1 Tr D + 2pD - ~ T H ,

= c t l Tr Y + ([j + y ) Y + (p - ;i)YT.


Since H is the difference between the angular velocity of the particle and the angular velocity of the region in which the particle is embedded, T is called the relative rotational viscosity. The coefficients a, /j, and ;I appearing in (2.13) are the viscosities of the gradient of total rotation. The coefficients p , I,, T , a, /j, and ;I are restricted by the inequalities

which follow from the Clausius-Duhem inequality. When the material coefficients p, 1, T, a, p, and ;I are assumed to be

constant and the constitutive equations (1.16) and (1.17) are substituted into the expressions for momentum conservation (1.14) and (1.q the following statements of momentum balance in terms of the velocity and angular velocity fields are obtained:

(jb + p)vv * v + pv2v + ~ T V X + pb - V/I = pi , (1.20)

(1.21) (. + /!j - ;1)VV - G + ( p + y)V% - 4 T H + pc = pk2G.

Equation (1.20) differs from the Navier-Stokes equation only by the term 2zV x H. If the polar fluid is assumed to be incompressible, then (1.13) reduces to

and (1.20) and (1.2 1) take the forms v - v = o (1.22)

( p + T ) V ~ V + 2zV x G + pb - V p = p i , (1.23)

(. + p - y)V(V - G ) + ( p + y ) X v2G - 4 T G + 2zv x v + pc = p k 2 G , (1.24)

where (1.10) has been used to eliminate H and where p is now not the thermodynamic pressure. The system of equations (1.22)-( 1.24) is a system of seven scalar equations in the seven scalar unknowns, v, G and 17.

The boundary conditions customarily employed in the solution of these equations are that the velocity v at the boundary must be equal to the velocity vo of the boundary and that the particle angular velocity G at the boundary must be equal to the angular velocity Go of the boundary, thus

v = vo on solid boundaries (1.25) and

G = Go on solid boundaries. ( I .26)

In the case when the boundary is stationary, neither translating nor rotating, both vo and Go are zero. When the boundary is the surface of a circular cylinder of radius h which is rotating with an angular velocity R about its axis of revolution vo and Go are given by

vo = Rhe, , Go = Re- , (1.27)

290 S. C. Cowiiq

where es is the azimuthal base vector and e, is the axial base vector. Thus vo represents the rigid-body translation of the boundary and Go represents the rigid-body rotation of the boundary.

The system of equations (1.22)-(1.24) together with the boundary conditions (1.25) and (1.26) characterize the typical boundary-value problem. In the event one should wish to proceed directly to the section on boundary- value problems, then note that this same system of equations has the numbers (2.1 17)-(2.119), (2.1 1 l), and (2.1 12), respectively, at that point in the text.

11. Foundations of the Theory

The purpose of this section is to present a full development of the theory of polar fluids. The foundations of the theory will receive particular attention. The first subsection develops the kinematics of Cosserat continua while the second subsection presents appropriate forms of the conservation of mass, energy, and linear and angular momentum, and the entropy inequality. The third subsection introduces the constitutive assumptions which are subsequently restricted by objectivity and thermodynamic arguments and, finally, linearized. The fourth subsection is concerned with the formulation of boundary-value problems while the fifth subsection presents the solutions to typical boundary-value problems.

A. KINEMATICS

The motion of a Cosserat continuum is characterized kinematically by a velocity field and an angular velocity field that are independent. Section II,A,l introduces these basic fields and Section II,A,2 is concerned with their gradients. In Section II,A,3 rules for the transformation of all kinematic variables under inversions and rigid-body rotations and translations are obtained and, finally, objective kinematic variables are identified.

1. The Cosserat Motion

A Cosserat body B consists of elements called Cosserat particles, denoted by P. A description of the translational motion of the Cosserat particle P coincides with the description of motion in an ordinary continuum:

x = x( P, t ) , (2.1) where t is time and x stands for the ordered set x, , x2, and x j of spatial

The Theory of’ Polur Fluids 29 1

d c7t

v = v(P, t ) = x(P, t ) = i (P , t ) . (2.5) p= co nst

292 S. C. Cobviii

and that, with (2.7), the orthogonality condition (2.3) can be written

dUidhi = d,, . (2.8)

Gij = duidu,j , (2.9)

(2.10)

(2.1 1 )

This shows that dui are the components of an orthogonal matrix. To measure the rotation of the director triad we introduce the tensor G,,

which is easily shown to be skew symmetric, G . . = L ) .it , = -L/ . ( j . = -G. .

d U I . = Gi,iduj.

I J U I a j U I u / J I '

by use of (2.8). A convenient representation for Gij in the index notation is

The rotation tensor G for the particle can be represented as an axial vector G ,

G = t(du x du). (2.12)

The field G is defined independently of the ordinary motion (2.1) of the particles and is called the particle angular velocity field. The translational motion (2.1) may be inverted and used to specify G as a function of spatial coordinates

G = G(x, 1). (2.13)

The two independent fields v(x, t ) and G(x, 1) constitute the basic fields of a Cosserat motion.

2. Glwdirnts of' the Coss~rat Motiori

The tensor of spatial velocity gradients L is a second-rank tensor defined

L. . = 1 ' . . (2.14)

and the tensor of particle angular velocity gradients 9 is an axial second- rank tensor defined by

y.. 1J = - G. 1 . J . . (2.15)

The rate-of-deformation tensor (strain-rate tensor) D and the spin tensor W are defined as the symmetric and skew-symmetric parts of the tensor of velocity gradients L,

w . = c (2.16a, b)

by I J - 1 . J 1

11 [ i . ~ l . D . . = 1'

1.1 ( i . j ) 5

Hence L may be written as their sum L. . = D . . + w.. (2.17)

The Theorj] qf Polar Fluids 293

The axial vector W associated with W is the usual angular velocity rather than the vorticity. The difference between the angular velocity W and the vorticity V x v is a factor of 2,

W = = + V X V or (2.18)

For Cosserat motions W is called the regional angular velocity field to distinguish it from the other angular velocity fields and to indicate that it measures theaverage rotation in a neighborhood. It is clear from the geometrical interpretations of vorticity (see Truesdell, 1954) that W is a measure of rotation computed by averaging (in several senses) over the neighborhood of the particle.

The relative angular velocity H is introduced to measure the differences between the particle angular velocity G and the average angular velocity W of the region in which the particle P is embedded,

(2.19)

In the special case when the relative angular velocity H vanishes, the motion is said to be a motion of constrained rotation in that the particles are constrained to rotate at the regional angular velocity W. This special type ofCoseratemotioncontainsal1 themotions possible inanordinarycontinuum.

Some motions which are possible Cosserat motions but which have no counterpart in ordinary continuum kinematics include irrotational particle motions and stationary particle motions. In the case of irrotational particle motions the total particle angular velocity G vanishes and H is the negative of W. This type of Cosserat motion is different from the more restrictive irrotational motions of an ordinary continuum which require that W vanish. Stationary particle motions are characterized by a vanishing of the velocity field, which requires that the regional angular velocity W vanish also, and a nonvanishing of the particle angular velocity field G which then coincides with 8. In this type of motion the particles are stationary, but rotating. The necessary and sufficient conditions for rigid-body motion of the Cosserat continuum are that the rate of deformation tensor D and the relative angular velocity H vanish

D = 0 , H = O . (2.20)

3. Equiculent Cosserat Motioiis

In later sections it will be necessary to require that the constitutive equations be invariant under rigid-body motions and inversions of the spatial reference frame. This is the objectivity requirement (Truesdell and Noll, 1965). The necessary kinematic formulas will be obtained in this section.

294 S. C. Cowirz

A change of spatial frame or reference is a one-to-one transformation of space and time that preserves distances, time intervals, and temporal order. It is expressed as a simultaneous change of position and time, x to x* and t to t*, such that

X* = c( t ) + Q(t)x, t* = t - a, (2.21)

where a is a constant, c ( t ) is a smooth vectorial function of time, and Q(t) is a smooth orthogonal tensorial function of time. A scalar that is invariant under the transformation (2.21) is said to be an objective scalar. Vectors and tensors that transform according to the equations

w*(f*) = Q(t)w(f), B*(t*) = Q(t)B(t)QT(r), (2.22a,b)

under the transformation (2.2 1) are called objective vectors and objective tensors, respectively. Finally, axial vectors and axial tensors that transform according to the rule

B(t*) = .sQ(t)B(t), Y*(t*) = sQ(t)YQ'(t), (2.23)

where

s = sign det Q(t) (2.24)

under the transformation (2.2 1 ) are called objective axial vectors and objective axial tensors, respectively. A function or field that is not objective is said to be relative.

The influence ofa change of frame upon the motion and upon the kinematical variables characterizing the motion is of primary interest. Consider two Cosserat motions .I/ and .&* where . K is given by (2.4) and .L/* by

N* = {x* = x * ( P, f*), d,T = S,*( P, t*) I dz - d; = huh). (2.25)

I f the two motions ./K and A/* are related by expressions

x* = x*(P, r * ) = c ( t ) + Q(t)X(P, f ) , t* = t - a (2.26a,b)

and

d,* = S,*(P, t * ) = Q(t)S,( P, t ) = Q(t)d, , (2.27)

then they are said to be equivalent Cosserat motions. The remainder of this section is devoted to determining which of the fields

v, G , c, L, D, and W are objective for all equivalent Cosserat motions and which are not. Objectivity of v, G , Y, L, D, and W is determined on the basis of their transformation law, that is to say, whether they transform according to (2.21), (2.22), or (2.23) or not. The transformation law for velocity v(x, f ) may be computed by taking the material time rate of (2.26a):

V* = X* = C + QX + Qv, (2.28)

The Theory of Polar Fluids 295

where the superposed dot denotes differentiation with respect to t or t*. A similar expression for the particle angular velocity field G(x, t ) is obtained by combining (2.27) and (2.9), thus

G* = sQG + K, (2.29)

where K is the axial-vector representation of the skew-symmetric tensor K defined by

K QQ' = -QQ' = - K T . (2.30)

Inspection of eqs. (2.28) and (2.29) shows that neither of the basic rate fields are objective.

Consider now the gradients of the basic fields v and G . From (2.14), (2.15), and (2.28)-(2.30) it can be shown that

(2.3 1 a,b)

which demonstrates that Y is an objective tensor while L is not. However, when L is decomposed into its symmetric and skew-symmetric parts and use is made of definitions (2.16), it follows from (2.31a) that

D* = QDQ", W* = QWQ' + K. (2.32a,b)

Hence the symmetric part of L is objective. The axial-vector equation associated with (2.32b),

W* = sQW + K, (2.33)

may be used in conjunction with (2.29) and (2.19) to show that the relative angular velocity H is objective:

A* = sQH. (2.34)

In summary, then, D, H, and 4 are objective while v, G, W, and L are not.

L* = QLQ' + K, 'P* = ,sQVQ',

B. DYNAMICS A N D THERMODYNAMICS

The purpose of this section is to present a development of the forms of conservation principles of mass, energy, linear and angular momentum, and the entropy inequality appropriate for Cosserat continua. The usual form of mass conservation applies for all Cosserat continua, and it is recorded here in the form of the continuity equation

i) + pV * v = 0, (2.35)

where p is the mass density. I n Section II,B,I an appropriate form ofenergy balance is argued, but even though it is heuristically motivated, the exact

296 S. C. Cowin

form of the energy balance is a postulate. In Section II,B,2 the conservation of linear momentum and angular momentum are obtained from the energy equation by requiring invariance of the energy under rigid-body translations and rigid-body rotations, respectively. This formal method of obtaining these momentum equations has the advantage of demonstrating the consistency of the structural form of the momentum balance with the structural form of the energy balance. In Section II,B,3 the entropy inequality is stated and an expression for the entropy production computed.

1. The Conservation of’ Eizergj,

An elementary form of the energy conservation law is

K + E = P + Q, (2.36)

where K is the kinetic energy, E is the internal energy, P is the mechanical power, and Q is the nonmechanical power of some body or part of a body. The specific internal energy will be denoted by r : ; thus for a body B

(2.37)

where dr is an element of volume. The nonmechanical power is supplied to the body either through its surface or in proportion to its mass; thus

(2.38)

where q is the heat flux across the surface per unit area, r is the heat supply function per unit mass and unit time, while aB means the surface of B, and da is an element of surface area.

The representations for E and Q are the same as in the ordinary continuum theories, but the representations for K and P will reflect the extra structure of the Cosserat continuum. The kinetic energy is written in the form

^ ^

K = 4 ). p ( u j u i + I j j G i G j ) d u , . n

(2.39)

where I , is called the inertia tensor of the Cosserat particle. The inertia tensor is required to be symmetric. The two expressions in the integrand for the kinetic energy correspond to the two kinds of kinetic energy associated with rigid-body motion. The first term is the usual translational kinetic energy; the second is the rotational kinetic energy. The mechanical power

Tlie Theory of Polar Fluids

supplied to the body is

297

(2.40)

where t i is the stress vector, hi is the body-force vector, Bi is a couple-stress vector acting on the surface, and t i is a body-couple vector. The first terms in each integral are classical; the second terms are associated with the director motion.

An integral form of the conservation of energy is obtained by substituting (2.37)-(2.40) into (2.36) and employing the transport theorem of Reynolds (see, e.g., Truesdell and Toupin, 1960). Thus

p(C + i i C i + +i i jG iG j + Z i j G i G j ) dc I l l

= (. (h,r., + c,d, + r ) p du + (. ( t l G , + B,G, - q ) da. (2.41)

In the following section this integral equation will be transformed into a field equation.

' S ' ,'B

2. The Conserration of' Linear iiizil Angular Moiiiriitum

In this section we use the method presented by Green and Rivlin (1964a,b,c) to obtain statements of momentum conservation from the statement of energy balance. This method, outlined concisely in Green and Rivlin (1964c), consists of requiring that the energy equation be invariant under superimposed rigid-body translations and rotations of the form (2.21).

First, the energy equation (2.41) is applied to a volume that is a tetrahedron bounded by one plane with an arbitrary unit norinal n and by three planes through the point x i parallel to the coordinate planes. I f du denotes the area of the surface of the tetrahedron normal to n and rlu, the area of the surface normal to the x i axis, then

LIUi = lli rla. (2.42)

Further, let T j , Ai j , and q j denote the values of t i , B i , and q, respectively, on the surface whose normal is in the direction of increasing x i . When the energy balance is applied to the tetrahedron and the volume of the tetrahedron is shrunk to zero in a way that preserves the orientation of its faces. we obtain an equation which has contributions from only the surface integrals:

( t i - Tj i i j ) r i + (Bi - Aijiij)Gi - q + q i i i i = 0. (2.43)

This equation is valid for all Cosserat motions. If we assume that B j , K i i , q, and qi are unaltered by constant superposed rigid-body velocities, c i in (2.43)

298 S. C. Cowin

is replaced by ci + c, where ci is an arbitrary constant velocity vector, then

( t i - T,j~~j)(~i + ci) + (Bi - Kijn,j)Gi - q + qini = 0. (2.44)

From the fact that ci is arbitrary and the fact that ti - T j n j is independent of (‘i 9

1 . I = T . n . J , t = Tn, (2.45)

and (2.44) reduces to

( B i - Aijnj)Gi - 4 + q,17~ = 0. (2.46)

Next, the resulting Eq. (2.46) is subject to a superposed uniform particle angular velocity field. If B i , hij , q, and qi are unaltered by a constant superposed particle angular velocity, and Gi in (2.46) is replaced by 6, + K i , where K i is an arbitrary constant axial vector, one has

(B i - A i , j ~ j ) ( G i + k i ) - q + qi/ l i = 0. (2.47) - -

Since ki is arbitrary and Bi - A i j n j is independent of xi, it follows that

B . = A.. /7 . J , B = An, (2.48)

and (2.46) reduces to

q = q;”; , q = q * 11. (2.49)

The basic results of this analysis are the relations (2.45), (2.48), and (2.49) relating the surface fluxes t, B, and q to tensors of higher order, T, A, and q, respectively, and to the unit normal n of the surface associated with the flux. Henceforth T will be called the (usual, but asymmetric) stress tensor, A the couple stress tensor, and q the heat flux vector. The unified derivation of these results shows that (2.48) follows from the same arguments as the more familiar results (2.45) and (2.49).

Field equations representing the conservation of energy and linear and angular momentum will now be obtained. The relations (2.45), (2.48), and (2.49) are substituted into (2.41). Then, assuming sufficient smoothness, one applies the divergence theorem to the last integral so that the entire equation is one volume integral. Since the volume integral must hold for all regions, its integrands must satisfy the same identity. Thus

p i + p v , ~ , + +piijGidj + pr i jGiGj

= p r - qi.i + pbici + pCiGi + T j r i , i + Tj.jUi + + Aij,jei, (2.50)

where the definition (2.15) has been employed. Previously we have assumed that T, A, and q are unaltered by uniform superposed rigid-body translations;

The Theorj, of' Polar Fluids 299

we now assume that ii, r, b, C, and 1 are similarly unaffected by these motions. Equation (2.50) is valid for all velocity fields and in particular for a velocity field 21, + c, , where ci is a constant vector. Thus

Since ci is arbitrary, it follows that

pi'; = 7 ] j , j + ph . I ? (2.52)

and as a result (2.50) reduces to * , .

pt + +piii Gi G j + pi,, Gi Gk = - y i , i + pci ci + 7;,ui,, + hLijYji + hjj,,Gi. (2.53)

Equation (2.50) is the usual form of the field equation for the conservation of linear momentum. Equation (2.53) must be valid for all particle angular velocity fields, in particular for a particle angular velocity G + K where K is a constant axial vector. When G + K replaces G , we must also replace L by L + K since L = D + W and W is increased by K so that H defined by (2.19) remains unaltered. Making these substitutions into (2.53) we obtain

. ,.- pi: + +pii,iGjGi + p i i j c i ~ , + + p i i j K i K j + prj ,GiGj + ~ I ~ ~ G ~ K ~

= / ) I ' - q,,; + p d , G , + p C ; K , + 7;j"i.j + 2 T p K p

+ Kii,,Gi + hij,jKi + K i , q i j , (2.54)

where we have employed the result

7 ; , j K i j = 2 T p K , , , (2.55)

which follows from (1.la). We assume now that C, I,,, I', and pi as well as 7;,, A,,, and q, are independent of superimposed rigid body rotations. I f K in (2.54) is replaced everywhere by -K and the resulting equation added to (2.54), then when one subtracts two times Eq. (2.53) from the result, we find that

(ii,i( + K) + iij( - K))K, K, = 0. (2.56)

Since this result must hold for all R i , we obtain the condition

( i i j ( + G ) + iij( - C))G, Gj = 0. (2.57)

The requirement (2.56) is new. I t is a consequence of the assumption that I is independent of superimposed rigid-body motions. The particle inertia tensor I i j can be transformed into the reference basis formed by the director

300 S. C. Cowin

triad : thus “fh = I f , (I‘,, 4, (2.58)

represents the components of the particle inertia tensor relative to the particle. When the condition (2.57) is transformed to the director frame it has the form

( j a b ( + e) + j a b ( - e))G, e b = 0. (2.59) I f we assume that I,, is independent of superposed rigid body motions, then

Equation (2.60) was postulated by Eringen (1964) and called the conservation of microinertia. It was derived here in the case of a rigid director triad by assuming that 1 and I were independent of superimposed rigid-body motions.

We return now to the consideration of Eq. (2.54). Equation (2.54) is true for nll K only if (2.57) is satisfied and only if

(2.61)

lo,, = 0. (2.60)

pli,j6,i = [ X i + 2T; + Ai j . i , in which case (2.54) reduces to

pi: = pr + T . D . . ‘ 1 1 . 1 + T . H . . 1 1 , I l + ,&..q.. 1 I 1.1 - q i , i , (2.62) where H is defined by (2.19). Equation (2.61) represents the conservation of angular momentum and (2.62) is the final form of the energy conservation. Generally it is assumed that the inertia tensor I is spherical, thus

l i . = k 2 & , I = k21, (2.63) where k 2 is independent of time and constant superposed motions, k being the radius of gyration of the Cosserat particle. We substitute (2.63) into (2.61) to obtain the final form of the conservation of angular momentum:

p k 2 G i = p f i + 2Ti + , & i j . j . (2.64)

3. The. Entropy Incqiralitj

The entropy inequality in the form of the Clausius-Duhem inequality is written

(2.65)

where is the specific entropy and 0 is temperature. We can convert this integral equation to a field equation if we assume sufficient smoothness to apply the divergence theorem and then use the fact that the resulting volume integral inequality must hold for any body or any part of a body. The result of these considerations is the inequality

ph + (qi/o),i - p(r /O) 2 0. (2.66)

T h e Theory of Polar Fluids 30 1

The set of terms

P’I + ( l / ’ Q ) q i . i - (l /’d)~r, (2.67)

appears in both the energy balance (2.58) and the entropy inequality (2.66). Elimination of these common terms yields

-(l/O)qiO,i - p$ - pbq + T j D i j + TjHji + f i i jP i j 2 0, (2.68)

where we have also employed the expression

$!l E - (2.69)

relating the specific free energy $!l to the specific internal energy I:. The form (2.68) is the form of the entropy inequality that we will employ in the following section.

C. CONSTITUTIVE RELATIONS

The linear constitutive equations defining polar fluids will be developed here. This development begins with rather general constitutive assumptions. Objectivity and thermodynamic restrictions are then used to limit the constitutive hypothesis, and finally, linearized constitutive relations are obtained.

1 . The Constitutive Assumptiori.s

Constitutive equations are needed for the specific free energy $!l, the entropy q, the heat flux q, the stress tensor T, and the couple-stress tensor A. The independent variables are the specific volume u (i.e., ( I - ’ ) , the temperature 0, the velocity v, the angular velocity G , and the gradients of tenipera- ture, velocity, and angular velocity. We denote this set by F,

3 = {u , 0, 0 ~ 1 , 0, 1 V,.] , G, , G.J. (2.70)

There are at least two points of view on how one should proceed at this point. The first school of thought holds that one should use physical and mathematical insight to specify which members of .F are to be taken as independent variables for each of the quantities for which constitutive equations are needed. The second school of thought views the method of the first as subjective and prefers the unprejudiced assumption that al l the constitutive equations should have the same set of independent variables unless a particular independent variable is in direct contradiction with the assumed symmetry of the material, with the principle of material objectivity or with the laws of thermodynamics. This position is described by Coleman and Mizel (1964) and is called the principle of equipresence. In the present situation we conform to the second school of thought and assume that $!l, q,

302 S. C. Cowiiz

q, T, and A all depend upon .Q,

$ = $(5), q = q(.P), q = q(,P), (2.71)

As no need or justification can be found for deviating from the set of unprejudiced assumptions (2.71), we take them as our starting constitutive assumptions.

T = T(.F), A = A(,F) .

2. Objectivity Restrictions

The principle or axiom of material objectivity is the statement that constitutive equations must be unaffected by rigid-body motions and the inversion of the spatial or Eulerian reference frame. It means, for example, that the stress in a material body is unchanged if the body is subjected to a superposed rigid-body motion or an inversion of the spatial reference frame. This is easy to see in the case of an elastic body because there is no deformation of the body in a rigid-body motion, and hence there is no strain and no stress. This idea is discussed in some detail by Truesdell and No11 (1965). I t will be used here to limit the form of the constitutive assumptions (2.71).

The set of eleven functions

v = V(X, t ) ,

A = A(x, t ) ,

G = G(x, t ) ,

$ = $(x, t ) ,

T = T(x, t ) ,

q = q(x, r ) , q = q(x, r ) , tj = O(x. t ) ,

(2.72)

r = r(x, r ) , b = b(x, t ) , C = C(x, t ) , (2.73)

constitutes a polar thermodynamic process if it is compatible with the balance of linear momentum (2.52), the balance of angular momentum (2.64), and the balance of energy (2.58). Actually, one need only specify the eight functions (2.72) to obtain a polar thermodynamic process because r(x, t ) is then given by the balance of energy (2.58), b by the balance of linear momentum (2.52), and C by the balance of angular momentum (2.64). The polar thermodynamic process specified by (2.72) and

V* = v*(x*, t*),

A* = A*(x*, t*),

q* = q*(x*, t*),

G* = G*(x*, t * ) ,

T* = T*(x*, t*), 4" = q*(X*, t*),

$* = $*(x*, t*),

8" = O*(x*, t * )

(2.74)

are said to be equivulent if under the transformation (2.21) the eight func-

The Theory of Polar Flitids 303

tions (2.72) are objective, that is to say if the eight functions transform according to the rules

V * ( t * ) = Q(t)v(t),

G*(t*) = .yQ(t)C(t),

A*(t*) = S Q ( ~ ) ~ Q " ' ( [ ) ,

Il*(t*) = Il(t)>

q*(t*) = Q(t)q(f),

T*(t*) = Q(t)T(t)Q"(t),

$*( t* ) = $(t) ,

O*(t*) = U ( t ) .

(2.75)

The principle of material objectivity is the requirement that the constitutive equations (2.7 1 ) be invariant under changes of the spatial reference frame of the form (2.21). Thus, if the constitutive equations are satisfied by a process (2.72), then objectivity requires that they be satisfied by all equivalent processes (2.74).

The subset of the set .F given by (2.70) consisting of functions in 3 that are objective is denoted by

C' = {u, 0, VO, D, H, 'Pi, (2.76)

where the objectiveness of these arguments is demonstrated by (2.31b), (2.32), and (2.34). That the gradient of a function is objective can be deduced from (2.21) and (2.22a). Since the set l( is objective, it is transformed by (2.2 1 ) into the set

C' Q = [v , 0, Q(t)VO, Q(t)DQ'(t), Q(t)HQ'(t), sQ(t)'PQ'(f)). (2.77)

In the particular case when Q is the inversion - 1 we introduce the notation

Cr -, = {c, 0, -VO, D, H, -'PI. (2.78)

From the principle of material objectivity it follows that the functions $, v, q, T, and A can depend upon 9 only through its objective subset 6; thus (2.71) reduces to

$ = (/(C), y~ = q(Q ), 4 = q(C'), T = T(C ), A = A(P). (2.79)

Furthermore, the restrictions (2.75) require that these six functions be isotropic in the set of variables Cf. The condition of isotropy requires that these functions obey the following identities for every orthogonal tensor Q:

$(C = $((' OX Yl((') = Il((' Q)? Q(t)q(l' ) = q(('oh

Q(t)T(I!.)Q'(t) 1 T(C'Q), .sQ(f)A(lf')QT(f) = A(l'Q). (2.80)

These identities follow, of course, from (2.76) and (2.80). This extends to polar fluids the well-known result that all homogeneous Newtonian fluids must be materially isotropic. The material isotropy of polar fluids is completely characterized by (2.80).

304 S. C. Cowin

Some interesting and important results can be obtained by considering Eq. (2.80) in the special case when Q is the central inversion - 1. In this case

I)((') = $(C q(C') = ~ I ( C - I ) , T(k) = T(C'-I), (2.81)

q(C) = - q ( L 1 ) , A((') = -A(('

Thus, for fixed values of u, 0, D, and H, the variables $, y~, and T must be even functions of VU and 9 while q and A must be odd functions of VU and 9. In particular then, it follows from (2.81) that q and A must vanish when V6' and 9 vanish.

3. Thernzotlynamic Restrictions

Thermodynamic restrictions on the constitutive functions (2.79) will be developed with the method refined in the papers of Coleman and Noll(1963) and Coleman and Mizel (1964). The method consists of employing the entropy inequality to restrict the functional dependence assumed for the constitutive relations. Recognizing the functional dependence of $ upon c, 0, VU, D, H, and 9 specified by (2.79), the entropy inequality (2.68) can be written in the expanded form

1 6'

- q i0 . i - pUq + 7;,iDij + 7],iHji + Ai,iYij 2 0. (2.82)

We now employ the argument initially outlined by Coleman and Noll(l963) and Coleman and Mizel (1964). They argue that there is at least one polar thermodynamic . . process satiqfying the constitutive relations (2.79) in which the values of0, u. , , H j k , i , and q j k . i can be specified independently of any other term in the equality. The inequality then implies the following restrictions:

D,,, H i j , qi j , v ~ , ~ , , ,

These restrictions show that $ is a function of L' and 0 only,

(1 = $ ( u , 0). (2.84)

From this result it also follows that q depends upon I ) and 0 only:

yI = - ( d $ / d 0 ) ( u , f l ) . (2.85)

Similarly, we introduce the notation

p = - (c?$/(;.L')(u, 0) (2.86)

The Theory of Polur Fluids 305

for the other derivative of $. The results (2.84)-(2.86) are the traditional thermostatic results for liquids. A polar fluid is therefore thermostatically equivalent to a liquid. When (2.83) is substituted back into the entropy inequality (2.82), it reduces to

- (l/O)q,O., 2 0, (2.87)

where a) is the mechanically dissipated power

0 = (Tj + pS . . )D . . 1 J l J + T . H . . 1 l i l + A i j Y i j . (2.88)

I n summary, the thermodynamic restrictions have reduced the functional dependence indicated for $ and q by (2.79) to that given explicitly by Eqs. (2.84) and (2.85). The functional dependence indicated for q, T, and A by (2.79) has not been reduced, but it must satisfy the reduced inequality (2.87) where is given by (2.88).

4. Linearizat ioli

A linearity assumption will now be introduced. Specifically, it is assumed that q, T, and A are linear and homogeneous in the variables VO, D, H, and 9. This assumption leads to a system of equations by which q, T, and A are related to VO, D, H. and 9 in a linear fashion, the coefficients of linearity being tensors of the appropriate rank. The conditions of isotropy (2.80) require that these coefficient tensors be isotropic tensors, thus we obtain the representations

(2.89)

(2.90)

(2.91)

for q, T, and A. The tensor 6 is introduced a s a shorthand for simple linear combinations of the temperature gradient VO,

q = -1iVO + (OV x G ,

T + pl = IJ. Tr D + 2,uD - 2sH,

A = l a Tr Y + (/j + ;)$ + (/j - ;I)$” + 06,

Oi,jk = o , i ( s j k - l ) , j6 j , , 0 . . 1.1 = e . . 1 j h . h 0 ; (2.92)

and /i, ( J ) , 2, 1.1, z, R, /I, 7 , and ca re coefficients independent ofVO, D, H, and 9 but possibly dependent upon 0 and I‘. The coefficient ti is the thermal conductivity, i. and ,LL are the usual viscosity coefficients, T is a relative rotational viscosity called, simply, the rotational viscosity, and K, p, and ;‘are viscosities associated with the total rotation gradient and are called, for simplicity, gradient viscosities. The coefficients (o and c are measures of an untraditional thermomechanical coupling.

The constitutive equation for the heat flux (2.89) reduces to the Fourier law of heat conduction when the thermomechanical coupling term COV x G

306 S. C. Cowin

vanishes. The constitutive equation (2.90) for the stress reduces to the New- tonian law of viscosity when the term 2zH vanishes. The thermomechanical coupling terms appearing in (2.89) and (2.91) are solenoidal or nondivergent if the coefficients co and 0 are constant. Since it is the divergence of the heat flux that appears in the expression for energy conservation (2.58) and the divergence of the couple-stress tensor that appears in the expression (2.64) for angular momentum conservation, it follows that the thermomechanical coupling terms do not enter directly into these conservation laws when the coefficients co and 0 are constant.

Grad (1952) discusses the magnitude of the viscosity coefficients a, p, y , and z relative to the magnitude of the Newtonian viscosity coefficients p and A. He employs some mean-free-path arguments to show that the gradient viscosities c(, /?, and 2’ are of the order k2p, where k is the radius of gyration introduced in (2.63), and that the rotational viscosity T is of the order k21; ’ p , where 1, is the mean free path. Grad concludes that the angular momentum effects “in a gas at any rate” are extremely small.

The coefficients K , o, A, p, z, a, p, 11, and cr are restricted by the requirements that q, T, and A obey the entropy inequality (2.87) for all values of the fields VB, D, H, and Y, To obtain these restrictions the constitutive equations (2.89)-(2.91) are substituted into (2.87) and (2.88), whereupon the entropy inequality takes the form

A(Tr D)2 + 2~1 Tr D2 + 4zH . H + a(Tr Y)’ + (p + y)Tr 9”’

+ (p - y)Tr 9’ + (K/B)VO . V 8 - ( w / O + o)VO . V x G 2 0. (2.93)

For the inequality (2.93) to hold for all Cosserat motions and all temperature fields it may be shown, by standard methods in the theory of quadratic forms, that the following conditions on the coefficients are necessary and sufficient:

p 2 0, 32 + 2p 2 0, ti 2 0, (2.94a,b,c)

3% + 2p 2 0, p 2 0, y 2 0, z 2 0, (2.95)

(w + Ho)2 I 47tio. (2.96)

The inequalities (2.94) are the usual inequalities restricting the viscosity coefficients and the thermal conductivity. Equations (2.95) restrict the gradient viscosities a, p, y and the rotational viscosity z, while (2.96) places a condition on the coefficients 0 and w. Cowin (1968a) gave a form of (2.96) containing two errors, a sign error and an omission of a factor of 4.

D. THE FORMULATION OF BOUNDARY-VALUE PROBLEMS

In the beginning of this section the basic system of equations for polar fluids are obtained and analyzed from a dimensional viewpoint. After a


discussion of the appropriate boundary conditions, the boundary-value problems for polar fluids are posed. Questions of uniqueness and stability of solutions of the boundary-value problems are considered in the closing subsection.

1. The Basic Systein of’ Equations

Expressions for the conservation of mass, linear momentum, angular momentum, and energy in a Cosserat continuum have been obtained and are repeated here in the indicated order:

ii + = 0, (2.35)

(2.52)

(2.64)

pL i = 7],i.j + phi ,

pk2Gi = /)ti + 2Ti + A i j , i , (2.62)

The Navier-Stokes equations for Newtonian fluids are derived by placing the constitutive relation for the symmetric part of the stress T, (i.e., the Newtonian law of viscosity) into the conservation of linear momentum, and subsequently rewriting the linear momentum balance in terms of the velocity vector. In the present theory there are six equations of motion. counting linear momentum balance as three and angular momentum balance as three, and constitutive relations for the couple stress A as well as for T. This system of equations reduces to six equations in the components of the velocity v and the particle angular velocity G. In deriving these equations it is useful to have the identity

G = H + + v x v, (2.97)

which follows from (2.18) and (2.19). Substituting the linear constitutive relations (2.89)-(2.91) into the conservation equations (2.52), (2.64) and (2.62) and assuming that the material coefficients i, ,LL, z, x, /j, and cr are constant, then

(2 + p)vv * v + pV2v + 2Tv x H + pb - v p = pV,

( M + f l - ;j)V(V * G ) + ([j + y)VzG - 4zH + pC = p k 2 G ,

(2.98)

(2.99)

pi; + p Tr D = /<V20 + pr + R(Tr D)2 + 2p Tr D2 + 47H * H + x(Tr 9)’ + (/I + ;i)Tr %‘7

+ (/) - ;$)Tr q2 - rrV0 * V x G . (2.100)

The velocity equation of motion (2.98) becomes the Navier-Stokes equation when the 2zV x H vanishes. Using the identity (2.97) it may be shown that

308 S. C. Cowin

the velocity equation of motion may also be written in the form

(i + ,LL - r)V(V - v ) + ( p + r)V2v + 2rV x G + pb - Vp = pv. (2.101)

Equation (2.100) expressing energy balance reduces to the corresponding equation for viscous fluids when H and Y vanish.

2. Di~izensio~iul Cori.sidrr.utioii.s \

Standard dimensional arguments will be used here to develop a character- ization of the polar fluid by dimensionless numbers. Before proceeding with these arguments, it is convenient to introduce two notations. First, a material property of the polar fluid having the dimension of length may be introduced by noting that the gradient viscosities s(, [j, and 7 are of dimension length squared times the dimension of viscosity. This property, called the material characteristic length and denoted by 1, is defined by

1 = [ ( B + y)/p]? (2.102)

Since the thermodynamic restrictions (2.94) and (2.95) require that /i, [j, and 7 all be positive, it follows that 1 is a real positive number. It should be noted that the definition of material characteristic length 1 used here and elsewhere is twice that employed by Cowin (1968a) and Pennington and Cowin (1969).

The second notation to be introduced is a symbol for the dimensionless ratio p/r . I t is convenient to introduce the symbol N :

h’ = [T/(,u + T ) ] ” ~ , 0 I N I 1, (2.103)

for this purpose. N is called the coupling number because it characterizes the coupling of the conservation of linear and angular momentum, as may be seen from (2.98) and (2.99). The coupling number N is a property of the fluid and is a measure of the degree to which a particle is constrained to rotate with the average angular velocity of the region in which it is embedded. N can also be considered as a ratio of viscous forces of relative rotation to the Newtonian or traditional viscous forces. The absence of viscous forces of relative rotation is characterized by the vanishing of r and N and corresponds to the case of completely unconstrained rotations. For vanishing r (2.98) and (2.99) are uncoupled and (2.98) reduces to the Navier-Stokes equation. As the coupling number tends to one the viscous forces of relative rotation become predominant and the particle will rotate with an angular velocity closer to the average angular velocity of the region in which it is imbedded. For N = 1, the total angular velocity G coincides with the average angular velocity W and the theory of polar fluids reduces to the couple- stress theory of fluids presented by Stokes (1966).

To convert the equations governing a polar fluid to their dimensionless

The Theory of‘ Polar Fluids 309

forms the dimensionless velocity v’, spatial coordinates x’, pressure p’, time t‘, and temperature 0’ are introduced:

v = V,V’, x = L” x’, p = Pop’,

t = (L,/V,)t‘, 0 = O,O‘, (2.104)

where V , , L o , P o , and 0, are dimension-bearing constants. In the development that follows a prime on a quantity will indicate that the quantity has been rendered dimensionless. When we substitute (2.104) into (2.98) and (2.99) and subsequently divide through by p V ; , we find that

where the dimensionless numbers R , F , Q. L, J , and C are given by

Po Q = - - ,I r/2 7

(2.105)

(2.106)

The numbers R, F , and Q are familiar from traditional viscous fluid theory; R is a Reynolds number, F is a Froude number, and Q is a pressure ratio. The dimensionless number J is a ratio of the radius of gyration k to the characteristic geometric length Lo while the number C plays a role analogous to that of the Froude number F. The dimensionless number L is called the length ratio; it is a ratio of the geometric characteristic length Lo to the material characteristic length 1.

Interesting properties of the polar fluid as L + a or L + 0 can be obtained for a certain broad class of polar-fluid motions. Specifically, as L --f 35 the system of Eqs. (2.98) and (2.99) reduce to the familiar Navier-Stokes equation for a viscous fluid:

(i + p)VV * v + pV2v + pb - V p = pi’. (2.108)

On the other hand as L --f 0 the system of Eqs. (2.98) and (2.99) again reduce

3 10 S. C. Cowiri

to the familiar Navier-Stokes equation for a viscous fluid.

(j” + p - z)VV * v + ( p + T)V’V + pb - Vp = pV, (2.109)

but with altered viscosities. The effect of L varying from infinity to zero is to increase the shear viscosity from 1-1 to 11 + z and to decrease the bulk viscosity from 3, + 41-1 to i + 4p - $T. Large values of L imply that the characteristic geometric length (of a given boundary value problem, e.g., the diameter of a tube) is much greater than the material characteristic length 1. Small values of L imply the converse. The class of flows for which these properties of the polar-fluid model hold are characterized by the conditions that (1) L o , z, and p all remain finite, (2) G vanishes on the boundary, (3) V x. (pC - p k ’ G ) = 0, (4) V2(V x G ) remains bounded as p + i‘ -+ 0, ( 5 ) p k ’ G - pC remains bounded as f l + 7 + K,, and (6) V - G is a constant. In spite of there being these six restrictive assumptions on the type of flow, a great many flows of interest satisfy these assumptions. The proof of these properties of the polar-fluid model follows that given by Cowin (196%).

An analysis of the energy equation (2.100) using the same procedure just employed on the linear and angular momentum equations (2.98) and (2.99) leads ‘to the familiar Eckert-Prandtl numbers and one new dimensionless number defined by

s = p vo Lo~CrOo , (2. I 10)

which involves the coefficient (T of thermomechanical coupling.

3. Boundary Corirfitiom

The same boundary conditions used in viscous fluid theory for the velocity field v(x, t ) can be used in polar-fluid theory. The most frequently used boundary condition in both viscous fluids and polar Huids is the condition which requires that the velocity of the fluid at the boundary equal the velocity vo of the boundary:

v = vo at all solid boundaries. (2.111)

There have been a variety of suggestions for the boundary condition to be used for the total angular velocity G(x, t) . The most frequently used is the condition that the total angular velocity of the polar fluid at the boundary equal the angular velocity of the boundary Go; thus,

G = Go at all solid boundaries. (2.1 12)

The axial vector G represents the total angular velocity of a fluid particle; hence, if a particle is inflexibly attached to a boundary, the total angular velocity of the particle is equal to the angular velocity of the boundary. This

The Theory qf Polar Fluids 311

condition is called the hyperstick condition or the “no- (relative) spin” condition in analogy with the “no-slip” condition expressed by (2.1 11).

Other suggestions in place of (2.112) have been made. Aero et a / . (1963) suggested that the couple stress on the boundary might be related to the difference between the total angular velocity of the fluid at the boundary and the angular velocity of the boundary by a friction factor, i.e.,

i n = F(G - Go) at all solid boundaries, (2.1 13)

where n is the unit normal to the boundary and F is a second-rank tensor representing the fluid-wall friction. In the special case when F vanishes, (2.1 13) contains the requirement that there be no couple stress on the solid boundary.

In a similar vein Condiff and Dahler (1964) have suggested that the skew- symmetric part of the stress T on the boundary might be related to the same difference in angular velocities by a friction factor, i.e.,

T . n = [F’(G - Go)] * n at all solid boundaries, (2.1 14)

where F’ is another second-rank tensor representing fluid-wall friction. A common characteristic of the boundary conditions (2.1 l2)-(2.114) is a

dependence upon the quantity G - Go at a solid boundary. The only exceptions to this are the cases when F and F’ vanish which correspond to no couple stress and no skew-symmetric stress on the boundary, respectively. Ariman et al. (1967), Ariman and Cakmak (1968), Rao et a/ . (1969), and Rajagopalan (1968) employ a boundary condition that does not have this characteristic; they require that

G = 0 at the solid rotating boundaries. (2.1 IS)

Cowin and Pennington (1970) have shown this boundary condition to be unreasonable because it requires a particle situated upon a rotating boundary to be itself not rotating, but rather to keep its orientation fixed in space. The difference between the boundary conditions (2.1 12) and (2.1 I S ) is illustrated in Figs. l and 2. The large concentric circles in the figures represent the wall of a fluid-filled cylindrical vessel rotating with a constant angular velocity R about its axis of revolution. In both figures the small circles tangent to the inner circle are enlarged representations of eight positions of a single fluid particle; the time interval between each of the eight positions is 7r/4R. Figure 1 represents the boundary condition (2.112) where the fluid particles do not rotate relative to the boundary; this is indicated by the arrowheads on the cylinder wall and on the particle representation always being contraposed. Figure 2 illustrates the boundary conditions (2.1 IS) in the same situation. In this case the particles at the boundary cannot rotate even though the boundary itself is rotating. Thus the arrowheads on the

312 S. C. Cowin

Flci . I . Illustration of the boundary condition (2.1 12) in thc case of a rotating boundary. From Cowin and Pennington (1970) by permission of Dr. Dietrich Steinkopff Verlag.

FIG. 2. Illustration of the boundary condition (2.1 15) in the case of a rotating boundary. From Cowin and Pennington (1970). by permission of Dr. Dietrich Stcinkopff Verlag.

The Theori> of Polar Fluid.\ 313

particle representations in Fig. 2 always maintain their same orientation. The objection to the boundary condition (2.1 15) is on physical grounds.

Unless the particle is acted upon by an outside force, it will tend to rotate with the system in which it is embedded and it will not maintain the fixed orientation in space required by (2.1 15).

4. The Boundury- Value Problems ,fbr the Compressible and Iiicon~pressihle Cases

In the general case of a compressible polar fluid the equations (2.35) (2.98), (2.99), and (2.100) expressing, respectively, the balance of mass, linear momentum, angular momentum, and energy are coupled and must be solved simultaneously. This system of equations forms a system of eight scalar equations in the thirteen scalar unknowns contained in v, G , H, 0, p, 11,

and E . The fields r, b, and C will be specified in a particular boundary-value problem. Five additional scalar equations are obtained by adjoining the kinematical identity (2.106) and the equations of state

i: = c(c, O), p = p(c , O), (2.1 16) which follow from (2.69), (2.85), and (2.86), to the eight scalar equations characterizing the conservation principles. Thus there is a system of thirteen equations in thirteen unknowns or ten equations in ten unknowns if we use (2.97) to eliminate H whenever it occurs and strike H from the list of unknowns.

To solve a boundary-value problem for a compressible fluid the system of Eqs. (2.35), (2.97)-(2.100) and (2.116) must be solved for the fields v, G , H, 8, p, c, and i: given the fields r, b, and C . The solution is subject to the restrictions (2.94)-(2.96) on the constant coefficients p , 1, K , SI, [j, 7 , 0, and w and subject to certain boundary and initial conditions. The boundary conditions on all the variables except G and H are the same as in classical theories for thermomechanical continua. The boundary conditions on G (or H) were discussed in the previous section.

When the constraint of incompressibility is applied to polar-fluid theory, the net effect is the same as in the theory for a Newtonian fluid. The substantial derivative of the density is zero and the conservation of mass (2.35) reduces to

v ' V = 0. (2.1 17) The constitutive equations (2.89)-(2.91) still have the same form but the pressure p occurring in (2.90) is no longer the thekmodynamic pressure (2.86) but is an arbitrary scalar. The most important 'consequence of the incompressibility requirement is, perhaps, the fact that the mechanical and thermal boundary-value problems uncouple in the sense that the mechanical problem can be solved independently of the thermal problem. Thus the system of

3 14 S. C. Cmvin

Eqs. (2.1 17) and ( p + z)V2v + 2zV x G + pb - Vp = pV, (2.1 18)

(LY + - y)V(V ' G ) + (B + y)V2G - 47G + 27V x v + pC = p k z G (2.1 19)

which follow from (2.98) and (2.99) through (2.96) and (2.1 17), form a system of seven scalar equations in seven scalar unknowns v, G, and p . The boundary conditions for this system are those discussed in the previous section.

5. Theorenis of Stabilitji arid Uiiiyiieriess

In this section two theorems for the universal stability of incompressible polar-fluid motions and two associated uniqueness theorems will be proved. This work is patterned after the treatment of the corresponding viscous fluid theorems given by Serrin (1959). The first three theorems are essentially due to Rao (1970, 1971), but the notation has been changed and some corrections have been made. All four theorems were obtained by Shahinpoor and Ahmadi (1972) in the special case of fluids with couple stresses (see Section III), neglecting the particle inertia. The results of Shahinpoor and Ahmadi (1972) were extended to include particle inertia by Cowin and Webb (1973). The two stability theorems give criteria for the stability of an arbitrary fluid motion in a bounded region aiid are, therefore, called universal stability theorems. The proof of these theorems involves the consideration of the kinetic energy K of a difference between the basic motion and another motion. If the kinetic energy K of the difference motion tends to zero as t --f cc, then the basic motion is said to be stable, or, more correctly, stable in the mean. In the following paragraph an expression for the time derivative of the kinetic energy of the difference motion is obtained. In the next paragraph certain identities and inequalities useful in the proof will be recorded. The two stability theorems and the two uniqueness theorems will then be proved.

We consider a region of space denoted by V = V ( t ) containing an incompressible polar fluid with velocity v aiid particle angular velocity G. The velocity and particle angular velocity of the perturbed motion are denoted by v* and G*. The velocity and particle angular velocity of the difference motion are denoted by u and A and defined as follows:

The translational and rotational kinetic energy, K , and K,, respectively, of the difference motion are given by

u = v * - v , A = G * - G . (2.120)

K , = (. t p u - u du, K , = [ i p k 2 A . A do. (2.121) ' V ' V

The total kinetic energy of the difference motion is

K = K , + K,; (2.122)


a result that follows from (2.39), (2.63), and (2.121). To compute an expression for the rate of change of kinetic energy we consider the equations of motion (2.1 18) and (2.1 19) both for the basic motion v, G and the altered motion v*, G* and subtract the two. Thus

p((?u/?t) + u - V v + v* * V U ) = V(p - p * ) + 2zV x A + ( p + 5)V%

(2.123)

pk2{ (?A/? t ) + u V G + V* * VA} = 25V x u - ~ T A

- (B + y)(V x V x A) + (LY + 2/1)V(V A), (2.124)

where the vector identity (1.8) has been employed. The scalar product of (2.123) with u and (2.124) with A gives

fp(;l/?t)(u * U) = - p u . DU - pV * [(f~ u)v*]

+ V - [ ( p - p*)u] + 2TU * v x A - ( p + .)(V x u) * (V x u)

+ ( p + 5)V - (u x V x u),

- pk2u * (‘PA) - fpk’V - [(A. A),*]

(2.125)

+ 2 5 A . v x u - 4 5 A . A

- ( p + y){V - [(V x A) x A] + (V x A) * (V x A)]

+ (. + 2B)[V - [(V - A)A] - (V - A)2], (2.126)

where the vector identities

V * (U x V x U ) = -(V x u ) . (V x U) + u.V’U - u . V ( V * u ) , (2.127)

A . (V x v x A) = (V x A) * (V x A) + V * [(V x A) x A], (2.128) A ’ V(V - A) = V[(V * A)A] - ( v * A)’ (2.129)

and the incompressibility condition (2.1 17) dn u have been employed. We now integrate (2.125) and (2.126) over the region V , use the divergence theorem, the transport theorem, and the boundary conditions

U = O , A = o on ?v. (2.130) The results of these manipulations are the expressions

- - d K T pu * Du d v + 25 (. u * V x A dr

d t ‘ V

- ( p + T ) (. (V x u) - (V x u) d c ‘ V

(2.131)

316 S. C. Cowin

and

- ( p + ; I ) 1' (V x A) - (V x A) t lc - (Y + 2p) (. (V A ) 2 ( 1 1 3 ,

' V ' V

(2.132)

where (2.121) has been employed. From (2.122), (2.131) and (2.132) we obtain an expression for the rate of change of the total kinetic energy

- 4s (. (A - qv x u) * (A - qv x u) dc ' V

- /1 (. (V x u) - (V x u) dt? - (p + ; I ) 1 (V x A) - (V x A) t ic ' V ' V

- (Y + 2p) 1. (V - A ) 2 d u , (2.133) ' V

where the integral relation

(. (u * v x A) dc = 1' (A ' v x u) tic, ' V ' V

(2.134)

which follows from the vector identity

V - (A x u) = u . v x A - A . v x u, (2.135)

the divergence theorem, and the boundary conditions (2.130) has been employed. Note also that L defined by (2.14) may replace D defincd by (2.16a) in (2.133) and vice versa. Another form of (2.133) may be obtained by using the integral identities

(2.136a)

and ^ ^

u 'PA LIU = - u - (VA)G t l r (2.136b) .JV

which are also dcrived using the divergence theorems and the boundary conditions (2.130). The second expression for the rate of change of kinetic

Tlzr Theory of Polar Fluids

energy is then

317

= pu - (Vu)v du + pk2u - (VA)G du dK

dt JV .ib - 42 [ (A - +v x u) - (A - +v x u) nu

‘ V

- (. + 28) ). (V * A ) 2 du. (2.137) ‘ V

The derivation of the results (2.133) and (2.137) depended on the divergence theorem and the transport theorem. These theorems are only true for bounded regions unless conditions on the asymptotic behavior of v and G are specified. If the asymptotic behavior of v and G is properly specified, the results (2.133) and (2.137) can be justified for infinite regions. These results can also be justified for infinite regions when the disturbances are assumed to be spatially periodic at each instant.

Identities and inequalities needed to bound the terms occurring in the expressions for the time rate of change of kinetic energy are summarized in this paragraph. Let d denote the diameter of a ball containing the spatial volume V ( t ) over which we integrated to find the kinetic energy K . A lower bound for the eigenvalues of the rate of deformation tensor D of the basic flow over the time interval 0 to t and over the spatial volume V ( t ) is denoted by - i n . Then

u * LU = u * DU 2 -WIU * U. (2.138)

Let M denote an upper bound for the magnitude (Tr $q7)1’2 of of the basic flow over the time interval 0 to t and over the spatial volume V ( t ) . Then

u - YA 2 -n(u - u ) y & * A)? (2.139)

Expanding the expression (u * u - dA . A)2 2 0 we obtain an inequality which permits us to write (2.139) in the form

u * YA 2 -+@A . A + (l/d)u * u). (2.140)

The maximum speed and the maximum total angular speed of the basic flow over the time interval 0 to t and over the spatial volume V(r ) , are denoted by uo and G o , respectively, so that

V - V < L $ , G . G < G i . (2.141)

318 S. C. Cowin

For any vector field w that vanishes on the boundary of V it can be shown that

1 [(V x w) . (V x w) + (V - w)’] do = Tr(Vw)(Vw)’ I IC. (2.142) ‘ V .J“

Thus if we define x* to be the minimum of (m + 2/1) and (/1 + ;,),

([j + y) (. (V x A) - (V x A) d v + (2 + 2p) 1’ (V - A ) 2 LI1’ ‘ V ‘ V

2 a* (. Tr(VA)(VA)’ dp. (2.143) ’ V

From Payne and Weinberger (1963) we know that

T ~ ( V U ) ( V U ) ~ dt. 2 (80/d2) (. u - u dc (2.144) ‘ V

and

Tr(VA)(VA)’ d r 2 (3n’/ri2) 1‘ A . A dr. (2.145) ib ‘ V

The identity

Tr(T - w 0 q)(T - w 0 q)T = Tr TTT + (w - w)(q * q) - 2q - T w 2 0, (2.146)

which holds for any tensor T and any vectors w and q can be used to show that

u * ( V U ) ~ I ( ,~~/2p)Tr(Vu)(Vu)~ + (p/2,u)(u - U)(V * v) (2.147)

and

u * ( V A ) G I ( , u / ~ ~ ) T ~ ( V A ) ( V A ) ~ + (p/2p)(u - u)(G . G). (2.148)

With these identities and inequalities recorded, we are now in a position to prove the theorems. The first two theorems involve the concept of stable motions. A motion v(t) and G ( t ) of a polar fluid is said to be stable if, for any arbitrary initial difference motion u(t,), A(to) with kinetic energy K(t,), there exists a t > to such that the kinetic energy K ( t ) of the difference motion at time t is arbitrarily small. In the first theorem the symbol J is introduced to denote the ratio of the radius gyration k to the ball diameter d:

J = kid. (2.149)

Theorem 1 (First universal stability criterion). Let V = V ( t ) be a bounded region of space contained within H ball of diameter d. Let v and G represent

The the or.^) of Polar. Fliiitls 3 19

the motion of an incompressible polar fluid in V and let them satisfy prescribed conditions at the boundary ?V. If the Reynolds number R satisfies the condition

R R", (2.150)

where

R" = min[80/(1 + J 'W), 3z2M/J2W], (2.151)

R = pmr12/p, M = a*/d2p, W = nd/2m, (2.152)

then the motion is stable. To prove theorem 1 the inequalities (2.138), (2.140), (2.143)-(2.145) are

substituted into (2.133) to obtain an inequality for dK/dt. When the notations introduced in (2.121), (2.149), and (2.152) are employed, this inequality can be rewritten in the form

d K / d t I { R - [80/(1 + J'W)])(l + J2W)(p/2/) d 2 ) K T

+ [ R - 3712(M/J2W)]2J2W(p/pk2)KR

- 4S (. (A - +v X U) * (A - $v X U ) dc . (2.153)

Since z must be positive i t is easy to see that the sufficient conditions for the right-hand side of (2.153) to be nonpositive are given by (2.150) and (2.151). This completes the proof of theorem 1.

Theorem 2 (Second universal stability criterion). Let V = V ( t ) be a bounded region of space contained within a ball of diameter (1. Let v and G represent the motion of an incompressible polar fluid in V and let them satisfy prescribed conditions at the boundary ?V. If the conditions

R 2 5 80/(1 + W2J2) , a" 2 S ( k 2 p ) , (2.154a,b)

' V

where

R = puo d/p , W = Go d/vo , (2.155)

are satisfied, the motion is stable. The proof of theorem 2 employs the inequalities (2.141), (2.142) (2.144),

(2.145), (2.147), and (2.148) in the expression (2.137) to obtain an inequality for d K / d t . The notations (2.121), (2.144), and (2.155) permit this inequality to be written in the form

d K / & 5 [ R 2 - [80/(1 + W2J2)]i(l + J 2 W 2 ) ( p / d 2 p ) K ,

+ [(k2p/2) - ~ * ] ( 6 z ~ / p d ~ k ~ ) K ,

- 42 1. (A - +v x u) * (A - +v x u) da. ' V

(2.156)

320 S. C. Cowin

Again, since z must be positive it is easy to see that the sufficient conditions for the right-hand side of (2.156) to be nonpositive are given by (2.154). This completes the proof of theorem 2.

The restriction (2.154a) of theorem 2 is not necessarily a restriction on the properties of the polar fluid. I t is simply one of a set of sufficient conditions for the inequality (2.156) to be nonpositive. It is a sufficient but not necessary condition for stability, a condition obtained by employing a collection of inequalities, some of which could possibly be sharpened to remove the restriction (2.154b) entirely from the statement of the theorem. The result (2.154b) is interesting in that Grad (1952) estimated a* to be of the magnitude p k 2 , as was mentioned in Section 11,C,4.

Theorem 3 (Uniqueness of steady motions). Let V = V ( t ) be a bounded region of space contained within a ball of diameter tl. Let v, G and v*, G* be two steady flows in V ( t ) subject to the same prescribed boundary conditions. Then if either of the conditions (2.150) or (2.154) is satisfied, the two flows are identical.

The proofof this theorem rests upon the fact that the kinetic energy of the difference motion v - v*, G - G* must be constant since the flows are steady. I t follows then from (2.153) or (2.156) that if (2.150) or (2.154), respectively, is satisfied, then the difference motion must be zero.

Theorem 4 (Uniqueness of unsteady motions). Let V = V ( t ) be a bounded region of space'contained within a ball of diameter d. If v, G and v*, G* are two motions subject to the same boundary conditions and the same initial conditions at t = 0, then the two motions will always be the same if either condition (2.150) or (2.154) is satisfied.

We will prove this only for the condition (2.150). Proof for the condition (2.154) is similar. From (2.122) (2.150), (2.151), and (2.153) it follows that

dK,Jdt I ( R - R * ) C K , (2.157)

where C is a positive number, with the dimensions of inverse time, given by

C = min[( 1 + ~ ~ ~ ) ( p / 2 p d 2 ) , 2 ~ ~ ~ ( p / y k ~ ) ] . (2.158)

The inequality (2.157) may be integrated, thus

K I K,e(R-R*)'' (2.159)

where K O is the kinetic energy of the system at t = 0. Since the two flows are everywhere identical at t = 0, it follows from (2.159) that K ( t ) = 0, and hence the two flows coincide for all time t > 0. A similar result is obtained by employing (2.154) and (2.156).

The universal stability theorems proved in this section are not as strong as those that can be obtained by considering particular situations such as

Poiseuille flow, Couette flow, etc. The four theorems of this section show that there are stable unique solutions to the basic system of differential equations for both steady and unsteady flows if such solutions exist.

E. TYPICAL BOIJNDAKY-VALUE PKOI~LEMS

A number of boundary-value problems have been solved for the iiicom- pressible polar fluid. The solutions reviewed in this section are for steady, fully developed laminar flows and are extensions of classical solutions of the Navier-Stokes equations. The names of Couette, Hagen, Poiseuille, and Stokes are used to describe the flows. Flows considered include plane Couette flow, plane Poiseuille flow, Hagen-Poiseuille or pipe flow, and Couette flow (flow between rotating cylinders).

Principal interest in these problems lies in the comparison of the predictions of polar-fluid theory with Newtonian fluid theory. One means of comparing these predictions is the concept of effective viscosity. Recall that for each traditional viscometer an experimental procedure based on Newtonian fluid theory and involving the measurement of certain parameters leads to the determination of the traditional shear viscosity of the fluid in that viscometer. For example, in a viscometer whose basic construction is a cylindrical tube in which the fluid flows axially under a pressure gradient (Hagen-Poiseuille flow), the traditional shear viscosity is determined by (2.198) from measurements of the volume flow rate Q v , the pressure gradient C, and the tube radius h. The effective viscosity of a polar fluid is that viscosity that would be determined by the procedure used for the Newtonian fluid. Thus, for example, for Hagen-Poiseuille flow the viscosity is obtained from the pressure gradient C, the tube radius h, and the volume flow rate as shown in (2.196).

A general conclusion that can be drawn with regard to effective viscosity of a polar fluid is that it is always greater than or equal to the Newtonian shear viscosity and that it depends strongly on the length ratio L. This result is only indicated here for no slip and no-spin boundary conditions, but it applies to any system of boundary conditions that does not supply energy to the flow at the boundary.

In some publications the value of N 2 defined by (2.103) is limited to the range between zero and one-half rather than to its full range, zero to one. The source of this difficulty is an error in Eringen (1966) which was later corrected by himself. Rather than recording the thermodynamic restriction (2.94a) that p be positive, Eringen wrote that the symbol he called ,uL, must be positive. From Table 1 one can see that this inequality requires, in our notation, that p be greater than or equal to T . This incorrect inequality leads to the false limitation that N 2 cannot exceed one-half. This error, which does

322 S. C. Cowirz

not influence any significant result in Eringen (1966), has crept into many subsequent papers, limiting particularly the plots of curves illustrating polar-fluid phenomena. Works containing this limitation include Ariman and Cakmak (1967, 1968), Ariman et ul. (1967), Rao er a/. (l969), and Raja- gopalan (1968).

1. Plane Couette Flow

Plane Couette flow is steady laminar flow between two infinite parallel flat plates, one of which is translating in its plane relative to the other. This flow for a polar fluid has been considered by Pennington (1966) Ariman and Cakmak (1967, 1968), Cowin (1968), Rajagopalan (l968), Hudimoto and Tokuoka (1969), Kirwan and Newman (1969) Pennington and Cowin (1969), and others. We follow closely here the presentation of Pennington (1966) and Pennington and Cowin (1969) because i t has several distinct advantages over other presentations.

A Cartesian coordinate system fixed relative to the bottom plate and centered between the plates will be used. The x1 axis is oriented along the direction of flow and the boundary plates are at x2 = F u . The following assumptions concerning the flow are made: (1) the flow is steady; (2) all variables are independent of x I and x,; (3) the body force b is the gradient of a scalar potential @ which is absorbed in the pressure by replacing p by p + / I @ ; (4) the body couple C vanishes; ( 5 ) the velocity components r2 and c, are zero; (6) the no slip and no-spin boundary conditions (2.1 1 1 ) and (2.112) apply and take the form

" , (a ) = 0 0 , v l ( - u ) = 0, G3(*u) = 0. (2.160) The system of Eqs. (2.1 17)-(2.119) for incompressible polar fluids are the ones to be solved. With little difficulty it can be shown that the assumptions made lead to the following solution of the system of equations: u, , u 3 , G , , G,, and p are all zero and u 1 and G3 are given by

sinh NLY c = c ' - 1 - 2(1 - P )

-

,. G - G , = - I). (2.161 b)

where Y = x,/a, 1 Y 1 I I , (2.162)

and P = (N/L)tanh NL, 0 I P i 1. (2.163)

The dimensionless velocity field u/uo is plotted in Fig. 3 for various values of

Tlie Tlieory of Polar Fluids 323

t’

Fici. 3 . Vclocitg distribution in Couctte llow of a polar fluid: :I dimensionless plot of rcprescntation (2.161a) for the velocity lield. The lines L = 0 and ‘x( --). L = 2(- ). and L = 5( ~ -) are for 1%’’ = 0.9. The line L = ,~/2(- ~ ) is for N’ = 0.5. From Pcnnington and Cowin (1969). by pcrmissioii of the Society of Rheology, Inc.

L and N . This representation of the velocity field does not lend itself to an easy interpretation. Thus, after a consideration of the stresses associated with this flow the fields c and G will be recast in a representation that lends itself to interpretation. This recasted representation is the key to the advantage in the formulation of this problem by Pennington (1966) and Penning- ton and Cowin (1969).

The stresses and couple stresses in plane Couette flow are determined by substituting the velocity and angular velocity fields (2.161) into the constitutive equations (2.90) and (2.91) and in the process using the fact that O,, , r 2 , r 3 , 6, , and 6, are all zero. From (2.15) and (2.97), the components of the stress tensor are all zero except for T, ,, T,,, and T33 which are all equal to - p and T,, and T,, which are given by

P O T - l 2 - 2 4 1 - P ) ’

241 - P )

(2.164) T2, =

324 S. C. Cowin

and the components of the couple-stress tensor are all zero except for A,, and A,, , which are given by

p + i ' - p c O P sinh NLY (1 - P ) sinh N L '

A,, = A,, = p - i' (2.165)

The plate moving with a velocity 1:0 transmits a shear stress to the fluid adjacent to the plate; this shear stress will be denoted by To. From (2.164) it can be seen that TI, is constant across the flow region, thus at the boundary

To = ~o/ '2~(1 - P) . (2.166)

This equation relates the shear stress To applied by the top plate to the velocity r0 of the top plate, to the distance 2u between the plates and to the dimensionless number P. The corresponding expression for Newtonian fluids in plane Couette flow is

r,, = pL'(J2u. (2.167)

Based on this Newtonian model it is customary to define an effective viscosity for other fluids by the formula

per, = 2uTo/uo . (2.168)

From (2.166) and (2.168) it follows then that the effective viscosity of a polar fluid in plane Couette flow is given by

/ 1 , f l = /i/( 1 - P) . (2.169)

Since P must lie between zero and one, it follows that pCff is always greater than or equal to the Newtonian shear viscosity for all values of N and L. The dimensionlcss ratio ,u/pc,ff is plotted in Fig. 4 as a function of length ratio L for various values of the coupling number squared N 2 . I f 11 and T are finite,

N z = O

N 2 = 0 2

N 2 = 0 4 N ' = 06 N 2 = 0 8

N 2 = 10

-~

0 I 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 LENGTH R A T I O L

The Theory of' Polar Fluids 325

then P + 0 and perf + p, as L + oc, but for L + 0, P + N2, and peff + p + 5. Thus as L + 0 the effective viscosity becomes the sum of the shear viscosity and rotational viscosity. This result is a special case of the property of L described in Section II,D,2.

The stresses T22, TI 2 , and A,, are the only stresses that act on the solid boundaries and which must be applied to maintain the flow. The action of A,, is to rotate the boundary plates about the x 3 axis. The value of A,, at the solid boundaries is obtained from (2.165) and (2.166),

K,,(*U) = _+2aT0P. (2.170)

Since P must lie between zero and one, the maximum value of A,, acting on the boundary is equal to the shear stress To applied to the top plate times the distance between the plates.

We return now to the consideration of the velocity and angular velocity fields (2.161). Substituting (2.166) into (2.161) a new representation for these fields is obtained by:

- I). (2.171b)

The regional angular velocity W 3 and the relative angular velocity 8, are given by

cosh NLY W = W - - 2 p " (N ' cosh N L - l j .

- To cosh NLY H - H - (1 - N 2 )

- 2p cosh N L ' (2.172)

where (2.97) and (2.171) have been employed. The distribution of the dimensionless velocity L',

(2.173) ~

v = pv/TO~z

is plotted against Y in Fig. 5 for a high value of the coupling number ( N 2 = 0.9) and various values of the length ratio L. For other values of N 2 the resulting graphs are similar, but the angle between the curves L = 0 and L = 3c decreases as N 2 decreases. More explanation is needed to see the significance of the representation (2.161a) plotted in Fig. 3, for the velocity distribution than to see the significance of the representation (2.171a) plotted in Fig. 5. The obscurity occurs in the case of the representation (2.161a) because the velocity uo of the top plate is the same for all values of L,

326

t’

s. c. co\t,;n

FIG. 5. Velocity distribution in Coucttc How; a dimensionless plot of the representation ). and

) arc fo r N’ = 0.9. From Pennington and Couin (l969), by permission of thc (2.171a) for the velocity field. The lines I, = O( L = x( ~

Society of Rheology, Inc.

~ - ~ ) . L : 2( --)? L = 5 ( -

and all velocity curves terminate at the same point on the top plate. In representation (2.171a) the applied shear stress To is the same for all values of L and this permits the velocity curves to “fan out.”

2. Plane Poiseiiille Flow

Plane Poiseuille flow is steady laminar flow between two stationary, infinite parallel flat plates where the motive force for the flow is provided by a pressure gradient. This flow for a polar fluid has been considered by Cowin (1962) Pennington (1966), Ariman and Cakmak (1967, 1968), Rajagopalan (1968), Kirwan and Newinan (1969) and others.

The same Cartesian coordinate system that was used for plane Couette flow in the previous section will be used here for plane Poiseuille flow. Also the same assumptions will be used with the following changes: (1) the pressure gradient in the z direction is not zero, but a constant denoted by

c = - p . I , (2.174)

and (2) the boundary conditions are replaced by

z , ( f a ) = 0, G, ( fa ) = 0. (2.175)

The Theory of Polur Fltriils 327

The solution of the system of Eqs. (2.1 17)-(2.119) obtained by using these assumptions is the following: u 2 , u 3 , GI, and 6, are zero and u1 and G3 are given by

, (2.176a) 2N C O S ~ N L - C O S ~ N L Y ' - 2p L ( sinh N L

sinh N L (2.176b)

The regional angular velocity p3 and the relative angular velocity H 3 are given by

sinh N L 1 ' sinh N L Y

- c u sinh N L Y H r H -~ ( N 2 - 1 ) .

- 2p sinh N L ' (2.177)

after (2.97) and (2.176) have been employed. Nondimensional forms of the velocity and angular velocities are defined by the following expressions

L. = 2pv/Ca2, G = 2pG/Cu, W = 2p W/Ca, H = 2pH/Cu. (2.178)

The velocity L. is plotted in Fig. 6 for N 2 = 0.9 and for various values of the length ratio L. The corresponding angular velocities are shown in Fig. 7.

The volume flow rate per unit width in the x3 direction is given by the integral of the velocity (2.176a) from the bottom plate to the top plate:

As N + 0 or as L + m ( N + 0 ) the volume flow rate takes on its value for plane Poiseuille flow of a Newtonian fluid,

Q = $(Cu3/p). (2.180)

As L + 0, Q has the form of (2.180), but with p replaced by p t- z. An effective viscosity analogous to that defined by (2.168) for plane Couette flow can be defined for plane Poiseuille flow in terms of Q. We define perf by (2.180) and use (2.179) to find per, for the polar fluid in plane Poiseuille flows:

perf = 2 Cu2Q-' =,/A ( 1 + 3 [j: - Ncosh N L ] ] - l . 3 L sinh N L

(2.181)

328

k y

S. C. Cowin

FIG. 6. Velocity distribution for plane Poiseuille flow and pipe flow. Thc lines L = 0(- - ~~ -), L = 2(-- - -), L = 5(- --), and L = z(-) arc for N Z = 0.9. From Pennington (1966).

FIG. 7. Angular velocity distributions for plane Poiseuille flow and pipe flow. The lines L = O(- - - -), L = 2(- -), L = 5(-- --), and L = m(-j are for N Z = 0.9. From Pennington (1966).


0 . J \ 0.8

0.7

0.6

< 0.5

3. 0 4

0.3

0 2

\

O.I 0 I 0 I 2 3 4 5 6 7 8 9 1 0 1 1 12 13 1 4 1 5

L E N G T H R A T I O L

FIG. 8. A plot of/(;pccf vs. Lfor plane Poiseuille Ilow. From Pennington and Cowin (197O), by permission of the Society of Rheology, Inc.

Figure 8 presents plots of the dimensionless ratio ,u/jieff versus the length ratio L for various values of the coupling number N .

3. Flow between Concentric Rotating Cylinders

The problem of flow between two concentric cylinders in relative rotation has been considered by Condiff and Dahler (1964), Pennington (1966), Stokes (1966) Ariman et a/ . (1967), Rajagopalan (1968), and Pennington and Cowin (1970). Condiff and Dahler (1964) consider a rotating electric field in conjunction with the polar effects. Stokes works the problem for the case N = 1 and uses a boundary condition requiring that the couple stresses vanish on the boundary. Ariman rt ul. (1967) and Rajagopalan (1968) employ the boundary condition (2.122) which is not reasonable on a rotating boundary.

A cylindrical coordinate system ( r , 4, z ) concentric with the cylindrical boundary surfaces is used. The inner radius b is denoted by a and the radius of the outer cylinder by b. The following assumptions are made: ( 1 ) the flow is steady, (2) all variables are independent of z and 4, (3) the body force b is the gradient of a scalar potential @ which is absorbed in the pressure by replacing p by p + pa, (4) the body couple (? vanishes, ( 5 ) u, and c, vanish, (6) the no-slip and no-spin boundary conditions (2.1 11) and (2.1 12) apply and may be represented by

(',,+,(a) = 0, c,,+,(h) = Rh, G_(u) = 0, Gz(h) = Q, (2.182)

where 0 is the constant angular velocity of the outer cylinder. The solution of the system of Eqs. (2,117)-(2,119) obtained by using these assumptions is

3 30 S . C. Cowin

the following: c r , u Z , G4, and G, are zero and r4 and Gz are given by

c4 = *'[C, I , [ N L ( D + R)] + C2 K , [ N L ( D + R)] co - D + R D l ~ ~ c4,,

D + 2 "- D + R (2.183)

GZ = RL(D ~ + 2 ) l C l I ,[NL(D + R)] - C,K,[NL(D + R ) ] 2 N C 0

2 N -

L ( D + 2 ) "1' where R , D, and L are the dimensionless numbers

(2.184) 2(r - u ) 2u h - N

I ' R = , D = , L =

h - u h - 0

subject to the restrictions O I R I 2 , O < D < K . , L > O , (2.185)

and the constants C , , C,, C, , C,, and C , are given by D2

D + 2 KO(NLD) C, = ( D + 2)K , [NL(D + 2)] -

D (2.186) - 2 N h , [ N L ( D + 2)] - + 11 ( N L D ) J ,

L I

C, = ( L / 2 N ) ( D + 2 ) [ C , I , (NLD) - C,Ko(NLD)] ,

C , = C , [ I , ( N L D ) - ( L D / 2 N ) I 0 ( N L D ) ]

+ C , [ K , ( N L D ) + ( L D / 2 N ) K o ( N L D ) ] ,

Co = ( L / 2 N ) ( D + 2){C , [ I , (NL(D + 2 ) ) - I ,(NLD)]

- C,[KO(NL(D + 2 ) ) - K o ( N L D ) ] ) .

In these expressions I , , I,, K O , and K , are modified Bessel functions of the

The Theory of Polar Flu ids 33 1

first and second kind. R is a dimensionless radius which ranges from zero to two as r goes from a to h. The constant D characterizing relative curvature is a dimensionless ratio of the inner cylinder diameter to the gap width. The limiting values of D, zero and infinity, are excluded because such geometries are meaningless. The length ratio L is a measure of the relative size of the gap to the material characteristic length. Thus for flow between concentric cylinders there are two dimensionless length ratios, one of which is characteristic of the geometry and the other of which relates the geometry to a property of the polar fluid.

The only stresses of interest are the shear and couple stresses acting on the inner cylinder. These stresses T,, and A=, are given by

2 p R D ( D + 2 ) T = - ~

4 r C,(D + R ) 2

x k, - 1

D + R ~~ [ C l Z , ( N L ( D + R ) ) - D

(2.187)

A=, = ( 2 p R h / C o ) [ C l I , ( N L ( D + R ) ) + C , K , ( N L ( D + R))] .

The torque per unit height M acting upon a cylindrical surface of radius I’ in the fluid is given by

M = 2 n r 2 q r + 2nrA,, = ~ x ~ R L ~ I ( C ~ / C O ) . (2.188)

Since the right-hand side of this equation is independent of r, it follows that the torque is a constant throughout the flow field. Equation (2.188) also shows that a linear relation exists between the torque M applied to the cylinder and the angular velocity R of the outer cylinder, as in Newtonian fluid theory. The effective viscosity for flow between concentric rotating cylinders is given by

M ( h 2 - a’) - 4(D + 1)c4 -

4na2 h2R D(D + 2)C0 ’ A l l = (2.189)

The dimensionless ratio p/peff for flow between concentric rotating cylinders is plotted in Figs. 9-11. The ratio ,u/pefr is plotted as a function of L for various values of N in each figure; the different figures are for different values of the constant D.

A comparison of Figs. 9-1 1 with Fig. 4 representing plane Couette flow indicates that flow between concentric rotating cylinders tends to plane Couette flow as D + a.

N = O N = 0.25 N = 050 N = 075 N = 1.00

o . , , , , , , , , , , , , , , . 0 I 2 3 4 5 6 7 8 9 1 0 1 1 12 1 3 1 4 1 5

L E N G T H R A T I O L

Pic;. 9. A plot of ~r ' ) I , , , vs L with D = 10 for flow between conccntric rotating cylinders From Peniiington and Cowin (1970). by permission of the Society of Rheology, Inc.

10'

0 9~

0 8-

0 7-

N = O

N = 025 N = 050

N = 075 N = 100

0 I 2 3 4 5 6 7 8 9 10 II 12 13 1 4 1 5

L E N G T H RATIO L

FIG. 10. A plot of ~ I ~ I I ~ , ~ vs L with D = 1 for llow betwecn concentric rotating cylinders. From Pennington and Cowin (l970), by permission of the Society of Rheology. Inc.

1.0-

0.9-

N = O N = 0.25 N = 0.50

N = 0.75 N = 1.00

0 I 2 3 4 5 6 7 8 9 1 0 1 1 12 1 3 1 4 1 5 L E N G T H R A T I O L

3- 0.4

A Z

:::y,, 0 , , , , , , , , , , , , 0 I 2 3 4 5 6 7 8 9 1 0 1 1 12 1 3 1 4 1 5

F I G . 1 1 . A plot of p/ / iCf f with D = 0.5 for How betwccn concentric rotating cylindcrs. From Pennington and Cowin (1970), by permission or the Society of Rhcology, Inc.


4. Hagen-Poiseuille Flow

Hagen-Poiseuille flow is pipe flow, or more precisely, a flow in which the fluid is driven through a circular cylinder by a pressure gradient. Hagen- Poiseuille flow of a polar fluid has been considered by Aero et al. (1963), Eringen (1966), Pennington (1966), Pennington and Cowin (1970), and others. The same cylindrical coordinate system that was used for the flow between concentric rotating cylinders in the previous section will be used for Hagen-Poiseuille flow. Also, the same assumptions will be used with the following changes: (1) u4 rather than ti, will be assumed to vanish; (2) the pressure gradient in the z direction is not zero, but a constant denoted by

c = - P , z ; (2.190) and (3) the boundary conditions (2.182) are replaced by

u,(h) = 0, G4(h) = 0, uJ0) and G,(0) are finite. (2.191) The solution of the system of Eqs. (2.1 17)-(2.119) obtained by using these assumptions is the following: v, , u,, G z , and G, are zero and u, and G4 are given by

(2.192b)

where R is a dimensionless radial variable, R = r/b. (2.193)

The characteristic geometrical length L, appearing in L is taken here to be 2b, the inner radius a having been set equal to zero. Substitution of (2.192) into (2.97) gives the regional angular velocity and the relative angular velocity :

A comparison of (2.192) and (2.194) for u r , e,, W 4 , and H , with the corresponding fields representing the solution for plane Poiseuille flow, namely lil and G 3 given by (2.176) and W , and H , given by (2.177), shows that the solutions are formally similar if I,, I , , and R are replaced by cosh, sinh, and Y , respectively. The plot of the dimensionless velocity distribution 4prz/Ch2 for Hagen-Poiseuille flow is indistinguishable from Fig. 6 which represents the dimensionless velocity distribution for plane Poiseuille flow.

334 S. C. Cowin

Actually, the maximum error is 4.3%, but at the scale of Fig. 6 this is nearly " indistinguishable." Similarly, Fig. 7 giving the plots of all three angular velocity fields for plane Poiseuille flow is indistinguishable from the plots of the same quantities for Hagen-Poiseuille flow.

The volume flow rate Q may be computed using (2.192a), thus

Therefore, the volume flow rate Q for a polar fluid is a linear function of the pressure gradient C, as in Newtonian fluid theory. Q is not, however, a simple function of the diameter of the pipe as in the corresponding expression for Newtonian fluids. The effective viscosity for pipe flow is given by

The dimensionless ratio p/peff is plotted in the paper of Pennington and Cowin (1970) as a function of Lfor various values of N 2 . The plot is so

nearly like the corresponding plot (Fig. 8) for plane Poiseuille flow that it is not repeated here.

The only stresses of interest are the ones that act on the solid boundary. These are T_, and A,, which may be computed substituting (2.192) into the constitutive equations (2.90) and (2.91) and using in the process the fact that

u,. , v, , G,, Gz are all zero. These stresses are evaluated on the boundary through the use of (2.15), (2.97) and (2.195), thus

Z,(h) = -iCh, (2.197a)

(2.197b) & r ( h ) = (- 16P/'nh2)(Qv - Q), where QV is the volume flow rate for a viscous fluid and is given by

QV = K h 4 / 8 p . (2.198) The result (2.197a) shows, as is to be expected, that the shear stress acting on the boundary is the same as the shear stress one would have if the fluid were Newtonian. The result (2.197b) shows that there is a couple applied to the pipe wall that is proportional to the difference in the volume rate between the Newtonian and polar fluid under the same conditions.

5. Steady Rotational Motion in LI Cplindrr

A polar fluid will undergo a pure rigid-body motion when subjected to a steady rotational motion in a cylindrical vessel. This corresponds identically to the classical result predicted by all theories of fluid behavior. The proof of this result for many physically reasonable boundary conditions will be

Tlir Theory of’ Polur. Fluids 335

sketched in this section. Also, a comment on an impossible extension of a classical result to polar fluids is made at the end of the section.

The same cylindrical coordinate system as employed in the previous two sections will be used here. The same assumptions as employed in Section II,E,3 for flow between rotating concentric cylinders will be used here with the following exceptions: ( I ) it will also be assumed that cr and 6, vanish, and (2) the boundary conditions (2.182) will be replaced by

r4 = Rh at R = 1 , (2.199)

where R is the angular velocity of the cylinder, and any OW of the three conditions

6, = R at R = I, Arz = ~ ( 6 ; - R) at R = 1, (2.200)

T,.= = F ’ ( e , - R) at R = 1,

where F and F‘ are components of the tensors F and F introduced by Eqs. (2.1 13) and (2.1 14) to represent frictional coefficients. The solution of the system of Eqs. (2.1 17)-(2.119) obtained by using these assumptions is the following: only vg and c4 are nonzero and they are given by

v4 = Rr, Gg = R, (2.201)

indicating that the fluid is performing a rigid-body motion. Note that if the boundary condition on c2 at I’ = h did not depend on the relative spin 6, - R, this result would not have been obtained. This problem is discussed further by Cowin and Pennington (1970) as an illustration of the different possible appropriate boundary conditions on G .

In Newtonian fluid theory the solution to the problem of steady axial flow in a rotating cylinder is obtained by superimposing the rigid-body rotation of the cylinder on the usual parabolic steady axial flow in a nonrotating cylinder, Hagen-Poiseuille flow. The steady axial motion and the steady rotational motion are uncoupled in that the speed of one motion is not dependent on the speed of the other. I t is shown by Cowin (1972b) that in the same problem involving polar fluids the axial motion and the rotational motion are not uncoupled and the rotational motion of the fluid is not a pure rigid-body motion, unless the radius of gyration k vanishes.

111. Related Theories

There are a number of theories of fluid behavior that are either specializa- tions or generalizations of the polar fluid. The names given these theories partially reveal the close structural connection between them. For example,

336 S. C. Cowin

in this section, we will consider fluids with deformable microstructure, micropolar fluids with stretch, dipolar fluids, and fluids with couple stresses.

From the point of view of the Cosserat continuum the basic idea underlying the most general of these theories is quite simple; allow the director triad, which the Cosserats kept rigid, to deform with the motion. The kinematics associated with the deforming triad as well as the appropriate form of the conservation equations for this situation are presented in Section II1,B. The theory of fluids with deformable microstructure, discussed in Section III,C, is based on the kinematical motion of a Cosserat continuum with a deformable director triad. If the triad is constrained to remain rigid, then the fluid with deformable microstructyre becomes a fluid with rigid microstructure, that is to say, a polar fluid. This relationship is indicated schematically in Fig. 12 as are the other relationships that will be discussed. If, rather than requiring the triad to remain perfectly rigid, one constrains the motion of the triad so that all three directors remain perpendicular during the deforma- tior, but experience identical rates of length change, then the theory of fluids with deformable microshucture reduces to the theory of micropolar fluids with stretch. Micropolar fluids with stretch are also discussed in Section III,C. The theory of micropolar fluids with stretch can then be reduced to the theory of polar fluids by requiring complete rigidity of the director triad as indicated in Fig. 12.

If the theory of fluids with deformable microstructure is constrained so that the Cosserat triad deforms in exactly the same way as the underlying continuum, then the theory of dipolar fluids is obtained. Dipolar fluids are discussed in Section III,D. Finally, if the theory of polar fluids is constrained so that the Cosserat triad rotates with the underlying medium, but remains rigid, then one obtains the theory of fluids with couple stresses. The theory of fluids with couple stresses is discussed in Section II1,A before the other theories because it is a specialization of the polar-fluid theory rather than a generalization as are the other theories. Dipolar fluids are related to polar fluids by the circuitious relationship shown in Fig. 12. However, in Section III,D it will be shown that all the viscometrically significant flows of polar and dipolar fluids are governed by the same differential equations, thus the solutions given in Section II,E for polar fluids are also the solutions for the corresponding problems in the theory of dipolar fluids. I t follows that the relationship between polar and dipolar fluids is closer than Fig. 12 indicates.

In this section all thermal effects are neglected for simplicity. Each of these theories can be developed along the lines of Section 11, but the results change only quantitatively, not qualitatively.

THEORIES O F FLUIDS WITH DEFORMABLE MICROSTRUCTURES

A theory of fluid behavior based on the idea of a Cosserat continuum with a deformable director triad. The deformation rate of the triad is characterized by the tensor G(i j ) .

L - - - - - _ _ _ _ _I L _ _ _ _ _ _ - - _ _ J

t MICROPOLAR FLUIDS

WITH STRETCH The director triad is still deformable, but only with respect to director volume changes.

I r - - - - - - - - - - - 1 I CONSTRAINT I I No dircwtor deforma- I I . I I tlon, G ( k m ) = 0. I

I POLAR FLUIDS I I DIPOLAR FLUIDS The triad is constrained to rotate at the regional angular velocity and to deform with the underlying medium.

The director triad is rigid.

I I 1 1

I CONSTRAINT I I No relative rotation I I No dependence of the I I between the rigid I I double stress tensor I

I I UPOH G(xm) D k m , , I

I triad and the medium I I I

t [ - COiusTRA&F - I

I m n i f f ( D k m ) . I

FLUIDS WITH COUPLE STRESS The triad is constrained to be rigid and to rotate at the regional angular velocity.

FK;. 12. A chart illustrating the rclationship between the various theories

338 S. C. Cowin

A. FLUIDS WITH COUPLE STRESSES

A theory of fluids with couple stresses was suggested by Stokes (1966). This theory is a special case of polar-fluid theory obtainable by constraining the director triad to rotate with the underlying medium, that is to say, at the regional angular velocity W. From (2.19) it is clear that this condition is equivalent to requiring that H vanish. This vanishing of H is effected by increasing the rotational viscosity z to infinity in a manner such that

l im2tH = P, (3.1) a-0 r - x,

where P is an axial vector. If the limit process indicated in (3.1) is applied to the constitutive equation (2.90), one sees that P i j represents the skew- symmetric part of the stress tensor,

P is not now given by a constitutive equation, i t is an indeterminate skew- symmetric second-rank tensor. It is the Lagrange multiplier associated with the workless constraint H = 0 and plays a role analogous to pressure in an incompressible fluid in the sense that the pressure p is a Lagrange multiplier associated with the workless constraint V * v = 0.

qj + p h i j = 6ij1Dkk + 2pDij - Pij; ( 3.2 )

Since H = 0 it follows from (2.19) and (2.97) that G = w = + v x v. (3.3)

v - G = v . W = 0. (3.4)

(3.5)

Thus

In view of (3.3) and (3.4) the constitutive equation (2.91) becomes

Aij = ( p + y ) q j + (f l - y)wj , i .

The constitutively indeterminate quantity P can be determined from the

P = J(pC + (/3 + y)V’W - pk2W), (3.6)

where (3.3) and (3.5) have been employed. Except for notational differences this equation is equivalent to (3.4) of Stokes (1966). I f the stress (3.2) is substituted into the conservation of linear momentum (2.52), with P given by (3.6) and W given by (3.3), the velocity equations of motion for a fluid with couple stresses are obtained:

balance of angular momentum (2.64), thus

(A + p)(1 - [(a + y)/2(i + p)]V’jVV * v

+ p{l + [(a + i,)/2p]V2)V2v + pb - V x pC - V p

= pv - v x pP[+V x v - (L’)], (3.7)

The Theorj- of Polar Fluids 339

where (L’) is the axial vector associated with the tensor L2. The solutions to problems presented in Section I1 for polar fluids are also

the solutions to the equivalent problems for a couple stress fluid if N is set equal to one.

B. THE COSSERAT CONTINUUM WITH A DEFORMABLE DIRECTOR TRIAD

If the Cosserat director triad deforms the relation (2.3) must be abandoned since it is the requirement of rigidity. In its place we write

d, * db = Lfaidhi = z,, , (3.8) where dUi are the covariant components of the directors which are related to their contravariant or reciprocal components dr by

d y b i = sg , (3.9)

and where Z,, is the covariant metric tensor of the director triad. The straining of the triad between times to and t would be measured by the difference between Z,, evaluated at to and z,, evaluated at t . The definition (2.9) of G i j is generalized to

G . . 1 &d“ J ’ (3.10)

and it follows from (3.8)-(3.10) that

2G(i j )d , idb~j = Z,, . (3.1 1 )

This equation shows that the symmetric part of the tensor defined by (3.10) is a measure of the rate of strain of the director triad and vanishes only when the triad is rigid. Thus, G defined by (3.10) is only skew symmetric when the director triad is rigid. The only additional variable that must be added to those considered in the theory of polar fluids in order to account for the deformation of the director triad is the symmetric part of the tensor G. I t is easy to show that the symmetric part of G is an objective variable.

This added degree of kinematic generality over that of polar-fluid theory requires a corresponding increase in generality for the conservation principles. The conservation of mass given by (2.35) and the conservation of linear momentum given by (2.52) are unchanged. To obtain the appropriate form of the balance of energy we replace the expression (2.40) for the mechanical power by

(3.12)

340 S. C. Cowiii

where B(i j ) is a surface stress tensor associated with the work done on the boundary by the director deformation and C(i,i) is a body tensor. When the manipulations leading to (2.48) are repeated, with (3.12) replacing (2.40) one obtains an additional equation

~ ( B ( l k ) 1 - A(,k)jn,jiG(l,) = q - qiMi 7 (3.13)

where is the double-stress tensor. Amnj is now composed of the double- stress tensor A,,,,,j and couple-stress tensor A,mnlj encountered in polar-fluid theory. The use of (3.13) and a repetition of the manipulations leading to (2.58) gives the conservation of energy in the form

p i = p r + T . D . . + T . H . . + A . . Y . . 1.1 l J 1J 11 1.1 1.1

- qi.i + + A ( i j ) k " ( i j ) k + MijG(ij) > (3.14)

where

M. . ij = - 2 ~ C ( i j ) + 'A 2 (ij)k.h (3.15)

and

(3.16)

which is a slight generalization of the definition (2.15).

angular momentum A similar development leads to the following form of the conservation of

(3.17) ~ l , r n n , r s G , v s , = ~Ccmn] + 27;mnI + A ~ r n n ~ j , j ~

where

1 m w . s = 3 I i j eimn ejn > (3.18)

and a restriction equivalent to (2.57) on the inertia tensor,

if one assumes that I,,,, is independent of superposed rigid-body translations and rotations.

For a deformable director Cosserat continuum there are twelve geometric degrees of freedom ; the six of the usual Cosserat continuum. characterized here by the velocity vector v and the total angular velocity G, and the six G(ij) associated with the deforming triad. The conservation of linear and angular momentum gives only six equations, hence six more equations would be needed to make any system involving such a continuum determinate. These additional degrees of freedom are associated with the double-stress tensor

In most theories with deformable microstructure some hypothesis concerning the quantity M,,,, is made.

The Theory of Polur Fluids 34 1

c. FLUIDS WITH DEFORMABLE MICROSTRUCTURE

In this section the constitutive relations for theories of fluids with deformable microstructure will be recorded. This is the theory that occupies the uppermost position in Fig. 12. In Section III,A we considered fluids with couple stresses which occupy the lowermost position in Fig. 12. The theory of polar fluids is in a position of intermediate generality.

To formulate a mechanical theory of fluids with deformable microstructure, constitutive equations are needed for T j , and When the developments of Section I1 are paralleled with the additional kinematic variable G(ijb, the assumptions of linearity and isotropy lead to the following constitutive relations for T,i and Ai jk:

Ti + P’Sij = ’ S i j A D k , + 2PD;j - 2TH;j + A1 G k k 8 i . i + 2,Ul G,ij,, (3.20)

‘ k n m = (‘1 y r m r + ~ 2 y m r r + M 3 y r r m ) ’ S k n

+ ( ‘ 4 y r n r + ‘ 5 y n r r + M 6 y r r n ) 6 k m (3.2 1)

+ (‘7 y r k r + M 8 ykrr + c19 yrr.k)6nm + + M 1 2 y k n r n + M 1 3 y k m n + M 1 4 y m n k + M I S y n m k 3

0 ynkm + I 1 ymkn

where is also a linear isotropic function of the same variables, a constitutive equation of the form

M(ij) = (J+2’kk + 2 3 G k k ) ’ S i j + 2 ~ 2 D i j + 2~3G(i j ) (3.22) is obtained where A 2 , A 3 , / i 2 , and p3 are material coefficients.

If G(i j , is zero, the constitutive relations (3.20) and (3.21) reduce to the constitutive equations for a polar fluid. Thus (3.20) reduces to (1.16) and (3.21) reduces to (1.17). This reduction is indicated by the line on the left- hand side of Fig. 12.

A number of special subtheories of the general case under discussion can be obtained by constraining the deformation of the triad in some particular fashion. For example, one could constrain the directors of the triad to remain orthogonal and allow their magnitudes to change. The theory called “micropolar fluids with stretch” requires that the directors all remain orthogonal and that the magnitudes of the directors all vary in the same way. Let M denote the magnitude of a covariant component of any director,

(3.23)

,u1, and c(1-c(15 are material coefficients. If one assumes that

M = (d,ida,)1’2 no sum on u, u = 1, 2, or 3,

and let G denote the instantaneous stretching of any director,

G = M M - ’ . (3.24)

342 S. C. Cowin

I t then follows from (3.23), (3.24), (3.1 l), and the orthogonality of the director triad that

G,ij, = Gd;, . (3.25)

Hence

G j j = G6,, + GIij, ,

and, from (3.15) and (3.26),

(3.26)

y m n j = 6 m n G . j + GLmn1.j . (3.27)

When (3.25) and (3.27) are substituted into (3.20) and (3.21), respectively, the constitutive relations for micropolar fluids with stretch are obtained. The intermediate relationship of this fluid model between that with deformable substructure and that with rigid substructure is illustrated in Fig. 12.

D. DIPOLAR FLUIDS

The theory of dipolar fluids was proposed by Bleustein and Green (1967). I t can be interpreted as a special case of fluids with deformable microstructure obtained by requiring that Gij coincide with the tensor of velocity gradients L i j ;

G.. IJ = L.. LJ . (3.28)

It follows from (2.19) that

H = O (3.29) for a dipolar fluid. When the restrictions (3.28) and (3.29) are imposed on the constitutive relations (3.20) it takes the form

7 ; j + j d i j = 6 j j A D k k + 2,~Di.j - Pjj , (3.30) where Pij is an indeterminate skew-symmetric second-rank tensor defined by (3.1). As was the case in fluids with couple stress, the indeterminacy is removed using the balance of angular momentum, hence from (3.17)

P m n = $ [ P C [ m n , + Arnin1j.j - P l , m n , r s *IrsJ. (3.3 1) The constitutive relation (3.21) upon application of the constraint (3.28), reduces to the expression

Aknrn = t C ( 8 U k . r r d n m + i('13 C I I Z ) U k , m n + t ( % 7 + S 1 9 ) U r , r k i S n m

M 6 ) ( U r . r n i j k n i + U r , r m i S k n ) + ;(@1 + ' 3 +

(3.32)

T h e Theory of’ Polar Fluids 343

The constitutive relations (3.30)-(3.32) are those obtained by Bleustein and Green (1967) when the dependence on the first and second gradients of density and temperature are neglected. In a similar discussion of the works of Bleustein and Green, Cowin (1968a) omitted P i j from (3.30) and failed to record (3.31). The relationship of dipolar fluid theory to the other fluid theories we have discussed is illustrated in Fig. 12. The theory of dipolar fluids also reduces to the theory of fluids with couple stresses when the dependence of Amnj upon D,, is eliminated. This completes our discussion of the various theories of fluids with deformable microstructure with the excep- tion of one point. That point is that there is a greater similarity between polar and dipolar fluids than is apparent from the structure of the theories as illustrated by Fig. 12. Specifically, the differential equations governing the velocity distribution for polar and dipolar fluids coincide under relatively weak conditions.

To show the similarity between these fluid theories we will obtain the differential equations for the velocity field in a dipolar fluid and compare them with those for a polar fluid. The inertia tensor I in both theories will be assumed spherical. When the constitutive relations (3.30) and (3.3 1) are substituted into the equations of motion (2.52) and the vector identity (1.8) is employed, then for a dipolar fluid

(1 + 2p)VV * v - p( l - 1 y ) V x v x v

= pv + Vp - pb + i V x (pk ’W - p C ) (3.33)

where

1; = (1/4p)(u5 + @ I 0 + u15 - c18 - @ I 2 - c(13)? (3.34)

and where u l -uI5 have been assumed to be constant. To obtain an equation corresponding to (3.33) for a polar fluid we use (1.10) and (2.99) to eliminate H from (2.98). Thus, using (1.8), we find that

(1 + 2p)(1 - I:vz)Vv * v - p(1 - p 7 2 ) V x v x v

= (1 - l IV2)(pv + V p - pb) + j, (3.35)

where I j = $V x ( p k z G - $2).

and

(3.36)

Comparison of (3.33) with (3.35) shows that the differential equations

344 S . C. Cowin

governing the velocity fields for polar and dipolar fluids will coincide if

0 = lfVz[(;2 + 2p)VV v - pV - V p + pb] + i V x p k 2 H . (3.38)

For many flow situations, in particular for most flow situations for which a solution to (3.33) or (3.35) is conceivable, (3.38) is identically zero and the differential equations governing the velocity distribution for polar and dipolar fluids coincide. For example, if the flow is incompressible and b is the gradient of a scalar function -@ and v, G, C, and VV2(p + p 0 ) all vanish, then both (3.33) and (3.35) reduce to

p(l - l:v2)v2v + V ( p + pa)) = 0. (3.39)

The no-slip boundary condition on velocity is the same for both polar and dipolar fluids, but the equivalence of the additional boundary conditions is a more difficult question. This question is discussed by Cowin (1968a) and by Pennington and Cowin (1969).

Many of the results of Section II,E are applicable to dipolar fluids. For example, both theories give the same solution for steady Poiseuille flow. Remarks to the contrary by Ariman (1968) were in error and due to notational differences and a notational inconsistency, as pointed out by Cowin (1968a). Polar-fluid and dipolar-fluid theory cannot be distinguished from one another on the basis of their velocity field in Poiseuille flow nor upon the basis of any quantity such as mass flow computed from the velocity field. It appears that it would be quite difficult to distinguish experimentally between the two theories. To the extent that this is true, polar-fluid theory has the advantage of simplicity over dipolar-fluid theory.

ACKNOWLEDGMENT

I thank F. M. Leslie, G. R. Webb, and C. S. Yih for criticisms of an earlier draft of this article. I also thank Janice Christy, Mary Jo Hathaway, and Mary Peckham for typing the manuscript. This work was partially supported by the National Science Foundation.

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N o t e cctltlrtl iri p r o o f : The paper of Klinc (1970) has j u s t come to my attention and I would have discussed it i n Scction 11, D, 5 had I known of its existence. Since the com- pletion of this article a review paper by Arinian L’r a/ . (1973) on essentially the same topic as this article, has appeared. The paper contains further references. I t also contains value judgments concerning the relevant literature which are considernbly different from my own.


Author Index

Numbers in italics refer to the pages on which the complete rcferences are listed

A

Abarbanel, S . S., 153, 191, 193, 236 Ackeret, J., 153, 171, 234 Acrivos, A., 202, 232, 234, 236 Adamson, T. C., I7 I , 200, 234 Aero, E. L., 282, 283, 287, 31 I , 333, 344 Afanasiev, E. F., 284, 347 Allen, S. J., 283, 284, 344, 345, 346 Amazigo, J. C., 2, 48, 55, 63, 64 Andreichikov, I. P., 261, 276 Ariman, T., 283, 284, 31 I , 322, 326, 329, 344,

Aroesty, J., 202, 207, 208, 235 Augusti, C. , 73, 81, 141

345

B

Batdorf, S. B., 68, 92, 98, 99, 141 Batterman, S . C. 73, 98, 105, 141 Bazant, Z. P., 96, 141 Becker, H., 68, 98, 99, 100, 142 Belcher, R. J., 179, 183, 234 Berger, S . A,, 220, 238 Bhatnagar, K . S., 283, 345 Bijlaard, P. P., 68, 98, 101, 104, 141 Birkhoff, G., 209, 234 Bleustein, J. L., 284, 342, 343, 345 Bolotin, V. V., 87, 141 Born, M., 282, 345 Bott, J. F., 210, 234 Bouthier, M., 268, 276 Brilliant, H. M., 171, 234 Brown, S . N., 146, 148, 151, 154, 177, 189,

225. 226. 230, 231, 232. 233. 234

Budiansky, B., 2, 48, 55, 63, 64, 86, 87, 92,

Bulygin, A. N., 282, 283, 287, 3 11, 333, 344 Burggraf, 0. R., 151, 179, 183, 216. 218, 232,

Bush, W. B., 153, 234, 235 Busse, F., 247, 252, 253, 256, 276

93, 98, 99, 116, 139, 141

233, 234, 236

C

Cakmak, A. S . , 283, 311, 322, 326, 329, 345 Calladine, C. R., 128, 131, 133, 141 Capell. K., 220, 234 Carrier, G. F., 35, 63, 147, 221, 234 Carter, J . E., 192, 234 Catherall, D., 177, 201. 234 Cebeci, T., 197, 235 Chang, G.-Z., 146, 149, 218, 221. 232, 235 Chapman, D. R., 153, 154, 171, 172, 186,

Chen, T. S . , 260, 262, 263, 276, 277 Cheng, S.-I., 214, 216, 235 Childs, M. E., 192, 237 Chilver, A. H., 59, 60, 64 Cicala, P., 101, 138, 141 Cochran, W. G., 222, 235 Cohen, G. A., 2, 64, 116, 141 Cole, J. D., 202, 207, 208, 235 Coleman, B. D., 281, 301, 304, 345 Condiff, D. W., 282, 283, 311, 329, 345 Considere, A,, 67, 141 Cosserat, E., 283, 345 Cosserat, F., 283, 345 Cowin, S . C., 282, 283, 284. 306, 308, 310,

191, 193, 235

311, 312, 314, 322, 323, 324, 326, 329, 332, 333,334,335,343,344,345,346,347

Cross, A. K.. 153. 235

349

3 50 Author Index

D

Dahler, J. S., 282, 283, 284. 31 I , 329, 345 Daniels, P. G., 219, 233, 234, 235 Danielson, D. A,, 55, 64 Davis, R . T., 147, 148, 235 Dennis, S. C. R., 146, 147, 149, 218, 221, 232,

Denny, V. I., 219, 237 De Silva, C. N., 283, 284, 344, 346 Dhawah, S.. 153, 236 Dijkstra, D., 147, 149, 152, 238 Drucker, D. C., 101, 138, 141, 143 Duberg, J. E., 68, 69, 85,133, 134, 141 Dunn, L. G., 70, 144 Dwight, J. B., 137, 142 Dwoyer, D. L., 233, 239

235

E

Eckhaus, W., 268, 276, 277 Emery, J . C., 192, 193, 238 Emmons, H. W., 201, 235 Engesser. F., 68, 142 Enlow, R. L., 231, 237 Erdogan, M. E., 284, 345 Ericksen, J. L., 283, 284, 291, 345, 346 Eringen, A. C., 282, 283, 284, 287, 300, 321,

322, 333, 346

F

Feldmann, F., 153, 171, 234 Feo, A., 162, 169, 237 Fernandez, F. L., 203, 236 Findley, W. N., 101, 143 Fishburn, B. D., 200, 234 Fitch, J. R., 2, 47, 55, 56, 64, 116, 142 Fitzhugh, H. A., 154, 189, 235 Fliigge-Lotz, I., 179, 189, 219, 237 Foster, R . M., 203, 238 Fox, H., 148, 236 Frauenthal, J. C., 55, 64 Friedrichs, K. O., 179, 235

Gerard, G., 68, 98, 99, 100, 142 Gere, J. M., 100, 102, 143 Goldburg, A., 214, 216, 235 Goldstein, S., 147, 148, 149, 177, 213, 222,

Gough, D., 267, 278 Grad, M., 282, 287, 306, 320, 346 Grange, J. M., 154, 235 Graves Smith, T. R., 137, 142 Greber, I., 153, 191, 193, 236 Green, A. E., 87, 142, 281, 284, 297, 342, 343,

Green, G., 346 Grossman, B., 148, 237 Grove, A. S., 232, 234 Guenther, W., 283, 346 Guiraud, J.-P., 233, 235

229, 235

345, 346

H

Haaland, S., 268, 269, 271, 273, 276 Hakkinen, R. J., 153, 191, 193, 214, 217, 220,

Hama, F. R., 199, 236 Hankey, W. L., 233, 239 Hartunian, R. A,, 210, 236 Heisenberg, W., 261, 277 Henniker, J. C., 280, 346 Hijman, R., 192, 237 Hill, L. R., 283, 31 I , 322, 329, 345 Hill, R., 68, 86, 87, 88, 90, 91, 101, 142 Hopf, E., 242, 261, 263, 277 Howard, L. N., 251, 253, 277 Howarth, L., 153, 154, 236 Huang, N. C., 133, 142 Hudimoto, B., 283, 322, 346 Hunt, G. W., 131, 143 Hunt, J. C., R., 233, 236 Hutchinson, J. W., 2, 55, 59, 61, 63, 64, 69,

70, 75, 77, 80, 82, 83, 86, 101, 104, 105, 112, 116, 128, 133, 137, 138, 139. 141, 142

235, 236

I

Inger, G. R., 210, 236 Iooss, G., 277

G J

Gadd, G. E., 201, 202, 235 Gaitatzes, G. A., 210, 236 Georgeff, M. P., 154, 189, 235

Jaunzemis, W., 345, 346 Jobe. C. E.. 151, 216. 218. 232. 233. -136

Airthor Index 35 I

Johns, K. C., 59, 60, 64 Johnson, B. G., 128, 133, 142 Johnson, K., 148, 237 Jones, C. W., 201, 202, 23.5 Jones, D. R., 151, 236 Jones. R . M., 98, 142 Joseph, D. D., 243, 253, 260, 261, 262, 263,

Jouquet, E.. 10, 64 276, 277

K Kadioglu, N., 284, 345 Kakutani, T., 268, 277 Kaloni, P. N., 284, 346 Kassoy, D. R., 202, 236 Keller, H. B., 146, 197, 232, 235, 238 Kirwan, A. D., Jr., 283, 322, 326, 346 Klemp, J. B., 202, 236 Kline, K. A., 283, 284, 344, 345, 346 Klineberg, J. M., 154, 235 Koiter, W. T., 2, 3, 10, 16, 36, 40, 52 , 59, 60.

61, 64. 69, 81, 90, 92, 93. 117, 127, 132, 139, 142, 143

Kuehn, D. M., 153, 154, 171, 172, 191, 193, 235

Kuiken, G . D. C., 61, 64 Kuvshinskii, E. V., 282, 283, 287, 311, 333,

344

L

Lanchon, H., 268, 277 Landau, L., 242, 261, 277 Larson. H. K., 153, 154, 171, 172, 191, 193,

Leckie, F. A,, 138, 143 Lee, L. H. N., 98, 133, 143, Lees, L., 153, 154, 189, 203, 235, 236 Leigh, D. C., 201, 235 Leslie, F. M.. 284, 346, 347 Leslie, L. M., 222, 236 Libby. P. A., 148, 236 Liepmann, H. W., 153, 236 Lighthill, M. J., 154, 155, 173, 195, 219, 236 Lin, C. C., 147, 221, 234, 261, 277 Lin, T. H.. 101, 143 Ling, C . H., 261, 268, 277 Liu, C. Y . , 284. 347 Liusternik, L. A,, 16, 65 Lugt, H. J., 147, 236

235

M

McIntire, L. V., 261, 277 Malvick, A. J., 133, 143 Mandel, J., 90, 143 Mangler, K W., 177, 234 Masson, B. S., 203, 238 Masur, E. F., 2, 3, 65 Matveeva, N. S., 198, 236 Mayers, J., 140, 143, 144 Meksyn, D., 261, 277 Melnik, R. E., 162, 169, 237 Messiter, A. F., 155, 162, 169, 199, 214, 216,

219, 231, 236, 237 Michno, M. J., 101, 143 Miller, D. S., 192, 237 Mills, R . H., 183, 237 Mizel, V. J., 281, 301, 304, 345 Moxham, K. E., 137, 142 Murphy, J . D., 189, 192, 193, 237, 239

N

Needham, D. A,, 192, 193, 237 Needleman, A., 103, 135, 136, 137, 143 Neiland, V. Ya., 155, 183, 198, 233, 236, 237 Newman, J. B., 100 Newman, N., 283, 322, 326, 346 Nikolaevskii, V. N., 284, 347 NOH, W., 281, 293, 302, 304, 345, 347

0

Olsson, G. R., 199, 237 Onat, E. T., 101, 138, 143 O’Neil, E. J., 214, 220, 236 Oswatitsch, K., 154, 237

P

Payne, L. E., 318, 347 Peddieson, J., Jr., 284, 347 Pekeris, C. L., 277 Pennington, C. J., 283, 308, 31 I , 312, 322,

323, 324, 326, 328, 329, 332, 333, 334, 335, 345,347

Peterson, E. E., 232, 234 Plotkin, A., 219, 237 Potter, M. C., 261, 277 Pretsch, J., 201, 237 Proudman, I. , 148, 237

352 Author Index

R

Rajagopalan, R., 283, 311, 322, 326, 329, 345,

Rall, L. B., 16, 65 Ramacharyulu, N. C. P., 283, 311, 322, 347 Rao, P. B., 283, 347 Rao, S. K. L., 283, 311, 314, 322, 347 Reeves, B. L., 154, 189, 236 Reissner, E., 30, 65 Reyhner, T. A,, 179, 189, 237 Reynolds, O., 252, 277 Reynolds, W. C., 261, 268, 277 Rice, J. R., 91, 142 Riley. N.. 223. 237 Rivlin, R. S., 281, 284, 297, 346 Rose, W. C., 192, 193, 239 Roshko, A., 153, 236 Rothfus, R. R., 242, 243, 263, 278 Rott, N., 153, 171, 217, 234, 235 Rubin, S. G., 148, 237

347

S

Sanders, J. L., 53, 65, 90, 92, 93, 98, 143 Sattinger, D., 260, 261, 263, 277 Schneider, L. I., 219, 237 Schubauer, G. B., 232, 237 Schwiderski, E. W., 147, 236 Scriven, L. E., 345 Seide, P., 2, 65 Serrin, J., 281, 314, 347 Sewell, M. J., 2, 59, 60, 65, 68, 73, 81, 86, 92,

Shahinpoor, M., 314, 347 Shanley, F. R., 68, 70, 143 Shkoller, B., 277 Smith, F. T., 197, 202, 204, 205, 209, 219,

233, 237, 238 Smith, S. H., 222, 238 Snowden, D. D., 232, 234 Sobolev, V. J., 16, 65 Spencer, D. J., 210, 236 Spiegel, E., 267, 278 Sterrett, J . R., 192, 193, 238 Stewartson, K., 146, 148, 151, 153, 154, 155,

157, 176, 177, 179, 183, 189, 195, 196, 197, 199, 201, 202, 204, 205, 209, 210, 212, 213, 214, 216, 219, 220, 223, 225, 226, 229, 230, 231. 232. 233. 234, 237. 23X. 272. 277

98, 105, 143

Stokes, V. K., 283, 284, 308, 329, 338, 347 Stowell, E. Z . , 98, 100, 143 Stratford, B. S., 223, 238 Stuart, J. T., 261, 272, 277 Sudria, J., 283, 347 Sun, C. T., 284, 347 Sychev, V. Ya., 224, 227, 230, 231, 238 Sylvester, N. D., 345

T

Takami, H., 146, 232, 238 Talke, F. E., 220, 238 Tatsumi, T., 268, 277 Taylor, T. D., 203, 238 Terrill, R. M., 148, 177, 238 Thomas, L. H., 261, 277 Thomas, T. Y., 252, 277 Thompson, J. M. T., 2, 65, 131, 143 Timoshenko, S. P., 100, 102, 143 Tokuoka, T., 283, 322, 346 Tollmien. W.. 261, 278 Toomre. J . , 267, 27X Toupin, R. A., 297, 347 Trilling, L., 153, 191, 193, 236 Truesdell, C., 283, 291, 293, 297, 302, 346,

Tsien, H. S., 70, 144 Turk, M. A,, 345 Tvergaard, V., 61, 65

34 7

V

Vainberg, M. M., 14, 16, 65 Valanis, K. C., 284, 347 van der Neut, A,, 61, 65 Van de Vooren, A. I . , 147, 149, 152, 232,

Van Dyke, M. D., 147, 213, 239 Veldman, A. E., P., 232, 233. 239 von KBrmBn, Th., 67, 68, 70, 72, 122, 143,

233, 238, 239

144, 222, 239

W

Walker, J. E., 242, 243, 263, 278 Walsh, J. D., 147, 235 Watson, E. C., 192, 193, 239 Watson, E. J., 201, 202, 235 Watson, J., 261. 278

353

Webb, G. R., 314, 345 Weinberger. H. F., 318, 347 Werle, M. J., 148, 233, 235, 239 Wesenberg, D. L., 140, 143, 144 Whan, A., 242. 243. 263, 278 Wieghardt, K. , 154, 237 Wilder, T. W.. 6X. 69, 141 Williams. P. G., 155, 176, 179, 183, 189, 201,

Woods, L. c., 231,232, 239

Y

Yih, C . S., 261. 278 Young, A. D., 212, 239 Yudovich, V. I., 261. -776. 278

2

234, 238 Zahn, J . , 267, 278 Zerna, W., X7, 142

Subject Index

A

Angular momentum, conservation of in polar fluid theory, 297-301

B

Bickley jet, in plane Poiseuille flow,

Bifurcating flow point, in Poiseuille flow,

Bifurcating solution, in plane Poiseuille

Bifurcation

269

256-261

flow, 261-263

post-buckling behavior and, 3-4 stress-strain relations in, 41-48 symmetric, 29

Bifurcation analysis buckling and, 17-19 for solids with corners on yield surfaces,

90-93 Birfurcation buckling

defined, 4 instability and, 10 stability and, 10

Bifurcation buckling mode, defined, 18 Bifurcation criterion

for Donnell-Mushtari-Vlasov theory of plates and shells, 93-97

plastic buckling and, 86-105 plasticity theories and, 97-105

Bifurcation predictions, 97- 105 Bifurcation theory, for nearly parallel

Boundary layers flows, 268-276

free-interaction, 153 multistructured. 145-233

Boundary-value problems, in polar-lluid

Buckling theory. 306-321

bifurcation analysia in, 17-19 bifurcation buckling mode in, 18 Donnell-Mushtari-Vlasov theory in, 52-55 of elastic structures, 1-63 energy approach in, 16-41 of Euler column. 30-32 Frechet derivatives in. 13 ~ 16 functional notation for, 11-16 in inextensional ring, 32-36 initial imperfections in, 19-22, 55-57 load-"shorteninf" relation in. 26 29 mode intcraction in, 58 ~ 6 3 plastic, see Plastic buckling post-buckling analysis in, 19-22 shallow-shell theor! and. 52 55 sinitillancotis niodcs in, 58- 60 stability of, 36-41 state variables for, 11-12 stationary functionals in, 30~-36 stress-strain relations in. 41-42 virtual-work approach in, 41-57

C

Catastrophic separation, in multistructured boundary layei-s. 229 231

Compi-e\siblc fluids. multistructured boundary layers and, 152-156

Compressive free interactions, 176- 184

Compressive interaction boundary layer, 156 Concentric rotating cylinders, flow

defined, 172

between, 329-332

354

Subject Index 355

Convex corners, in supersonic flow, 194-200 Cosserat continuum

with deformable director triad, 339-340 in polar-fluid theory, 283, 336

equivalent, 293- 295 in polar-fluid theory, 290-292

Cosserat motion

Couette flow, in polar-fluid theory, 322-326 Couple stresses, in polar-fluid theory,

Cylinders 338-339

flow between rotating, 329-332 rotational motion in, 334-335

D

Dipolar fluids, theory of, 342 344 Donnell-Mushtari-Vlasov theory (of plates

and shells), 52-55 bifurcation criterion and, 93--97 boundary-layer terms in, 10% 113 circular plate under radial compression in,

eigenvalue problem in, 106-108 initial post-bifurcation behavior for,

two-column problems in, 119-122

123-127

105 I32

E

Elastic structures buckling and post-buckling behavior in,

energy approach in, 16 41 Frechet derivatives in. 13-16 functional notation in, 11-16 functions and functionals for, 1 I ~ I2 Gateaux derivatives in, 13-15 imperfection sensitivity and, 7-8 initial imperfections in, 4 7, 22 26 inner products in, 12-13 linear operators in, 12-13 load-shortening relation in, 8-9 mode interaction, 58-63 multilinear operators in, 12-13 norms in, 12-13 simple models of. 2- I 1 stability in. 10 1 I . 36 41

1-63

state variables for, 1 1 12 strain energy in, 26-27 stress-strain relations in. 41 42 variational calculus in, I I - 16 variations in, 13-15 virtual-work approach in, 41-57

Energy, conservation of in polar-fluid

Entropy inequality. in polar-fluid theory,

Euler column, post-buckling behavior of.

Expansive free interactions, 174- 176

Expansive interaction boundary layer, I56

theory. 296- 297

300-30 I

30-~32

defined, 172

F

Flat plates multistructured boundary layers on

symmetrically disposed, 213-221 145- 233

Fluctuation motion. in Poiseuille flow,

Fluids 249-250

compressible, I52 156 with couple stresses, 338 339 with deformable microstructure. 341-342 dipolar, 342-344 incompressible, 146- I5 I polar, .see Polar fluids

13-16 Frechet derivative\. in elastic structures.

Free-interaction boundary layer, I53

G

Gateaux dcriiatives. i n elohtic structures. 13-15

H

Hagen-Poiseuille flow. in polar-fluid theory, 333-334

1

Imperfections, in plastic buckling, 127-132 Imperfection sensitivity, in elastic structures,

7-8

356 Subject 1ndc.x

Incompressible fluids, multistructured boundary layers and. 146-151

Inextensional ring, post-buckling behavior of, 32-36

Instability, in elastic structures,

L

Laminar cornparkon theorem? flow. 251 252

Laminar Poiseuille flow basic equations for, 245-246 global stability of. 246-249

Lift. viscous corrections to. 225

10- 1 1

i n Poiseuillc

239 Limit-point buckling, defined, 6 Linear momentum, conservation of in

polar-fluid theory, 297-300 Load-shortening relation, in elastic

structures. 8-9

triple deck in, 158-167 and viscous corrections to lift in,

wall temperature in. 161 162 225-229

N

Navier-Stokes equations in Poiseuille flow, 250-251 solutions to in multistructured boundary

layers, 147, 152, 161, 188 190

0

Oil canning, defined, 6 Orr-Sommerfeld theory, in plane

Poiseuille flow, 268, 275 276

P M

Mass-flux discrepancy theorem, in Poiseuille

Moderate blowing, in supersonic boundary

Momentum conservation, in polar-fluid

Multistructured boundary layers, 145-233

Bow, 252

layer, 201

theory, 297-300

basic flow in, 156-158 catastrophic separation in, 229-231 comparison with experiment, 188- 194 compressible fluids and, 152- 156 compressive free interactions in, 176- 184 and convex corners in supersonic flow,

expansive free interactions in, 174-176 and free interactions in supersonic flow,

incompressible fluids and, 146-151 injection into supersonic boundary

layer. 200-2 12 plateau in, 184-188 Sychev theory in, 231-233 trailing edge in, 2 13-224 and trailing-edge flows for bodies with

transsonic free interaction in. 169 171

194-200

171-174

infinite thickness, 222-224

Parallel flows, bifurcdtion theory and,

Plane Couette flow, in polar-fluid theory,

Plane Poiseuille flow

268--276

322-326

see ulso Poiseuille flow bifurcating solution, 261-263 false-problems method in, 272 friction-factor discrepancy in, 260 inferences and conjectures in, 263-267 Orr-Sommerfeld theory and, 268, 275-276 in polar-fluid theory, 326-329 response curves for, 241-276

bifurcation criterion in, 86-105 in circular plate under radial compression,

Plastic buckling, 67- 140

123- 127 in column under axial compression,

132-135 comparison model in, 78 continuous model in, 76-86 determination of /j and i., in, 113- I19 and Donnell-Mushtari-Vlasov theory of

initial imperfections in, 73-75, 127 132 lowest-order boundary layer terms in,

plates and shells, 93-97

108- 1 I ?

Subject Index 357

Plastic buckling (cont.) numerical examples in, 132-135 perfect model in, 71-73 post-buckling analysis in, 82-84 Ramberg-Osgood stress-strain relation

Shanley-type model of, 70-75 simple models of, 70-85 in spherical and cylindrical shells,

theory of, 68-69

in, 81

137-140

Plasticity theory, bifurcation predictions

Plateau, in multistructured boundary layer,

Platc injection, in supersonic boundary

Poiseuille flow

in, 97- 105

184- 188

layer, 206-2 12

see u/so Plane Poiseuille flow bifurcation point in, 256-261 bifurcation theory for nearly parallel

equal-shear-stress theorem in, 252 fluctuation motion and mean motion in,

laminar, 242-244 laminar and turbulent comparison

mass-flux discrepancy theorem in, 252 Navier-Stokes equations in, 250--25 1 Orr-Sommerfeld theory and, 268, 275- 276 pressure-gradient discrepancy theorem

response curves for. 241-276 response function near bifurcation point

steady causes and stationary effects in,

turbulent plane, 253-256

see also Polar-fluid theory boundary-value problems in, 283 dipolar fluids and, 342-344 historical development in study of,

notation for, 284-286 Polar-Ruid theory, 279-344

boundary-value problems in, 306-321 Clausius-Duhem inequality and, 300 concentric rotating cylinders in, 329 -332

flows in, 268-276

249-250

theorems in, 25 1-252

in, 252

in, 256-261

250-25 1

Polar fluids

282-284

conservation of cnergy in, 296- 297 constitutive assumptions in, 301-302 constitutive relations in, 301-306 Cosserat continuum and, 282-283, 336 Cosserat motion in, 290- 292 dynamics and thermodynamics of 295-301 entropy inequality in, 300-301 fluids with couple stresses and, 338-339 foundations of, 290-335 Hagen-Poiseuille flow and, 333-334 kinematics in, 290-295 linearization in, 305-306 momentum conservation in, 297 301 objectivity restrictions in, 302-304 plane Coiiette flow in, 322 326 plane Poiseuille flow in, 326 329 related theories and, 335-344 and rotational motion in cylinders,

stability and uniqueness theorems in.

synopsis and, 286-290 thermodynamic restrictions in, 304 305 typical boundary-value problems in,

334-3 35

314-321

321-335

Donnell-Muslitari-VIasov theory, 105-132

Post-bifurcation behavior. for

Post-buckling analysis, 19-22, 58-60 stress-strain relation in, 41 ~ 4 2

Post-buckling behavior bifurcation and. 3-4 defined. 4 in elastic structures. 1-63 of Euler column, 30- 32 of inextensional ring, 32-36

Potential energy, stationary, 16- 17

R

Reynolds number, friction factor and, 244

S

Shallow-shell theory, 52-55 Shanley-type model, in plastic

buckling, 70-75

Subject Index

Slot injection, in supersonic boundary layer, 203-206

Snap buckling, 5-7 Snapping, defined, 6 Stationary potential energy, principle of,

Strain energy, in elastic structures, 26 Stress-strain relation

16-17

buckling and. 41-42 shallow-shell theory and, 52-55

Strong blowing, in supersonic boundary

Supersonic boundary layer injection into, 200-212 plate-injection in, 206-2 12 slot injection in, 203-206 strong and weak blowing in, 201

convex corners in, 194200 free interactions in, 171-174

boundary layers, 231-233

layer. 201

Supersonic flow

Sychev separation theory, in multistructured

Symmetrically disposed flat plate, trailing edge of, 213-224

T

Tollmien-Schlichting waves, in Poiseuille flow, 272

Trailing edge, of symmetrically disposed

Trailing-edge flows, for finite-thickness

Transsonic free interaction, in

flat plate, 213-224

bodies, 222-224

multistructured boundary layer, 169- I7 1

Triple deck, 158-167 fundamental equation of in

multistructured boundary layer, 167- I69

Turbulent plane Poiseuille flow, 253-256

Two-column problems, in Donnell- see also Poiseuille flow

Mushtari-Vlasov theory, 119-122

V

Variational calculus, in elastic structures,

Virtual-work approach and equations, 11-16

for buckling in elastic structures, 41-57

W

Weak blowing, in supersonic boundary layer, 201

A 4 e15 C 6 D 7 E 8 F 9 G O H 1 1 2 1 3

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