199 Jounua! of Non-Newtonian Florid Mechunks, 7 (1980) 199-212 @ Elsevier Scientific Publiiig Company, Amsterdam - Printed in The Netherlands DIR SWELL OF A MAXWELL FLUID: NUMRRICAL PREDICTION M.J. CROCHET and R. KEUNINGS Unit4 de Mhmique Appliqu4e, Bdtiment St&in, Place du Lewnt 2, B-l 348 Louvain-la-Neuve (Belgium) (Received January 10,198O) Numerical results are presented on the calculation of die swell at the exit of slit, circular and annular dies. The material is an upper-convected Maxwell fluid (rubberlike liquid); the numerical method is of the mixed finite ele- ment type. 1. Introduction The present paper is devoted to a theoretical study of die swell of an upper-convected Maxwell fluid. The calculation is based on recent progress in numerical methods for solving noneNewtonian flow; in particular, the results shown here represent a further extension of the mixed finite element method for fluids with implicit constitutive equations introduced by Kawa- hara and Takeuchi [l] and further studied by Crochet and Rezy [ 21. The method used here differs from [2] in view of the choice of shape functions. We consider the two classical problems of a slit die and a circular die, where we assume that the upstream flow is fully developed Poiseuille flow, while the contact forces vanish on the free surface. The method of calcula- tion has been adapted for calculating two free surfaces simultaneously, and we have been able to study die swell at the exit of an annular die. While most problems of practical interest occur at a very low Reynolds number, we are also considering the case of circular die swell where inertia and visco- elastic forces occur simultaneously. Over the last few years, numerical results have been published on slit and circular die swell of a viscous inelastic fluid [3,4] and of a second-order fluid [ 51. Recent results [ 61 have been published on circular die swell of an implicit fluid; they do not, however, agree with the expected hehaviour of
Transcript
199
Jounua! of Non-Newtonian Florid Mechunks, 7 (1980) 199-212 @ Elsevier Scientific Publiiig Company, Amsterdam - Printed in The Netherlands
DIR SWELL OF A MAXWELL FLUID: NUMRRICAL PREDICTION
M.J. CROCHET and R. KEUNINGS
Unit4 de Mhmique Appliqu4e, Bdtiment St&in, Place du Lewnt 2, B-l 348 Louvain-la-Neuve (Belgium)
(Received January 10,198O)
Numerical results are presented on the calculation of die swell at the exit of slit, circular and annular dies. The material is an upper-convected Maxwell fluid (rubberlike liquid); the numerical method is of the mixed finite ele- ment type.
1. Introduction
The present paper is devoted to a theoretical study of die swell of an upper-convected Maxwell fluid. The calculation is based on recent progress in numerical methods for solving noneNewtonian flow; in particular, the results shown here represent a further extension of the mixed finite element method for fluids with implicit constitutive equations introduced by Kawa- hara and Takeuchi [l] and further studied by Crochet and Rezy [ 21. The method used here differs from [2] in view of the choice of shape functions.
We consider the two classical problems of a slit die and a circular die, where we assume that the upstream flow is fully developed Poiseuille flow, while the contact forces vanish on the free surface. The method of calcula- tion has been adapted for calculating two free surfaces simultaneously, and we have been able to study die swell at the exit of an annular die. While most problems of practical interest occur at a very low Reynolds number, we are also considering the case of circular die swell where inertia and visco- elastic forces occur simultaneously.
Over the last few years, numerical results have been published on slit and circular die swell of a viscous inelastic fluid [3,4] and of a second-order fluid [ 51. Recent results [ 61 have been published on circular die swell of an implicit fluid; they do not, however, agree with the expected hehaviour of
free jets. Instead of using an implicit model, Caswell and Viriyayuthakorn have devised a new algorithm for fluids of the integral type [7]; the same method has been used for studying die swell [ 81, and provides resulti similar to ours.
It must be pointed out, however, that no existing numerical algorithm has been able to predict die swell for large Deborah (or Weissenberg) numbers. All iterative procedures “blow up” before reaching “finite” values of the Deborah number; a serious comparison with theoretical results [9,10] is pre- mature in view of the limited range of convergence of the numerical calcula- tiOIlS.
Sections 2 and 3 explain the numerical method and the procedure for determining the free surface. In sections 4,5 and 6 we consider the case of slit, circular and annular dies.
2. Numerical method
We consider the motion of an upper-convected Maxwell fluid which has the following constitutive relations,
a=-pI+T, T+d=2@, (2.1)
where p denotes the pressure, a is the Cauchy stress tensor, T the extra-stress tensor and D the rate of def rmation tensor; p is the constant shear viscos- ity, h is a natural time, and P is the upper-convected stress derivative defined by
~=C-LT-TL~=~$+~~VT-LT-TLT. (2.2)
Here u is the velocity, L is the velocity gradient and LT its transpose. The momentum equations for steady flow are given by
-Vp+V. T+f=pv* Vv (2.3)
where f is the body force per unit volume, and the incompressible character of the fluid imposes that
v-0=0. (2.4)
The method used here is a mixed finite element method with six unknown fields in plane flow and seven unknown fields in axisymmetric fiow; details on the numerical method have been given in [ 23 and [ 111. Briefly, the velocity, the extra-stress and the pressure are approximated by
T’ = i$l Tiri, P* =,$lP'6i, (2.5)
where d, T', pi are nodal values and &, ri, 3/{ are shape functions. The set of field and constitutive relations is discretized by means of the
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Gale&in method and becomes
(ri, T' +X~"-~/JD')=O, l<i<M, (2.6)
((v$~)~,--p*I+ T*> --<c#~~,f> +b$,, jw* l Vu*)
= ((@i, 5>>, l<i<L, (2.7)
(J/i, v l 0:) = 0, l<i<N, (2.6)
where single and double angular brackets denote L2 scalar products over the domain of integration and its boundary, respectively, while t is the imposed contact force on the boundary.
The stream function is calculated a posterion’ from the velocity compo- nents by solving a Poisson equation with either Dirichlet or Neumann boun- dary conditions.
A central problem in the elaboration of a mixed method is the selection of shape functions; a detailed study of various possible choices is given in [ll]. Here we take the option of a conforming element which requests C!’ continuity for the velocity and the extra-stress field; moreover, we select a continuous approximate pressure field. We shall consider triangular elements in which the pressure is linear, while the extra-stresses and the velocity com- ponents are represented by complete second-order polynomials (Fig. 1). Standard solvers for banded non-symmetric systems are inappropriate here in view of the large number of nodal values. The frontal method of solution [12] with diagonal pivoting leads to a drastic reduction of core storage; it is effective provided that the variables are eliminated in the right order, i.e., the extra-stress components followed by the velocity components and finally by the pressure. Double-precision arithmetics (64 bits) is essential for the type of problem we are treating here. The code may be tested on plane and axisymmetric Poiseuille flow, which is calculated exactly in view of the choice of shape functions; for a random grid containing 63 elements and 944 unknowns, and with unit mean velocity and unit shear viscosity in a
u,v,p,Txx.Tyy,Txy 4a
/ \ 1 -xx9 yysxy
I \
u 9 v *Trr .Tw ,Tzz .Trz
Plane flow Axisymmetric flow
Fig. 1. Triangular finite element.
202
tube of unit radius, the order of magnitude of the error is lo-l3 for the velocity, lo-‘* for the stresses and lo-l1 for the pressure.
3. Boundary conditions and selection of grid
The three problems to be discussed below deal with the free surface flow at the exit of a long plane, tubular or annular region with upstream fully developed Poiseuille flow (Fig. 2), and lead to four types of boundaries, i.e. solid wall, free surface, upstream boundary and axis of symmetry.
On solid walls, we impose the condition that the fluid st&ks to the wall, while on free surfaces we impose vanishing contact forces t appearing on the right-hand side of (2.7). For a mixed method, the boundary conditions are less obvious on the upstream sections and on axes of symmetry. In the up stream section, we impose the fully developed velocity field; the extra-stress components are not imposed in that section.
The Gale&in form (2.6) of the constitutive relations allows for the calcu- lation of the extra-stresses at the inlet on the basis of the flow downstream; the comparison between the calculated stresses and their values in fully developed flow provides a good check of the validity of the solution.
On an axis of symmetry, we impose a vanishing normal velocity compo- nent; the second condition may consist of either a vanishing tangential con- tact force on the right-hand side of (2.7), or a vanishing shear stress on the axis. We found that a vanishing contact force may lead to oscillatory shear stresses, while a vanishing shear stress may produce an oscillating axial velo- city. In the present paper, we will impose a vanishing contact force on the axis of symmetry.
The initial undeformed grid consists of a set of rectangles divided into four triangular elements; such a layout provides the possibility of a dense grid with a minimal active front. The size of the elements in the neighbour- hood of the lips results from a compromise: elements of moderate size reduce the peak values and smooth the solution, while small elements enhance the singularities and may generate spurious stress oscillations in neighbouring elements.
The shape of the free surface is calculated by means of an iterative proce- dure. Let us consider the case of an annular die in Fig. 3 with two free sur-
0 v-0, F,-0
Fig. 2. Boundary conditions for free surface problem.
203
Fig. 3. Description of the free surface.
faces S1 and Sa described by the equations
r = Fl@), r= F2CG, ZfJQZ<Zl.
:1 --i
(3.1)
Since the free surface is also a stream surface, we must have
% F,(z) = f Ez, F&)1 9 Fi(zO) = 4 9 i=l,2
where u and u are the radial and axial velocity, respectively, and ri are the fixed radii at the exit. The iterative procedure starts from cylindrical surfaces on which vanishing contact forces are imposed; new surfaces are defined by the equations
k, F?ld(~)l, Fine" (zo) = f; , i=l,2 (3.3)
which are integrated by means of Simpson’s rule. Four iterations are suffi- cient in general. Since our program provides the possibility of including iso- parametric elements which allow for a description of the surface by means of parabolic segments, we verified whether it would offer a smoother free surface. The answer is definitely negative; while subparametric elements with straight edges produce a smooth surface, it was found that isoparametric ele me& lead to bad-looking corners on the free surface.
4. Slit die swell
In the present problem, we consider a slit of length 2 and halfwidth 1, in which a Maxwell fluid flows with a unit mean velocity, while the length of the extruded sheet is 3; a deformed grid is shown in Fig. 4. The maximum shear rate on the upstream wall, which is denoted by i, is 3 in the present problem; solutions are obtained for various values of the Deborah number, which is defined by
De=Xi. (4.1)
P
b-0.
\L
P
Fig
. 4.
Fin
ite
elem
ent
grid
, st
ream
line
s (D
e =
0.6
),
pres
sure
, sh
ear
stre
ss
and
long
itud
inal
st
rem
fi
elds
(D
e =
0 a
nd D
e -
0.5)
fo
r
Txx
die.
206
TABLE 1
Swelling and exit loam as a function of the Deborah number (slit die)
De 0 0.25 0.60 0.60 0.75 SW (96) 18.8 16.9 18.9 20.6 26.6 Ea 0.18 0.20 0.26 (0.26) -
The exit losses Ex are evaluated by comparing the difference Ap between the upstream value of the pressure and the preasum lolie that we would encounter in the slit without free surface, to twice the maximum shear stress at the wall in fully developed Poiseuille flow, i.e.
Ex = Ap/2r,. (4.2)
The swelling SW is the relative increase of the thickness of the aheet. Table 1 gives the values of EX and SW as a function of De for a Maxwell fluid flowing through a slit die.
The shape of the free surface is shown in Fig. 5. It is intere&ing to note that the width of the free stream decreases with respect to its Newtonian counterpart for small valuea of De. The free mrfkce and the streamlines remain smooth up to the largest value of De; the latter are shown in Fig. 4 for De = 0.6. Figure 4 shows contour linee of the prkure, the shear strees Txy and the lcmgitudinal stress TX, for Newtonian and non-Newtonian flow; the values of the variablea have been divided by 7, (‘3 in the present prob- lem) for the sake of ndrmalization. It is found that, while the &ear stress is relatively little affected by the elabicity of the fluid, high extensional stresses arise at the lip when De increases. It is conjectured that these high
Fig. 5. Shape of the free eurface (slit die).
206
extensional stresses, proper to an upper-convected Maxwell fluid, are causing the eventual lack of convergence of the numerical procedure. The program was not run for De values higher than 0.7 in view of the poor quality of the stress field.
5. Circuhu die swell
We consider now a circular cylindrical tube of length 4 and radius 1, in which a Maxwell fluid flows with unit mean velocity, while the length of the extruded rod is 4; a deformed grid is shown in Fig. 6. The mnrimum shear rate on the upstream wall is 4. The symbols EX and SW have the same mean- ing as in Section 4, except that SW now refers to the relative increase of the radius of the rod. Table 2 gives the values of& and SW as a function of De for a Maxwell fluid flowing through a circular tube; table 2 also shows the values of SW obtained from Tanner’s theory [9], where we have adjusted the swelling ratio from 1.100 in [9] to 1.126; the correspondence between nu- merical and theoretical values is rather good.
We note also the good correspondence.between the Newtonian values SW = 12.6 and BX = 0.28 and those found in [3] which were SW = 12.8 and Ex = 0.265.
Figure 7 shows the shape of the free surface when De increases. Figure 6 shows the normalixed values of the pressure, shear stress T,, and longitudinal stress T,, for Newtonian and non-Newtonian flow. It is found again that T,, reaches a peak value at the exit. For values of De larger than 2/3, we find that the pressure field loses its uniformity across the section, and the solu- tion degenerates.
It is interesting to examine the axial velocity profile upstream and down- stream the exit of the tube; in Fig. 8, we have indicated the velocity profiles in various sections and compared them with the fully developed parabolic profile. We note that the actual profile remains close to the parabolic profile up to a distance of one-half radius from the exit, while the difference between the profiles is notable in the exit section.
The program has also been run when the Reynolds number defined by
Re = !i?RpV/p (5.1)
TABLE 2
Swelling and exit 10-s function of the Deborah number (circular die)
De 0 0.5 1 SW (%) -7.8 --6.9 -4.7 Ex -0.30 -0.13 -il.03
De-
a D
e-
1.
-1,7,-l.
p -7
. -5
. -3.
-1.
-1.
51.7
'l.,
___
52
a.
Fig
. 9.
Fin
ite
elem
ent
grid
, st
ream
line
s (D
e =
l),
pr
eesu
re,
shea
r st
ress
an
d ax
ial
stre
am fi
elds
(D
e =
0 a
nd D
e =
1)
for
circ
ular
di
e w
ith
Re
= 4
0.
P
210
has the value 40. Table 3 gives the values of SW and Ex for that case; the swelhng is negative and decreases in absolute value when De increases. The inertia effects tend to lower the peak values of the axial stress component, and the approximate stress field is much smoother; we have been able to reach a Deborah number of one. This may be observed on Fig. 9, where the normalized peak value of T,, is 8 at the exit for De = 1, while on Fig. 6 it was 9 for De = 0.5; for Re = 40, the stress contour lines stretch out further downstream than in creeping flow at the same Deborah number.
6. Annular die swell *
Let us consider the case of an annular die with inner radius 1 and outer radius 4/3. The domain contains two free surfaces, and the iterative proce- dure must be performed on both simultaneously. The swelling is now described by three parameters, only two of which are independent; SW, SWi, SW,, will denote the relative increases of the thickness of the tubular region, the inner radius and the outer radius respectively. Table 4 gives the values of SW, SwI, Swc and Ex for various values of De; Ex is the ratio between the exit pressure loss and twice the shear stress on the inner wall in the fully developed flow.
Figure 10 shows the (deformed) finite element grid and the streamlines for De = 0.7. Also shown on Fig. 10 are the normalized pressure and stress contours for De = 0 and De = 0.7. It is found that the pressure contours are normal to the axis of symmetry in the annulus, and the development of the normal stresses T,, is quite visible for De = 0.7. The pressure singularity at the lips loses its intensity when De increases, while the opposite is true for !I’,, . A closer look at these singularities may be obtained on the perspective view of Fig. 11, where we show a three-dimensional view of the pressure field for De = 0, and of the extra-stress T,, for De = 0.7.
* We wish to thank Professor J.M. Dealy for suggesting the problem.
k4t
r= I.
r=o.
De4
7 0800.
J,
Fig
. 10
. F
init
e el
emen
t gr
id,
stre
amli
nes
(De
= 0
.7),
pr
essu
re,
shea
r st
ress
and
axi
al s
tres
s fi
elde
(D
e =
0 a
nd D
e =
0.7
) fo
r an
nula
r di
e.
0
Fig. 11. Pressure and stress singularities in annular die.
References
1 2 3 4
5 6 7
M. Kawahara and N. Takeuchi, Comp. Fluids, 6 (1977) 33-45. M.J. Crochet and M. Bezy, J. Non-Newtonian Fluid Mech., 5 (1979) 201-218. R.E. Nickeii, R.I. Tanner and B. Caswell, J. Fluid Mech., 66 (1974) 18Q-206. R.I. Tanner, R.E. Nickel1 and R.W. Biiger, Comput. Meth. Appl. Mech. Eng., 6 (1975) 165-174. K.R. Reddy and R.I. Tanner, J. Rheol., 22 (1978) 661-665. P.W. Chang, Th.W. Patten and B.A. FinIayson, Comp. Fluids, 7 (1979) 267-293. M. Viiyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mach., 7 (1980) 246- 267.
8 M. Viriyayuthakorn and B. Caswell, to be pubbshed. 9 R.I. Tanner, J. Polym. Bci., 8 (1970) 2067-2078.
10 J.R.A. Pearson and R. Trottnow, J. Non-Newtonian Fluid Mech., 4 (1978) 196-216. 11 M.J. Crochet, J. De Canni&e and R. Keunings, to be published. 12 D.M. Irons, Int. J. Num. Meth. Eng., 2 (1970) 5-32.
