Three-Dimensional Stagnation Flow and Heat Transfer on a...

Three-Dimensional Stagnation Flow and Heat Transferon a Flat Plate with Transpiration

Ali Shokrgozar Abbassi∗ and Asghar Baradaran Rahimi†

Ferdowsi University of Mashhad, 91775-1111 Mashhad, Iran

DOI: 10.2514/1.41529

The existing solutions of Navier–Stokes and energy equations in the literature regarding the three-dimensional

problem of stagnation-point flow, either on a flat plate or on a cylinder with orwithout transpiration, are only for the

case of axisymmetric formulation. In this study, the nonaxisymmetric three-dimensional steady viscous stagnation-

point flow and heat transfer in the vicinity of a flat plate are investigated when suction and blowing are also

considered in themodel.An externalfluid, along zdirection,with strain rate a impinges on thisflat plate andproduces

a two-dimensional flow with different components of velocity on the plate. This external flow, as an example, can be

generated in the formofmultiple jet streams in a row inwhich the jets are in equal distances fromeach other along the

x axis. A similarity solution of the Navier–Stokes equations and energy equation is presented in this problem. A

reduction of these equations is obtained by use of appropriate similarity transformations. Velocity profiles and

surface stress tensors and temperature profiles along with pressure profiles are presented for different values of

velocity ratios and Prandtl number for sample cases of transpiration.

Nomenclature

a = constantF, G = inner-region functionsf, g = similarity functionsPr = Prandtl numberp = pressureS = nondimensional transpirationT = temperatureU, V,W = inviscid flow componentsu, v, w = velocity componentsW0 = rate of suction or blowingx, y, z = Cartesian coordinates� = thermal diffusivity��, ��, ��, �, �� = constants� = boundary-layer thickness" = perturbation parameter� = similarity variable� = nondimensional temperature� = velocity ratio in x and y directions = viscosity� = kinetic viscosity

= inner variable� = density� = shear stress� = inner-region variable

I. Introduction

T HERE are many exact solutions for Navier–Stokes and energyequations regarding the problem of stagnation-point flow andheat transfer in the vicinity of a flat plate or a cylinder. Removing thenonlinearity in these problems is usually accomplished by super-position of fundamental exact solutions that lead to nonlinearcoupled ordinary differential equations by separation of coordinate

variables, but in all the three-dimensional cases, only axisymmetricformulation of the problem has been considered. Fundamentalstudies in which flows are readily superposed and/or the axisym-metric case were considered include the following papers presentedin the literature: uniform shear flow over a flat plate in which the flowis induced by a plate oscillating in its own plane beneath a quiescentfluid [1]; two-dimensional stagnation-point flow [2]; the flowinduced by a disk rotating in its own plane [3]; flow over a flat platewith uniform normal suction [4]; three-dimensional stagnation-pointflow [5]; and axisymmetric stagnation flow on a circular cylinder [6].Further exact solutions to the Navier–Stokes equations are obtainedby superposition of the uniform shear flow and/or stagnation flow ona body oscillating or rotating in its own plane, with or withoutsuction. The examples are as follows: superposition of two-dimensional and three-dimensional stagnation-point flows [7];superposition of uniform suction at the boundary of a rotating disk[8]; also the solution for a fluid oscillating about a nonzeromean flowparallel to aflat platewith uniform suction given [9]; superposition ofstagnation-point flow on a flat plate oscillating in its own plate, andalso consideration of the case where the plate is stationary and thestagnation stream is made to oscillate [10]; uniform shear flowalignedwith outflowing two-dimensional stagnation-pointflow [11];uniform flow along a flat plate with time-dependent suction andincluded periodic oscillations of the external stream [12]; heattransfer in an axisymmetric stagnation flow on a cylinder [13];unsteady laminar axisymmetric stagnation flow over a circularcylinder [14]; nonsimilar axisymmetric stagnation flow on a movingcylinder [15]; transient response behavior of an axisymmetricstagnation flow on a circular cylinder due to time-dependent free-stream velocity [16]; unsteady viscous flow in the vicinity of anaxisymmetric stagnation point on a cylinder [17]; shear flow over arotating plate [18]; radial stagnation flow on a rotating cylinder withuniform transpiration [19]; suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations [20];axisymmetric stagnation flow toward a moving plate [21]; axisym-metric stagnation-point flow impinging on a transversely oscillatingplate with suction [22]; axisymmetric stagnation-point flow andheat transfer of a viscous fluid on a moving cylinder with time-dependent axial velocity and uniform transpiration [23]; axisym-metric stagnation-point flow and heat transfer of a viscous fluid on arotating cylinder with time-dependent angular velocity and uniformtranspiration [24]; and similarity solution of nonaxisymmetric heattransfer in stagnation-point flow on a cylinder with simultaneousaxial and rotational movements [25]. The only three-dimensionalnonaxisymmetric solution in aforementioned studies is the one in [7],

Received 10 October 2008; revision received 4 April 2009; accepted forpublication 16 April 2009. Copyright © 2009 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved. Copies of this papermay be made for personal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0887-8722/09 and $10.00 incorrespondence with the CCC.

∗Graduate Student.†Professor, Faculty of Engineering; [email protected] (Correspond-

ing Author).

JOURNAL OF THERMOPHYSICS AND HEAT TRANSFERVol. 23, No. 3, July–September 2009

513

http://dx.doi.org/10.2514/1.41529

which is with large deviations from the correct solution because ofthe approximation methods used, where, in a later work [26], thesedeviations have been discussed.

In this study, the nonaxisymmetric three-dimensional steadyviscous stagnation-point flow and heat transfer in the vicinity of a flatplate are investigated in the presence of suction and blowing bysolving Navier–Stokes equations. The importance of this researchwork is encountered in solutionswhere theflowpattern on the plate isbounded in one of the directions, for example, x axis, because ofwhatever physical limitations. The external fluid, along z direction,with strain rate a impinges on this flat plate and produces a two-dimensional flow with different components of velocity on the plate.This external flow, as an example, can be generated in the form ofmultiple jet streams in a row, in which the jets are equal distancesfrom each other along the x axis and interact with each other on theplate. A similarity solution of the Navier–Stokes equations andenergy equation is derived in this problem. A reduction of theseequations is obtained by use of these appropriate similarity transfor-mations. The obtained coupled ordinary differential equations aresolved using numerical techniques. Velocity profiles and surfacestress tensors along with temperature profiles are presented fordifferent values of impinging fluid strain rate, different forms of jetarrangements, Prandtl number, and sample values of suction andblowing parameters.

II. Problem Formulation

Flow is considered in Cartesian coordinates �x; y; z� withcorresponding velocity components �u; v; w�, see Figs. 1 and 2. InFig. 2, a general three-dimensional stream surface along with itsboundary-layer thickness is shown, which is produced because of anstagnation-point flow impinging on a flat platewhere the flow patternin x direction is bounded because of whatever physical limitation.This limitation can be generated, for example, in the form ofmultiplejet streams in a row in which the jets are in equal distances from eachother only along the x axis and there is no limitation along the y axis.Obviously, the boundary-layer thicknesses are different in x and ydirections, contrary to the case of axisymmetric flow when they arethe same. Figure 2 depicts the same situation but for a certain rates ofsuction and blowing and their comparison with the case of notranspiration. We consider the laminar steady incompressible flowand heat transfer of a viscous fluid in the neighborhood of stagnationpoint on a flat plate located in the plane z� 0. An external fluid,along z direction, with strain rate a impinges on this flat plate andproduces a two-dimensional flow with different components ofvelocity on the plate. This external flow, as an example, can begenerated in the form ofmultiple jet streams in a row inwhich the jetsare in equal distances from each other along the x axis. The steadyNavier–Stokes and energy equations in Cartesian coordinatesgoverning the flow and heat transfer are given as

Mass

@u

@x� @v@y� @w@z� 0 (1)

Momentum

u@u

@x� v @u

@y�w@u

@z�� 1

�

@p

@x� �

�@2u

@x2� @

2u

@y2� @

2u

@z2

�(2)

u@v

@x� v @v

@y� w@v

@z�� 1

�

@p

@y� �

�@2v

@x2� @

2v

@y2� @

2v

@z2

�(3)

u@w

@x� v @w

@y�w@w

@z�� 1

�

@p

@z� �

�@2w

@x2� @

2w

@y2� @

2w

@z2

�(4)

Energy

u@T

@x� v @T

@y�w@T

@z� �

�@2T

@x2� @

2T

@y2� @

2T

@z2

�(5)

where p, �, �, and � are the fluid pressure, density, kinematicviscosity, and thermal diffusivity, respectively.

III. Self-Similar Solution

A. Fluid Flow Solution

The classical potential flow solution of the governing Eqs. (1–4) isas follows, in which we have exerted the parameter � in x directioncomponent of velocity to introduce three-dimensionality:

U� a�x; 0< � � 1 (6)

V � ay (7)

W ��a�� 1�z�W0 (8)

P1 � P0 � 12�a2��2x2 � y2 � �� 1�2z2 �W20 � 2aW0�� 1�z�(9)

where p0 is stagnation pressure and � is the coefficient whichindicates the difference between the velocity components in x and ydirections. The velocity components in these directions are the sameif �� 1, indicating that each of the two adjacent single jets are farenough from each other and therefore there are no interactionsbetween them.W0 is the suction or blowing rate in z direction.

A reduction of the Navier–Stokes equations is sought by thefollowing coordinate separation, in which the solution of the viscousproblem inside the boundary layer is obtained by composing the

x

y

z

Fig. 1 Three-dimensional stream surface and velocity profiles.

X

Y

Z

S = 2

S = 0

S = -2

Fig. 2 Stream surface for selected values of transpiration.

514 ABBASSI AND RAHIMI

inviscid and viscous parts of the velocity components as thefollowing:

u� a�xf0��; 0< � � 1 (10)

v� ay�f0�� g0�� (11)

w��a�p�g�� 1�f�� W0 (12)

��a=�

pz (13)

in which the terms involving f�� and g�� in Eqs. (10–12) comprisethe Cartesian similarity form for steady stagnation-point flow and theprime denotes differentiation with respect to �. Note, boundary layeris defined here as the edge of the points where their velocity is 99%oftheir corresponding potential velocity. Transformation Eqs. (10–13)satisfy Eq. (1) automatically and their insertion into Eqs. (2–4) yieldsa coupled system of ordinary differential equations in terms of f��and g�� and an expression for the pressure

f000 � �� 1�f� g � S�f00 � ��1 � �f0�2� � 0 (14)

g000 � �� 1�f� g� S�g00 � �g0 � 2f0�g0 � �1� ��f0�2 � 1� � 0(15)

p�x; y; z� � P0 ��a2

2��2x2 � y2� � 1

2�a3S�S� � 2

��a�p�� 1�z�

� �a�f12�� 1�f� g�2 � �f0 � g0� � �f0 � �� 1��

� �a�� 1�� aS�g (16)

Relation (16), which represents pressure, is obtained by integratingEq. (4) in z direction and by use of the potential flow solutionEqs. (6–9) as boundary conditions. The function g�� outside theboundary-layer region is independent of the variable � and equal to aconstant value �. Therefore,

� � lim�!1

g�� Const:and

S� W0��ap S > 0 �suction� S < 0 �blowing�

The boundary conditions for the coupled differential Eqs. (14) and(15) are

�� 0: f� 0; f0 � 0; g� 0; g0 � 0 (17)

�!1: f0 � 1; g0 � 0 (18)

Note that, when �� 1, the case of axisymmetric three-dimensionalresults are obtained (Homman [5]). When �� 0, the results are thesame as the two-dimensional problem.

B. Heat Transfer Solution

To transform the energy equation into a nondimensional form forthe case of defined wall temperature, we introduce

�� T�� T1Tw � T1

(19)

Making use of transformation Eqs. (10–13), the energy equationmaybe written as

�00 � Pr �0�g� �� 1�f � S� � 0 (20)

with the boundary conditions as

�� 0 �� 1 (21)

�!1 �� 0 (22)

where Pr� �=� is the Prandtl number and the prime indicatesdifferentiation with respect to �.

Note that, forPr� 1, the thickness of the fluid boundary layer andheat boundary layer become the same, and therefore this concept isproved by reaching Eq. (20) from Eq. (14) through substitution of�� f0.

Equations (14), (15), and (20) are solved numerically using ashooting method trial and error and based on the Runge–Kuttaalgorithm, and the results are presented for selected values of � andPr in following sections. Because Eqs. (14) and (15) are coupled, weguess a value for g�� function first and solve Eq. (14) for f��. ThenEq. (15) is integrated and a new value of g�� is obtained, which isused to solve Eq. (14) again. This procedure is repeated until thedifference of the results is less than 0.00001.

IV. Shear Stress

The shear stress at the wall surface is calculated from

� � �@u

@zex �

@v

@zey

�z�0

(23)

where is the fluid viscosity. Using the transformation Eqs. (10–13),the shear stress at the flat plate surface becomes

� � ��12a32��2x2f002 � y2�f00 � g00�2�12 (24)

This quantity is presented for different values of � in later sections.

V. Asymptotic Analysis

Asymptotic results for large values of Prandtl numbers are given inthis section. Assuming

"� 1=Pr (25)

as a perturbation parameter, energy Eq. (20) can be written as

"�00 � �g� �1� ��f � S��0 � 0 (26)

Since " appears in front of the highest-order term, an inner and outeranalysis is needed to obtain a composite solution for all the values ofPrandtl numbers. A perturbation expansion in outer region isassumed as

��; "� � �0�� "�1�� O�"2� (27)

Substitution of this expansion into Eq. (26) and collecting thecoefficients of the powers of " and setting them equal to zero givestypical equations as

�00�� 0 (28)

which solves to �0�� constant. Therefore, the outer solution isintroduced as �0�� constant.

To obtain the inner solution, we consider the following stretchingof the variables:

� �=" ��; F�� f��=" ��

G�� g��=" ��; �� =" �� (29)

Substitution of these new variables into Eqs. (14), (15), and (26) andtaking the distinguished limits (Nayfeh [27]) would result in

�� 3=5; �� 2=5; �� 0 (30)

Therefore, the energy equation governing the inner region (inside theboundary layer) would be

�00 � �G� �� 1�F � S��0 � 0 (31)

in which the prime indicates differentiation with respect to .Intersection of the solutions of this equation with the outer solution

ABBASSI AND RAHIMI 515

would bring about uniformly valid solutions throughout the regionfor all values of Prandtl numbers.

VI. Presentation of Results

In this section, the solution of the self-similar Eqs. (14), (15), and(20) along with the surface shear stresses for different values ofvelocity ratios and Prandtl numbers are presented.

The boundary-layer thickness in the two directions on theflat plateversus the velocity ratio is presented in Fig. 3 for selected values ofsuction and blowing. This thickness is larger in x direction comparedto y direction because of the difference of the velocity components inthese directions, and the difference of the boundary-layer thicknessin these directions decreases as � increases until the value of unitywhere these two layers meet each other, which is a validation of ourresults compared to the axisymmetric problem case. From thisfigure,the following relations can be obtained for the boundary-layerthickness versus the ratio of the velocities in potential flow:

For S� 0,

�x ��0:75�� 2:75 �y ��0:35�� 2:35

For S� 2,

�x ��0:1�2:7 � 0:014�� 2:99 �y ��0:1�1:95 � 2:96

And for S��2,

�x � 0:1�2:1 � 0:45�� 1:61 �y ��0:15�1:2 � 0:01�� 1:49

Figures 4–7 present the profiles of f0, g0, and f0 � g0 for differentvalues of velocity ratio � and selected values of transpirations. Thesmaller the �, the bigger g0, and therefore the difference between thevelocity components is larger. For �! 1, then g0 ! 0, and the twovelocity components become the same, which is again a validation ofour result compared to the axisymmetric problem case.

λ

Bo

un

dar

y-L

ayer

Th

ickn

ess

0 .2 0 .4 0 .6 0 .8 10

0 .4

0 .8

1 .2

1 .6

2

2 .4

2 .8

3 .2

X B .L. T hickne ssY B .L. T hickne ss

S = 0

S = -2

S = 2

Fig. 3 Boundary-layer thickness versus variation of velocity ratio and

selected values of suction and blowing.

f',g

',f'

+g

'

0 1 2 30

0.2

0.4

0.6

0.8

1

f 'f ' + g '

g '

g '

g '

f 'f ' + g '

f 'f ' + g '

η

S = -2S = 0S = +2

Fig. 4 Typical u and v velocity components for �� 0:05 and selectedvalues of suction and blowing.

f',g

',f'

+g

'

0 1 2 30

0.2

0.4

0.6

0.8

1

f 'f ' + g '

g '

g '

g '

f 'f ' + g '

f 'f ' + g '

η

S = -2S = 0S = +2


f',g

',f'

+g

'

0 1 2 30

0 .2

0 .4

0 .6

0 .8

1

f 'f ' + g '

g 'g '

g '

f 'f ' + g '

f 'f ' + g '

η

S = - 2S = 0S = + 2


f',g

',f'

+g

'

0 1 2 3

0

0 .2

0 .4

0 .6

0 .8

1

f 'f ' + g '

g '

g 'g '

f 'f ' + g '

f 'f ' + g '

η

S = -2S = 0S = + 2



Figures 8–11 depict the f and g profiles, and therefore the wcomponent of velocity for different values of velocity ratio and forselected values of suction and blowing. The bigger the �, the largerthe absolute value of the w component of the velocity, as expected.This component of velocity, which is the penetration of momentuminto the boundary layer in the z direction, changes abruptly as� increases because the boundary layer increases faster as thisparameter gets larger, and therefore there is need formore penetration

of the momentum and hence this component of velocity gets bigger.The effect of blowing is in the direction of increasing the wcomponent of velocity and suction in the direction of decreasing it, asexpected. It is interesting to note that, as �! 0 (for example,�� 0:05), the flow governing differential equations appears in thenew form of

f000 � �f� g�f00 � 0g000 � �f� g�g00 � �g0 � 2f0�g0 � �1 � �f0�2� � 0

Which are definitely different with the governing equations of thetwo-dimensional problem case [2], because � tends to zero graduallyand the basic governing equations remain three-dimensional. Notethat the existence of the physical limitation in x direction is the causeof the gradual change of � from one to zero.

The temperature profiles for different values of velocity ratio andselected values of Prandtl numbers and transpiration are presented inFigs. 12–23. Increase of velocity ratio and increase of Prandtlnumber both cause the decrease of the temperature profile. It is alsonoted that, for �! 1 and Pr� 1, the temperature boundary layer isobtained the same as the velocity boundary layer and is also avalidation of the nonaxisymmetric temperature compared to theaxisymmetric problem case.

In presenting the uniform value of temperature profiles for higherPrandtl numbers (Pr� 10:0), an asymptotic method has been used.These uniform values of temperature profiles have been obtainedfrom matching an outer solution with an inner solution.

η0 1 2

-2

-1

0

1

2

f, g,

f

g

f

fg

g

w

a ν√

w

a ν√w

a ν√

w

a ν√w

a ν√

w

a ν√

S = -2S = 0S = + 2

Fig. 8 Typicalw component of velocity for�� 0:05 and selected valuesof suction and blowing.

η0 1 2

-2

-1

0

1

2

f, g,

f

g

f

fg

g

w

a ν√

w

a ν√

w

a ν√

w

a ν√

S = -2S = 0S = +2

Fig. 9 Typical w component of velocity for �� 0:1 and selected valuesof suction and blowing.

η0 1 2

-2

-1

0

1

2

f, g,

f

g

f

f

g g

w

a ν√

w

a ν√

w

a ν√

w

a ν√

S = -2S = 0S = + 2


η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 0.5

S= 2

S = 1

S = 0

S = -1

S = - 2

θ

Fig. 12 Temperature profile for �� 0:01 and Pr� 0:5 for selectedvalues of suction and blowing.

η0 1 2

-2

-1

0

1

2

f, g,

f

g

f

f

g

g

w

a ν√

w

a ν√

w

a ν√

w

a ν√

S = - 2S = 0S = +2



η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 0.5

S = 2

S = 1

S = 0

S = - 1

S = - 2

θ


η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 0.5

S = 2

S = 1

S = 0

S = - 1

S = - 2

θ


η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 0.5

S = 2

S = 1

S = 0

S = - 1

S = - 2θ


η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 1

S = 2

S = 1

S = 0

S = -1

S = -2

θ


η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 1

S = 2

S = 1

S = 0

S = -1

S = -2

θ


η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 1

S = 2

S = 1

S = 0

S = -1

S = -2

θ



η0 1 2 3

0.2

0.4

0.6

0.8

1

Pr = 1

S = 2

S = 1

S = 0

S = -1

S = -2

θ


η0 1 2 3

0

0.2

0.4

0.6

0.8

1

Pr = 10

S = 2

S = 1

S = 0

S = -1θ


η0 1 2 3

0

0 .2

0 .4

0 .6

0 .8

1

Pr = 10

S = 2

S = 1

S = 0

S = -1θ


η0 1 2 3

0

0.2

0.4

0.6

0.8

1

Pr = 10

S = 2

S = 1

S = 0

S = -1

θ


η0 1 2 3

0

0.2

0.4

0.6

0.8

1

Pr = 10

S = 2

S = 1

S = 0

S = -1θ


λ0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

S = -1

S = 0

S = +2

S = -2

S = +1

X DirectionY Direction

τρν a

1/2 3/2

_______

Fig. 24 Shear stress in x and y directions versus� and selected values ofsuction and blowing.


Figure 24 presents the change of surface shear stress on the flatplate in terms of velocity ratio �. The following relations can bededuced from this plot:

For S� 0,

�x � �1:55 � 0:3� �y � 0:1�1:1 � 0:03�� 1:235

For S� 2,

�x � 1:5�1:32 � 1:25�� 0:02 �y � 0:0951�� 2:67

For S��2,:

�x � 0:45�2:08 � 0:05� �y � 0:01�2:5 � 0:006�� 0:476

As �! 0, the stress tensor in x direction tends to zero, but note that�� 0 does not represent a physical situation.

Pressure profiles inside the boundary layer are shown in Fig. 25 forselected values of �. From these profiles, it can be seen that, withincrease of velocity ratio in x and y directions and tending toward thesymmetric situation, variation of pressure inside the boundary layerincreases, because � affects velocity directly and pressure changeswith velocity in power form.

VII. Conclusions

The effects of suction and blowing have been presented in thisproblem using a similarity solution of the Navier–Stokes equationsand energy equation for nonaxisymmetric three-dimensionalstagnation-point flow and heat transfer on a flat plate. This task hasbeen accomplished by choosing appropriate similarity trans-formations and reduction of these governing equations to a system ofcoupled ordinary differential equations and subsequent numericalintegration. Velocity components, temperature profiles, pressurechange, surface stress tensor, and asymptotic solution of temperatureprofile for a high Prandtl case have been presented for selectedvalues of velocity ratios. Increase of velocity and thermal boundary-layer thicknesses are encountered in all cases for blowing andtheir decrease in the case of suction. These differences have beencompared with the case of no transpiration for selected values ofthese parameters. This problem represents many physical situationsincluding the stagnation-point problem, in which the impinging flowis generated in the form ofmultiple jet streams in a rowwhere the jetsare equal distances from each other along one axis.

References

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[2] Hiemenz, K., “Boundary Layer for a Homogeneous Flow Around aDropping Cylinder,” Dinglers Polytechnic Journal, Vol. 326, 1911,pp. 321–324.

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[4] Griffith, A. A., and Meredith, F. W., “The Possible Improvement inAircraft Performance Due to the Use of Boundary Layer Suction,”Royal Aircraft Establishment Rept. No. E 3501, p. 12.

[5] Homman, F. Z., “Der EINFLUSS GROSSER Zahighkeit bei derStrmung um den Zylinder und um die Kugel,” Zeitschrift fuerAngewandte Mathematik undMechanik, Vol. 16, No. 3, 1936, pp. 153–164.doi:10.1002/zamm.19360160304

[6] Wang, C. Y., “Axisymmetric Stagnation Flow on aCylinder,”Quarterlyof Applied Mathematics, Vol. 32, 1974, pp. 207–213.

[7] Howarth, L., “TheBoundary Layer in Three-Dimensional Flow, Part II:The Flow Near Stagnation Point,” Philosophical Magazine, Ser. 7,Vol. 42, 1951, pp. 1433–1440.

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[23] Saleh, R., and Rahimi, A. B., “Axisymmetric Stagnation-Point Flow

η

∆

0 1 2 3

-14

-12

-10

-8

-6

-4

-2

0P B

.L.

∆ λ = 10%∆ λ = 50%∆ λ = 90%

P at B.L. forP at B.L. forP at B.L. for

Fig. 25 Pressure profiles for selected values of � and S� 0.


http://dx.doi.org/10.1002/zamm.19360160304http://dx.doi.org/10.1093/qjmam/7.4.446http://dx.doi.org/10.1098/rspa.1955.0160http://dx.doi.org/10.1017/S002211205600007Xhttp://dx.doi.org/10.1093/qjmam/18.3.287http://dx.doi.org/10.1007/BF00385923http://dx.doi.org/10.1016/0020-7225(78)90029-0http://dx.doi.org/10.1016/0020-7225(79)90009-0http://dx.doi.org/10.1007/BF00420004http://dx.doi.org/10.1023/A:1004243728777http://dx.doi.org/10.1002/aic.690190540http://dx.doi.org/10.1023/A:1004211515780

and Heat Transfer of a Viscous Fluid on a Moving Cylinder with Time-Dependent Axial Velocity and Uniform Transpiration,” Journal ofFluids Engineering, Vol. 126, No. 6, 2004, pp. 997–1005.doi:10.1115/1.1845556

[24] Rahimi, A. B., and Saleh, R., “Axisymmetric Stagnation-Point FlowandHeat Transfer of a Viscous Fluid on a Rotating Cylinder with Time-Dependent Angular Velocity and Uniform Transpiration,” Journal ofFluids Engineering, Vol. 129, Jan. 2007, pp. 106–115.doi:10.1115/1.2375132

[25] Rahimi, A. B., and Saleh, R., “Similarity Solution of Unaxisymmetric

Heat Transfer in Stagnation-Point Flow on a Cylinder withSimultaneous Axial and Rotational Movements,” Journal of HeatTransfer, Vol. 130, No. 5, 2008, pp. 054502.1–054502.5.doi:10.1115/1.2885173

[26] Shokrgozar, A. A., and Rahimi, A. B., “Non-Axisymmetric Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate,”Journal of Fluids Engineering, Vol. 131, No. 7, 2009, pp. 074501.1–074501.5.doi:10.1115/1.3153366

[27] Nayfeh, A. H., Perturbation Techniques, Wiley, New York, 1985.

http://dx.doi.org/10.1115/1.1845556http://dx.doi.org/10.1115/1.2375132http://dx.doi.org/10.1115/1.2885173http://dx.doi.org/10.1115/1.3153366

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