momentum transfer on power law stretching plate with free stream pressure gradient

Momentum transfer on power law stretching plate withfree stream pressure gradient

Noor Afzal

Faculty of Engineering, Aligarh Muslim University, Aligarh 202002, India

Received 19 August 2002; accepted 13 November 2002

Abstract

The similarity solution of laminar boundary layer driven by the stretching surface boundary and

pressure gradient, each proportional to the same power law of the downstream coordinate, based on

composite reference velocity (sum of the velocities of stretching boundary and free stream) has been for-

mulated by single set of equations, containing two parameters: b measuring the stretch rate of the movingboundary, and � the ratio of free stream velocity to composite reference velocity. The closed form exact and

asymptotic solutions in special cases, approximate integral solution and general numerical solutions havebeen obtained. For b > 0 and 06 �6 1 solutions are unique. For b < 0 solutions are dual for 0 < � < �0ðbÞ,unique solution for � ¼ 0 and � ¼ �0ðbÞ and no solution exists � > �0ðbÞ.� 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction

In many engineering systems both the moving boundary surface with speed Uw are parallel tothe ambient fluid speed U1. Examples are the cooling of films or sheets, conveyor belts, metallicplates and cylinders. In the absence of stretching in the bounding surface subjected to zero fluidpressure gradient, Abdulhafez [1] and Chappidi and Gunnerson [2] studied the laminar boundarylayer in two cases Uw > U1 and U1 > Uw separately and formulated two sets of boundary valueproblems. Afzal et al. [3] formulated a single set of equations by employing the composite ref-erence velocity U (defined later), irrespective of whether Uw > U1 or Uw < U1. Later, Lin andHaung [4] published the problem in same variables as Afzal et al. [3], but the case of reversemoving surface boundary condition was separately considered, leading to another second setof equations, and the duality of solution could not be explored. On other hand, the work of

E-mail address: [email protected] (N. Afzal).

0020-7225/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0020-7225(03)00002-8

International Journal of Engineering Science 41 (2003) 1197–1207www.elsevier.com/locate/ijengsci

mail to: [email protected]

Afzal et al. [3] deals with a single set of equations that remain valid both for the forward or reversemoving surface.The similarity solution corresponding to the Falkner–Skan equation subjected to slip velocity k

(¼ �Uw=U1) at boundary wall, had been studied by several workers, by adopting the referencevelocity UðxÞ ¼ U1ðxÞ [5]. Adopting reference velocity UðxÞ ¼ UwðxÞ the flow on a continuouslystretching sheet had been studied by Afzal and Varshney [6] as described in Aziz and Na [7] andAfzal [8,9]. This motivates the need to unify two approaches, following Afzal et al. [3], that dealwith the stretching of bounding surface subjected to free stream fluid pressure gradient. Thenumerical solutions along with some closed form exact and asymptotic solutions have beenpresented. The approximate integral solution are summarized in Appendix A.

2. Analysis

The two dimensional laminar boundary layer equations of an incompressible fluid subjected topressure gradient are

ouox

þ ovoy

¼ 0 ð1Þ

uouox

þ vouoy

¼ � 1

qp0ðxÞ þ m

o2uoy2

; p0ðxÞ ¼ �qU1dU1

dxð2Þ

Here, u is the stream wise velocity in x-direction and v is the normal velocity in y-direction, p0ðxÞ isthe static pressure gradient, q is fluid density, m is molecular kinematic viscosity and U1ðxÞ is thevelocity at the edge of the boundary layer of thickness d. The boundary conditions on stretchingsurface velocity UwðxÞ and at the edge of boundary layer with pressure gradient free streamvelocity U1ðxÞ are

y ¼ 0; u ¼ UwðxÞ; v ¼ 0;yd! 1; u ! U1ðxÞ ð3Þ

The similarity solution exist if stretching plate velocity UwðxÞ and velocity at the edge of boundarylayer U1ðxÞ obey the following power law relations

Uw ¼ U0wxm; U1 ¼ U01xm; b ¼ 2m1þ m

ð4Þ

The similarity transformation based on the composite reference velocity UðxÞ, following Afzalet al. [3], is given by

u ¼ UðxÞf 0ðgÞ; g ¼ y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ mÞUðxÞ

2mx

r; UðxÞ ¼ U1ðxÞ þ UwðxÞ ð5Þ

and the laminar boundary layer equations (1)–(4) reduce to

1198 N. Afzal / International Journal of Engineering Science 41 (2003) 1197–1207

f 000 þ ff 00 þ bð�2 � f 02Þ ¼ 0 ð6Þ

f ð0Þ ¼ 0; f 0ð0Þ ¼ 1� �; f 0ð1Þ ¼ � ð7a;b; cÞ

� ¼ U1

U1 þ Uw; bðb; �Þ ¼

Z 1

0

�� f 0dg ð8Þ

Here �, b and bðb; �Þ are the constants. Eqs. (6) and (7) for � ¼ 1 are classical Falkner–Skanequations and for � ¼ 0 are the equations for the stretching sheet proposed by Afzal and Varshney[6] and Afzal [8,9]. For 0 < � < 1 both the free stream and wall are moving in positive x-direction,� > 1 the wall is moving in negative x-direction towards the origin and � < 0 the free stream is innegative x-direction. For b ¼ 1 and � ¼ 0 the solution [10] is f ðgÞ ¼ 1� e�g, and other closedform solutions are described below.

2.1. b ¼ �1 solution

Eq. (6) for b ¼ �1 can be integrated twice to get

f 0 þ 1

2f 2 ¼ 1

2�2g2 þ �dg þ 1� �; d ¼ �b ¼

Z 1

0

f 0 � �dg ð9Þ

Eq. (9) is a Riccati type equation and its solution for � ¼ 0 and � 6¼ 0 are

f ðgÞ ¼ffiffiffi2

ptanhðg=

ffiffiffi2

pÞ; � ¼ 0 ð10Þ

f ðgÞ ¼ �g þ d �d expð�d�g � 1

2�2g2Þ

1�ffiffiffiffiffiffi2p

pd4�exp d2

2

� �erf �gþdffiffi

2p

� �� erf dffiffi

2p

� �h i ; � 6¼ 0 ð11Þ

The axial velocity gradient at the wall from Eq. (9) is given by

f 00ð0Þ ¼ �d ¼ �2� 1

2

��

1=2

; d2 ¼ 2ð1� 2�Þ ð12Þ

show dual solutions exist for � < 0:5 and no solution for � > 0:5. The differentiation of solution(12) with respect to � give

df 00ð0Þd�

¼ � 2� 6�ffiffiffiffiffiffiffiffiffiffiffiffiffi2� 4�

p ;ddd�

¼ � 4ffiffiffiffiffiffiffiffiffiffiffiffiffi2� 4�

p ð13Þ

The nature of duality with respect to �, in the solution may further be explored from relations (13),which show the following. (i) At � ¼ 0, df 00ð0Þ=d� ¼ �

ffiffiffi2

p, which represent the dual solution,

physically the wedged-shape curve of included angle 2 tan�1ffiffiffi2

p¼ 109:47� for �P 0. (ii) At

� ¼ 1=3 the result df 00ð0Þ=d� ¼ �0, shows that the dual solutions are two tangents, on the f 00ð0Þcurve, parallel to �-axis. (iii) At � ¼ 1=2, df 00ð0Þd� ¼ �1, which show that the dual solutions are

N. Afzal / International Journal of Engineering Science 41 (2003) 1197–1207 1199

the tangents on the curve parallel to f 00ð0Þ-axis and the solution turns back to imply non-existenceof the solution for � > 1=2.

2.2. b ! 1 asymptote

The asymptotic solution for large b is analysed in terms of following variables

f ðgÞ ¼ffiffiffia

phðzÞ; g ¼

ffiffiffia

pz; a ¼ 6=b ð14Þ

and the boundary layer equations (6) and (7) become

h000 þ 6ð�2 � h02Þ ¼ �ahh00 ð15Þ

hð0Þ ¼ 0; h0ð0Þ ¼ 1� �; h0ð1Þ ¼ � ð16a;b; cÞ

The asymptotic expansion hðzÞ ¼ h1ðzÞ þOðaÞ may be expressed in the closed form

h1ðzÞ ¼z

1þ z; � ¼ 0 ð17aÞ

h1ðzÞ ¼ffiffiffiffiffiffiffiffiffiffiffi1þ �

pþ �z�

ffiffiffiffiffi3�

ptanh z

ffiffiffiffiffi3�

p"

þ tanh�1ffiffiffiffiffiffiffiffiffiffiffi�þ 1

3�

r #; � 6¼ 0 ð17bÞ

The velocity gradient at the wall is given by

f 00ð0Þ ¼ ð2�� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3bð1þ �Þ

r; b ¼ 3

ffiffiffiffiffi2�

ph�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ð�þ 1Þ

p i ffiffiffib

pð18Þ

The higher order perturbations for stretching sheet � ¼ 0 have been studied [9] and the skinfriction along with its Euler transform are

f 00ð0Þ ¼ �ffiffiffiffiffiffi2b3

r1

�þ 1

5bþ 18

125b2þ

ð19aÞ

f 00ð0Þð2þ bÞ3=4 ¼ � 4

9Y

� 1=4

1

�� 41

20Y þ 801

4000Y 2 þ

; Y ¼ 1=ðb þ 3Þ ð19bÞ

For b ¼ 1, 0 and )1 the improved series (19b) predicts )1.024, )0.6496 and 0.0435 whereas exactvalues are 1, )0.62755 and 0 respectively. Further, in the case of the stationary plate (� ¼ 1) thehigher order solution and its Euler transform [11] yield

f 00ð0Þ ¼ b1=2ð1:1547þ 0:0746=b þ 0:00509=b2 � 0:00182=b3 þ Þ ð20aÞ

f 00ð0Þ ¼ Y �1=2ð1:1547� 0:5027Y � 0:1019Y 2 � 0:0384Y 3 þ Þ; Y ¼ 1=ð1þ bÞ ð20bÞ


predicted good results not only for positive values of but for negative also, and for b ¼ 1 it givesf 00ð0Þ ¼ 1:2345 where as exact result is 1.2326. Further, for small b ! 0 asymptotic solution forvarious values of � are of also interest and described in Appendix B.

2.3. � � 1=2 asymptote

The asymptotic solution is considered, around the exact solution f ðgÞ ¼ g=2 at � ¼ 1=2. In-troducing the following change of variables

f ðgÞ ¼ �g þ DgðgÞ; D ¼ 1� 2� ð21Þ

the boundary layer equations (6) and (7) yield

g000 þ 1

2gg00 � bg0 ¼ D

1

2g

� � g

g00 � bð1� g0Þg0

�ð22Þ

gð0Þ ¼ 0; g0ð0Þ ¼ 1 g0ð1Þ ¼ 0 ð23Þ

The leading order asymptotic gðgÞ ¼ g1ðgÞ þOðDÞ has the solution

g01ðgÞ ¼ /

�� b;

1

2;� 1

4g2� W g/

1� 2b2

;3

2;

�� 1

4g2

ð24Þ

where /ða; b; xÞ is the confluent hypergeometric function and CðaÞ is the gamma function. Thevelocity gradient at the wall yields

f 00ð0Þ ¼ ð2�� 1ÞW ; b ¼ ð2�� 1Þ 2W1þ 2b

; W ¼ Cð1þ bÞC 1þ2b

2

� � ð25Þ

3. Results and discussion

The numerical solutions of Eqs. (6) and (7) for velocity profile f 0ðgÞ against g are displayed inFig. 1, with non-dimensional parameter 06 �6 1 for each value stretching parameter b ¼ 0, 0.5, 1,1.5, 2. For each value b the velocity profile shows gradual variation as velocity ratio � increasesfrom 0 to 1. The gradient f 0ðgÞ is positive for � > 1=2 and negative for � < 1=2 and all lines crosseach other in the neighborhood of a value g0ðbÞ, say; b ¼ 0, then g0 � 1 and b > 0, then g0 < 1.The case � ¼ 0:5 corresponds to parallel flow solution where f 0ðgÞ ¼ 0:5.For parameter � ¼ 1, 0.5 and 0 the numerical solutions f 00ð0Þ have been displayed in Fig. 2

against b, in the range �26 b6 2. The asymptote (18) for large b are also displayed in samefigure. The case � ¼ 1 corresponds to solution of classical Falkner–Skan equation and no com-ment is needed. Eqs. (6) and (7) for the stretching sheet (� ¼ 0) were first proposed by Afzal andVarshney [6]. The solution f 00ð0Þ increases with b and becomes zero at b ¼ �1. For b < �1, f 00ð0Þchanges sign (becomes positive) and as b ! �2, f 00ð0Þ approaches infinity.


For b positive, and 06 �6 1 the numerical solutions f 00ð0Þ displayed against � with parameterb ¼ 0, 0.5, 1.0, 1.5 and 2, in Fig. 3 are unique. The perturbation solution (23) shows that f 00ð0Þ is alinear function around � ¼ 1=2 and compares well with the numerical solution strictly around� ¼ 1=2. At b ¼ 0, it predicted f 00ð0Þ ¼ �0:5642ð1� 2�Þ, when compared with exact solutionf 00ð0Þ ¼ �0:62755 at � ¼ 0 and f 00ð0Þ ¼ 0:4696 at � ¼ 1 and therefore next order perturbation isneeded. In case � > 1 and (a) b ¼ 0 there are dual solutions for 1 < � < 1:548 and no solution for

Fig. 1. The velocity distribution f 0ðgÞ against g for non-dimensional velocity parameter 06 �6 1 for various values ofthe prescribed boundary stretching (equal to pressure gradient) parameter b ¼ 0, 0.5, 1.0, 1.5, 2.0.


Fig. 2. The velocity gradient at wall f 00ð0Þ against b, the stretch rate of the moving boundary surface. � ¼ 0 purely

stretching surface in the absence of free stream, � ¼ 1 purely non-stretching sheet subjected to pressure gradient, and

f eAA ¼ 0:5, f 00ð0Þ ¼ 0. Asymptote b ! 1 for � ¼ 0 and � ¼ 1 from relation (18).

Fig. 3. Favorable stretching surface boundary bP 0: The velocity gradient at wall f 00ð0Þ against � in the domain

06 �6 1, for specific values of parameter b in the range 06 b6 2.


� > 1:548 [3], (b) 0 < b < 0:15 there are triple solutions for 1 < � < �0ðbÞ and unique solution for� > �0ðbÞ, (c) 0:14 < b < 1 unique solutions for 1 < � < 1.For b ¼ �1 the closed form exact solutions (12) for f 00ð0Þ displayed against � in Fig. 4, possess no

solution for � > 1=2, dual solutions for 0 < � < 1=2, and unique solution at � ¼ 0 and � ¼ 1=2.Further, the exact solution (13) around � ¼ 0, predicts df 00ð0Þ=d� ¼ �

ffiffiffi2

pand the dual solution is

represented by wedge-shape curve of included angle 109.47� around �P 0, thereby leading to theunique solution at � ¼ 0. For other values of b negative, the numerical solutions f 00ð0Þ displayedagainst �, with parameter range �16 b6 0 in Fig. 4 which show dual solutions for 0 < � < �0ðbÞand that at b ¼ �0:5, f 00ð0Þ6 0 for all values of �6 �0ðbÞ. Further, the dual numerical solutions onthe lower branch of have also been displayed in Fig. 4. Based on exact solution (12) for b ¼ �1 thewedge-shaped curve around �P 0 has encouraged to analyse the similar behavior for other valuesof b in the domain �1 < b < 0 on the lower branch (around �P 0) of dual numerical solutions.The results for 0 < � < 0:4, have been displayed in Fig. 4 by dotted lines due to slow numericalconvergence for relatively large values of g1 needed for numerical integration and that it may beresolved by analytical means. Therefore, b < 0 there exists �0ðbÞ such that the solutions are dual for0 < � < �0ðbÞ, unique solution for � ¼ 0 and � ¼ �0ðbÞ, and there exists no solution for � > �0ðbÞ.

Appendix A. Approximate solution

An integral of the momentum equation (4) under the boundary conditions (5) gives

Z 1

0

�f 0 � f 02 dg þ bZ 1

0

�2 � f 02 dg ¼ f 00ð0Þ ðA:1Þ

Fig. 4. Adverse stretching surface boundary b < 0: The velocity gradient at wall f 00ð0Þ against � in the domain

06 �6 �0ðbÞ, for specific values of parameter b in the range �16 b6 0.


Introducing a function UðfÞ such that

UðfÞ ¼ u� U1

Uw � U1¼ f 0ðgÞ þ �� 1

2�� 1; f ¼ g

AðA:2Þ

Uðf ¼ 0Þ ¼ 0; Uðf ¼ 1Þ ¼ 1 ðA:3Þ

the momentum integral (A.1) yields

ð1þ bÞðB1 � B2Þ � �½B1 � 2ð1þ bÞB2� ¼ U0ð0Þ=A2 ¼ K ðA:4Þ

B1 ¼Z 1

0

1� U0 df; B2 ¼Z 1

0

U0 � U02 df ðA:5Þ

The axial velocity gradient at the wall is given by

f 00ð0Þ ¼ ð2�� 1ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU0ð0ÞK

q; b ¼ ð2�� 1ÞB1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU0ð0Þ=K

qðA:6Þ

The relations (A.4) and (A.6) show that the solution does not exist if K becomes negative for� > �0 where

�0 ¼ ð1þ bÞ 1

þ 1þ 2bH � 2ð1þ bÞ

�; H ¼ B1

B2ðA:7Þ

where shape factor H > 2.If one considers the trial velocity profile

UðfÞ ¼ F ðfÞ þ bGðfÞ ðA:8Þ

F ðfÞ ¼ 2f � 2f3 þ f4; GðfÞ ¼ fð1� fÞ3=6 ðA:9Þ

then B1 and B2 from relation (A.5) are given by

B1 ¼3

10� b120

; B2 ¼37

315� b945

� b2

9072ðA:10Þ

The axial velocity gradient at the wall is given by relation (A.4) with

U0ð0Þ ¼ 2þ b6; K ¼ ð1þ b � �Þ 3

10

�� b120

þ ð1þ bÞð�1þ 2�Þ 37

315

�� b945

� b2

9072

ðA:11Þ

In case of another trial velocity profile the relation (A.8) becomes

F ðfÞ ¼ ð3f � f3Þ=2; GðfÞ ¼ ðf � 2f2 þ f3Þ=6 ðA:12Þ


B1 ¼3

8� b48

; B2 ¼39

280� b560

� b2

1680ðA:13Þ

the axial velocity gradient at the wall is given

U0ð0Þ ¼ 3

2þ b4; K ¼ ð1þ b � �Þ 3

10

�� b48

þ ð1þ bÞð�1þ 2�Þ 39

280

�� b560

� b2

1680

ðA:14Þ

In acceleration flows (b > 0), the skin friction (A.6), with parameters (A.10)–(A.12) from velocityprofile (A.8) and for decelerating flows (for �1=2 < b < 0) the skin friction (A.6) with parameters(A.13) and (A.14) from velocity profile (A.12) are, in general, good enough for purpose of skinfriction calculations. Although, relation (A.4) predicts dual solutions, but trial velocity profiles(A.9) and (A.11) are not appropriate to describe the lower branch of dual solution as well as �0ðbÞfrom relation (A.7).

Appendix B. Small b series

The asymptotic solution for b ! 0 for various values of � is also on interest in exploration ofthe singularities associated with the solution. The case � ¼ 1 has been studied [7,12] and as-ymptotic series solution have been improved by extracting the nearest square root singularity atb ¼ �b0 ¼ �0:198838 to get [12]

f 00ð0Þ ¼ 0:88219ffiffiffiffiffiffiffiffiffiffiffiffiffib þ b0

pþ 0:07622þ 0:30973b � 0:27843b2 þ 0:43550b3

� 0:84152b4 þ 1:8539b5 � 4:40762b6 þ 10:70649b7 � 24:63554b8

þ 44:59949b9 � ðB:1Þ

and predictions are good for �b0 < b < b0. The convergence of the series for a general value of bmay also be improved by recasting the series in terms of Euler variable Z, which maps the sin-gularity at infinity, to obtain [12]

f 00ð0Þðb þ b0Þ�1=2 ¼ 1:05312þ 0:052648Z þ 0:023004Z2 þ 0:011757Z3 þ 0:0064405Z4

þ 0:0036471Z5 þ 0:0020899Z6 þ 0:0011915Z7 þ 0:00066396Z8

þ 0:00035297Z9 þ 0:00017154Z10 þ ;Z ¼ b=ðb þ b0Þ

ðB:2Þ

which for b ! 1 predicts 1.15509 against exact value 1.15470. The series (24) predicts f 00ð0Þ tofour significant digits for all values of b from zero to infinity.


References

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Mass Transfer 32 (1989) 1383–1386.

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(1980) 289–293.

[7] A. Aziz, T.Y. Na, Perturbation Methods in Heat Transfer, Hemisphere, New York, 1986, p. 182.

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(1983) 217–219.

[9] N. Afzal, Heat transfer from a stretching surface, Int. J. Heat Mass Transfer 36 (1993) 1128–1131.

[10] L. Crane, Flow past a stretching plate, Z. Angew. Maths. Phys. 21 (1970) 645–647.

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momentum transfer on power law stretching plate with free stream pressure gradient

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