falkner–skan equation for flow past a stretching surface with suction or blowing: analytical...

Applied Mathematics and Computation 217 (2010) 2724–2736

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Applied Mathematics and Computation

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Falkner–Skan equation for flow past a stretching surface with suctionor blowing: Analytical solutions

Noor AfzalFaculty of Engineering, Aligarh Muslim University, Aligarh 202 002, India

a r t i c l e i n f o

Keywords:SuctionBlowingDual solutionsStretching surfaceSimilarity solutionsFalkner–Skan equation

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.07.080

E-mail address: [email protected]

a b s t r a c t

The simultaneous effects of suction and injection on tangential movement of a nonlinearpower-law stretching surface governed by laminar boundary layer flow of a viscous andincompressible fluid beneath a non-uniform free with stream pressure gradient is consid-ered. The self-similar flow is governed by Falkner–Skan equation, with transpirationparameter c, wall slip velocity k and stretching sheet (or pressure gradient) parameter b.The exact solution for b = �1 and three closed form asymptotic solutions for b large, largesuction c, and k ? 1 have also been presented. Dual solutions are found for b = �1 for eachvalue of the transpiration parameter, including the non-permeable surface, for each pre-scribed value of the wall slip velocity k. The large b asymptotic solution also dual withrespect to wall slip velocity k, but do not depend on suction and blowing. The critical valuesof c, b and k are obtained and their significance on the skin friction and velocity profiles isdiscussed. An approximate solution by integral method for a trial velocity profile is pre-sented and results are compared with the exact solutions.

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1. Introduction

Many industrial processes involves continuous stretching surfaces cooled by an external stream along the production line.Boundary layer behavior over a moving continuous solid surface is an important type of flow occurring in several engineer-ing processes. The examples are the thermal processing of sheet-like materials is a necessary operation in the production ofpaper, linoleum, polymeric sheets, wire drawing, drawing of plastic films, metal spinning, roofing shingles, insulating mate-rials, fine-fiber matts, cooling of films or sheets, conveyor belts, metallic plates and cylinders. In virtually all such processingoperations, the sheet moves parallel to its own plane. The moving sheet may induce motion in the neighboring fluid or, alter-natively, the fluid may have an independent forced-convection motion that is parallel to that of the sheet. Both the kinemat-ics of stretching and the simultaneous heating or cooling during such processes have a decisive influence on the quality ofthe final products. In virtually all such processing operations, the sheet moves parallel to its own plane.

In many engineering systems laminar flow on the moving boundary surface with speed Uw is subjected to an ambientfluid speed Ue. For Uw > Ue or Uw < Ue these two problems are physically different and cannot be mathematically transformedinto one another. The analysis may be considered in two cases separately, when Uw < Ue (with basic scale Ue) and Uw > Ue

(with basic scale Ue), and thus two sets of boundary conditions have to be formulated (Abdelhafez [1]). The first set Ue isthe basic velocity (for Uw < Ue) the first set of boundary conditions was studied by Klemp and Acrivos [2] and Hussainiet al. [3].

Afzal [4,5] considered a reference velocity Ur as Ur = Uw + Ue, and proposed a single set of boundary layer equation alongthe boundary conditions, irrespective of whether Uw > Ue or Uw < Ue. Abraham and Sparrow [6] considered the reference

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N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736 2725

velocity is the velocity difference Ud = jUw � Uej which uses the magnitude of the relative-velocity in conjunction with thedrag formula for the case in which only one of the participating media is in motion. They found that the results of exactsolutions demonstrate that this model is flawed and under predicts the drag force, and thus the use of the relative-velocitymodel can lead to gross errors in the drag force. The extent of the error increases as the two participating velocities approacheach other in magnitude. The solution depends not only on the velocity difference jUw � Uej but also on the velocity ratioUw/Ue.

In view of these applications, Sakiadis [7] initiated the study of boundary layer flow over a continuous solid surface mov-ing with a constant speed in an otherwise quiescent fluid medium. Due to entrainment of ambient fluid, this boundary layerflow is quite different from that over a semi-infinite flat plate (Blasius [8] problem). An important class of similarity solutionscorresponding to the boundary layer on nonlinear stretching impermeable wall was first presented by Afzal [9–11], andwork [9] has been described in the book by Aziz and Na [12]. The resulting ordinary differential equation of Afzal [9,10]which contains a parameter has been discussed in detail by Brighi and Hoernel [13] and Guedda [14], in connection withsimilarity solutions arising during free convection in porous media. The effects of suction and injection on momentumand thermal boundary layers over a two-dimensional or axisymmetric nonlinear stretching surface in a stationary fluidhas been studied by Afzal [15].

It is well-known that the effects of injection on the boundary layer flow are of interest in reducing the drag force (seeSchlichting [16], Rosenhead [17]. The boundary layer problem of a semi-infinite flat plate moving in a free stream with masstransfer (suction or injection) has been recently discussed by Ahmad [18] and Weidman, Kubitschek and Davis [19] where asmass transfer on a stationary plate was studied by Watanabe [20] and Yih [21]. Yang and Chien [22] presented analytic solu-tion of the Falkner–Skan equation when b = �1 for suction and injection on a stationary surface. Kudenatti and Awati [23]have studied the effects of suction in Falkner–Skan flows by series solution and method of stretching of variables. Thenumerical solutions of the Falkner–Skan equation with suction and injection were presented by Koh and Hartnett [24]. Rileyand Weidman [25] have studied multiple solutions of the Falkner–Skan equation for flow past a stretching boundary whenthe external velocity and the boundary velocity are each proportional to the same power-law of the downstream distance.The solution of Falkner–Skan equation [26] has been extensively studied for a stationary surface with out transpiration in[27,28]. The uniqueness of flow of a Navier–Stokes fluid due to a stretching boundary has been considered by McLeodand Rajagopal [29]. The effects of non-Newtonian fluids past a porous plate with suction or injection has been studied byMansutti et al. [30] and second grade fluids by Massoudi et al. [31].

The simultaneous effects of transpiration through and tangential movement on a sheet gives the self-similar boundarylayer flow driven by far field velocity has been considered here. The analytical solutions including mass injection as wellas suction on the walls are considered for Falkner–Skan equation for flow past a stretching boundary when the externalvelocity and the boundary velocity are each proportional to the same power-law of the downstream distance. An approxi-mate minimum error solution for a trial velocity profile is also presented by minimizing the error in the square of the integralof the boundary layer equations over entire domain by the least square method.

2. Analysis of self-similar flow

The boundary layer equations for incompressible two-dimensional mean turbulent flow subjected to pressure gradient, instandard notation are

@u@xþ @v@y¼ 0; ð1Þ

u@u@xþ v @u

@y¼ Ue

dUe

dxþ m

@2u@y2 : ð2Þ

The similarity solution of the laminar boundary layer equations subjected to a stretching boundary surface with velocity Uw

and pressure gradient with free stream velocity Ue is considered where

UwðxÞ ¼ U�wxm; UeðxÞ ¼ U�exm; ð3Þ

and co-ordinate system is shown in Fig. 1. The boundary conditions of the flow at the wall and free stream are

y ¼ 0; u ¼ UwðxÞ; v ¼ VwðxÞ; y=d!1; u ¼ UeðxÞ: ð4Þ

The similarity transformation is given by

u ¼ UeðxÞF 0ðgÞ; f ¼ y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þmÞUeðxÞ

2mx

rð5Þ

and the laminar boundary layer equations reduce to

F 000 þ FF 00 þ bð1� F 02Þ ¼ 0; ð6Þ

Fð0Þ ¼ c; F 0ð0Þ ¼ k; F 0ð1Þ ¼ 1: ð7a;b; cÞ

Fig. 1. Physical model and co-ordinate system in the boundary layer with pressure gradient over a stretching wedge surface with suction or blowing.

2726 N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736

The flow parameters b and k are constants, and c is also a constant (positive for suction and negative for blowing) given byrelation

b ¼ 2mmþ 1

; k ¼ Uw

Uec ¼ �VwðxÞ ðmþ 1Þ mUeðxÞ

2x

� ��1=2

: ð8Þ

The boundary layer displacement thickness d is given by relation

d ¼ �K ¼Z 1

01� F 0 df: ð9Þ

Under the transformation

FðfÞ ¼ gðzÞffiffiffikp

; f ¼ zffiffiffikp ; ð10Þ

the Eqs. (6) and (7) may be expressed aa

g000 þ gg00 � bg02 ¼ � b

k2 ; ð11Þ

gð0Þ ¼ gw; g0ð0Þ ¼ 1; g0ð1Þ ¼ 1k; ð12Þ

where gw ¼ c=ffiffiffikp

. For large wall velocity k ?1 the Eqs. (11) and (12) yield

g000 þ gg00 � bg02 ¼ 0; ð13Þgð0Þ ¼ gw; g0ð0Þ ¼ 1; g0ð1Þ ¼ 0: ð14Þ

The Eqs. (13) and (14) for a power-law stretching of a continuous sheet on a impermeable wall were first proposed by Afzaland Varshney [9] and Afzal [10].

2.1. Solution for b = �1

The Eq. (6) for b = �1 may be integrated twice subject to the boundary conditions (7a,b,c) yield

F 0 þ 12

F2 ¼ 12

f2 þKfþ kþ 12c2 ð15Þ

and F00(0) = K � kc. Under the following transformation

FðfÞ ¼ fþKþ 2K 0ðfÞKðfÞ ; ð16Þ


the Eq. (15) yields a Riccati type equation as K00 + (f + K)K0 = 0 whose solution becomes

Fig. 2.various

K ¼ Ap2

� �1=2erf

fþKffiffiffi2p

� �þ B: ð17Þ

The relations (16) and (17) provided the solution

FðfÞ ¼ fþK�2 exp � 1

2 ðfþKÞ2h iffiffiffip2

perf fþKffiffi

2p� �

þ BA

; ð18Þ

The boundary condition (7a) yields

BA¼ � 2

K� cexp

K2

2

!� p

2

� �1=2erf

Kffiffiffi2p� �

: ð19Þ

The relations (18) and (10) yield the solution

FðfÞ ¼ fþK�ðK� cÞ expð�Kf� 1

2 f2Þ1� ðK� cÞ

ffiffiffip8

pexp K2

2

� �erf fþKffiffi

2p� �

� erf Kffiffi2p� �h i : ð20Þ

The boundary layer displacement thickness K and axial velocity gradient at the wall F00(0) from solution (15) yield

K ¼ �½2ðk� 1Þ þ c2�1=2; ð21Þ

F 00ð0Þ ¼ �kcþK ¼ �kc� 2ðk� 1Þ þ c2� �1=2: ð22Þ

In relations (21) and (22) the square root sign indicates that the solution exists for k P 1 � c2/2. The shear stress f00(0) vs k isshown in Fig. 2 for c = 1,0,�1. The solutions are dual for k > 1 � c2/2 and unique for k = 1 � c2/2. The dual velocity profiles areshown in Fig. 3(a) and (b).

The solution of Eqs. (13) and (14) for b = �1 yield

gðzÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ g2

w

qtanh

z2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ g2

w

qþ hc

� �; hc ¼ �tanh�1 gwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ g2w

p !

; ð23Þ

g00ð0Þ ¼ �gw; gð1Þ ¼ ð2þ g2wÞ

1=2: ð24Þ

2.2. Large b asymptote

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

FðfÞ ¼ b�1=2HðnÞ; f ¼ b�1=2n; ð25Þ

and the boundary layer Eqs. (4) and (5) become

H000 � H02 þ 1 ¼ �b�1HH00: ð26Þ

Exact solution for b = �1 in analytical closed form: the dual solutions of the wall shear stress F00(0) versus non-dimensional slip velocity of sheet k, forvalues of the wall suction velocity parameter c = 0.5, 0 and �1.

Fig. 3. Exact solution b = �1: the dual solutions of the velocity profiles F0(f) versus f for prescribed slip velocity parameter (say, k = 2) for (a) suction/injection parameter c = 1, 0, �1, (b) suction/injection parameter c = 2, 0, �2.


The asymptotic expansion for b ?1 is H(n) = H1(n) + b�1H2(n) + � � �. The leading order equation and boundary conditions

H0001 � H021 þ 1 ¼ 0; ð29ÞH1ð0Þ ¼ Hw; H01ð0Þ ¼ k; H01ð1Þ ¼ 1: ð30Þ

An integral of relation (29) subject to the boundary conditions (30) yields

H0021 �23

H031 þ 2H1 ¼43

ð31Þ

and shear stress on the surface become

H001ð0Þ ¼ �ðk� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23ðkþ 2Þ

r: ð32Þ

A further integral of (29) yields

H01ðnÞ ¼ �2þ 3tanh2 nffiffiffi2p þ /

� �; ð33Þ

H1ðnÞ ¼ Hw þ nþ 3ffiffiffi2p

tanh /� tanhnffiffiffi2p þ /

� � ; ð34Þ

where

/ ¼ � tan�1

ffiffiffiffiffiffiffiffiffiffiffiffikþ 2

3

r: ð35Þ


(i) Case 1: In terms of original variables the solution becomes

Fig. 4.suction

F 0ðfÞ ¼ �2þ 3tanh2 f

ffiffiffib2

rþ tan�1


3

r !; ð36Þ

FðfÞ ¼ fþ cþ 3

ffiffiffi2b

s ffiffiffiffiffiffiffiffiffiffiffiffikþ 2

3

r� tanh f

ffiffiffib2

rþ tan�1


3

r !" #: ð37Þ

The velocity gradient at the wall is given by

F 00ð0Þ ¼ ðk� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b3ðkþ 2Þ

r; ð38Þ

K ¼ cþ 3

ffiffiffi2b

s ffiffiffiffiffiffiffiffiffiffiffiffikþ 2

3

r� 1

!: ð39Þ

(ii) Case 2: In terms of original variables the solution becomes

F 0ðfÞ ¼ �2þ 3tanh2 f

ffiffiffib2

r� tan�1


3

r !; ð40Þ

FðfÞ ¼ fþ cþ 3

ffiffiffi2b

s�


3

r� tanh f

ffiffiffib2

r� tan�1


3

r !" #: ð41Þ

The velocity gradient at the wall is given by

F 00ð0Þ ¼ �ðk� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b3ðkþ 2Þ

r; ð42Þ

K ¼ cþ 3

ffiffiffi2b

s�


3

r� 1

!: ð43Þ

The square root sign indicates that the solution for large b exists for k P �2, that to is dual. The shear stress f 00ð0Þ=ffiffiffibp

vs kbased on relations (40) and (42) have been shown in Fig. 4. The solutions are dual for �2 < k < 1 describing the aiding andopposing flows regimes. The dual solutions coincide for k = �2 and k = 1. The solution for k > 1 on lower branch, shown bysolid line, is regarded as unique as the solution of corresponding upper branch, shown by dotted line, is physically unreal-istic. The solution are shown in Fig. 5.

The solution of Eqs. (13) and (14) large slip velocity (b ?1) becomes

gðzÞ ¼ gw þzb1

zþ b1; b1 ¼ �

ffiffiffiffiffiffiffiffi6=b

p; ð44Þ

g00ð0Þ ¼ �ð2b=3Þ1=2; gð1Þ ¼ gw �

ffiffiffiffiffiffiffiffi6=b

p: ð45Þ

Asymptotic solution for b ?1: the wall shear stress F 00ð0Þ=ffiffiffibp

versus non-dimensional slip velocity of sheet parameter k, does not depend on/injection parameter c.

Fig. 5. Velocity distribution F0(f) vs fffiffiffibp

for various values the non-dimensional slip velocity parameter k, based on lowest order solution for large b ?1,which does not depend on wall transpiration.


2.3. Asymptotic for k ? 1

The exact solution of Eqs. (6) and (7) for k = 1 is F(Z) = f + c. The asymptotic series solution for k ? 1 is considered aroundthis exact solution by expanding the stream function in the power of parameter k � 1. Introducing the following change ofvariables

FðfÞ ¼ fþ cþ ðk� 1ÞGðfÞ; ð46Þ

the boundary layer Eqs. (6) and (7) yield

G000 þ fþ cð ÞG00 � bG0 ¼ ðk� 1Þð�GG00 þ bG02Þ; ð47ÞGð0Þ ¼ 0; G0ð0Þ ¼ 1 G0ð1Þ ¼ 0: ð48Þ

The asymptotic expansion for G is

GðfÞ ¼ G1ðfÞ þ ðk� 1ÞG2ðfÞ þ ðk� 1Þ2G3ðfÞ þ � � � ð49Þ

The first order equations are

G0001 þ fþ cð ÞG001 � bG01 ¼ 0; ð50Þ

G1ð0Þ ¼ 0; G01ð0Þ ¼ 1; G01ð1Þ ¼ 0: ð51Þ

The solution to first order Eqs. (50) and (51) may be expressed in closed form

G01ðfÞ ¼ AU �b;12;�1

2ðfþ cÞ2

� �þ Bðfþ cÞU 1� 2b

2;32;�1

2ðfþ cÞ2

� �; ð52Þ

where U(a,b,x) is the confluent hypergeometric function. It is well-known that U(a � b,0) = 1 and as x ?1we have (Abrom-witch and Stugen [32])

Uða; b;�xÞ ¼ CðbÞCðb� aÞ x

�a þ � � � ð53Þ

where C(a) is the gamma function. The constants A and B estimated from the boundary conditions (51) yield

A ¼ U �b;12;�1

2c2

� ��WcU

1� 2b2

;32;�1

2c2

� � �1

; B ¼ �AW ; ð54Þ

where

W ¼ffiffiffi2p Cð1þ bÞ

Cð1þ2b2 Þ

: ð55Þ


The solution (52) based on relations (54) and (55) yields the velocity distribution

G01ðfÞ ¼U �b; 1

2 ;� 12 ðfþ cÞ2

� ��Wðfþ cÞU 1�2b

2 ; 32 ;� 1

2 ðfþ cÞ2� �

U �b; 12 ;� 1

2 c2� �

�WcU 1�2b2 ; 3

2 ;� 12 c2

� � : ð56Þ

The solution in terms of F(f) yields

F 0ðfÞ ¼ 1þ ðk� 1ÞU �b; 1

2 ;� 12 ðfþ cÞ2

� ��Wðfþ cÞU 1�2b

2 ; 32 ;� 1

2 ðfþ cÞ2� �

U �b; 12 ;� 1

2 c2� �

�WcU 1�2b2 ; 3

2 ;� 12 c2

� � ð57Þ

and friction parameter becomes

F 00ð0Þ ¼ �ðk� 1ÞW U �b;12;�1

2c2

� ��WcU

1� 2b2

;32;�1

2c2

� � �1

: ð58Þ

(i) Special case c = 0: The solution (57) for no transpiration from the surface (c = 0) becomes

F 0ðfÞ ¼ 1þ ðk� 1Þ U �b;12;�1

2f2

� ��WfU

1� 2b2

;32;�1

2f2

� � : ð59Þ

The velocity gradient at the wall yields

F 00ð0Þ ¼ �ðk� 1ÞW: ð60Þ

(ii) Special case b = 0: The solution (57) for moving sheet subjected to a uniform oncoming parallel stream of velocity(b = 0) becomes

F 0ðfÞ ¼ 1þ ðk� 1Þ1�

ffiffiffi2p

qðfþ cÞU 1

2 ;32 ;� 1

2 ðfþ cÞ2� �

1�ffiffiffi2p

qcU 1

2 ;32 ;� 1

2 c2� � : ð61Þ

Using the relation (Abromwitch and Stugen [28])

U12;32;�z2

� �¼

ffiffiffiffipp

2zerfðzÞ ð62Þ

and the solution Eq. (57) is simplified as

F 0ðfÞ ¼ 1þ ðk� 1Þerfc fþcffiffi

2p� �

erfc cffiffi2p� � ; ð63Þ

where erfc(x) = 1 � erf(x) is the complimentary error function. The velocity gradient at the wall yields

F 00ð0Þ ¼ �ðk� 1Þffiffiffiffi2p

r exp � c2

2

� �erfc cffiffi

2p� � : ð64Þ

2.4. Large suction

For large suction, introducing the variables

FðfÞ ¼ cþ hðgÞc

; f ¼ gc; ð65Þ

into (4) and (6) the boundary layer equations yield

h000 þ h00 ¼ �c�2½hh00 þ bð1� h02Þ�; ð66Þhð0Þ ¼ 0; h0ð0Þ ¼ k; h0ð1Þ ¼ 1: ð67Þ

The asymptotic expansion for h is

hðgÞ ¼ h0ðgÞ þ c�2h2ðgÞ þ c�4h4ðgÞ þ � � � ð68Þ

Fig. 6. Large suction asymptotic solution (77) for wall shear stress F00(0) versus suction/injection parameter c (for typical no-slip condition k = 0) for variousvalues of stretching sheet parameter b = 2, 1 and 0, and comparison with minimum error solution (85a) and exact solution for b = 1.


The first and second order equations are

h0000 þ h000 ¼ 0; ð69Þh0002 þ h002 ¼ �h0h000 � bð1� h020 Þ; ð70Þh0004 þ h004 ¼ �h0h002 þ 2bh00h02 � h000h2: ð71Þ

The solution to first and second order problems are

h0 ¼ gþ ð1� kÞðe�g � 1Þ; ð72Þ

h2 ¼ �ð1� kÞ ðkþ 2bþ 1Þfðgþ 1Þe�g � 1g þ 12g2e�g � 1

4ð1� kÞð1� bÞðe�2g � 2e�g þ 1Þ

ð73Þ

The wall shear stresses h00ð0Þ and boundary layer thickness dh ¼R1

0 ð1� h0Þdg are given by relations

h00ð0Þ ¼ 1� kþ 1� kc2 2bþ 1� 1

2ð1� kÞð1þ bÞ

þ � � � ð74Þ

dh ¼ 1� k� 1� kc2 2bþ 2� 1

4ð1� kÞðbþ 3Þ

þ � � � ð75Þ

The solution in terms of original variables f becomes

FðfÞ ¼ cþ fþ1� kcðe�cf�1Þ� ð1� kÞ

c3 ½ðkþ2bþ1Þfðcfþ1Þe�cf�1gþ12c2f2e�cfþ1

4ð1� kÞð1�bÞðe�2cf�2e�cfþ1Þ�

ð76Þ
The wall shear stress and displacement thickness becomes
F 00ð0Þ ¼ ð1� kÞcþ 1� kc

2bþ 1� 12ð1� kÞð1þ bÞ

þ � � � ð77Þ

K ¼Z 1

0ðF 0ðfÞ � 1Þdf ¼ c� 1� k

cþ ð1� kÞ

c3 2bþ 2� 14ðk� 1Þðbþ 3Þ

þ � � � ð78Þ

The large suction asymptotic prediction (77) for friction factor F00(0) versus transpiration parameter c (for typical no slip con-dition k = 0) is shown in Fig. 6 for various values of stretching parameter b = 2, 1 and 0. A comparison with minimum errorsolution of Section 3, and exact numerical solution for b = 1 are also shown in same figure. The leading term in Eq. (77) due toWeidman et al. [18] provides accurate values of the shear stress parameter for all k for c > 8. The second order relation (77)decribes the results for lower values of c, as shown in Fig. 6.

3. Minimum error solution

An approximate solution is developed for an adopted velocity profile which contains some free constants but satisfies allthe boundary conditions. The Karman momentum integal coupled with Karman–Pohlhausen method has been commonlyused [15,16]. Afzal [33] proposed alternate approach of minimum error solutions obtained by minimizing the integral of


the square of the error by Euler–Lagrange differential equation (fundamental equation of calculus of variations) used for esti-mation of a good approximate solution of the partial differential equations of the boundary layers in several cases: flat plate,stagnation point flow, moving plate and non-Newtonian fluids. The minimum error predictions were found better when com-pared with traditional Karman–Pohlhausen method [29]. If u/ U1 = f(g), g = y/d is the approximating functions satisfying theboundary conditions, involves some unknown parameters, then the error of residual in the boundary layer Eq. (2) becomes

eðx; y; mÞ ¼ m@2u@y2 � U1

dU1dx� u

@u@x� v @u

@y: ð79Þ

If the integral of the error is taken as zero as
Z d
0eðx; y; mÞdy ¼ 0; ð80Þ

then we get traditional Karman momentum integral, from which unknown parameter in the trial velocity profile is esti-mated. It is shown in Afzal [33] that various local potential methods, used by Venkateswarlu and Deshpande [34], Prigogineand Glansdroff [35], Lebon and Lambermont [36] and Doty and Blick [37], correspond to
Z d
0eðx; y; mÞ/dy ¼ 0; / ¼ @u

@d; ð81Þ

which is equivalent to a moment of function / of the error in the boundary layer equation, from which unknown parameterin the trial velocity profile is estimated. It is also possible to choose another moment function but then the method becomesarbitrary. Hsu [38] employed the Galerkin method to study a class of two-dimensional boundary layer problems. It can beshown that the local potential or the Galerkin method for a given problem, will yield same result when the approximatingfunction are the members of a complete set. When the solutions are not based on complete set, the functional has to satisfycertain additional conditions in terms of unknown parameters to get results equivalent to Galerkin’s technique (MacDoanald[39]). It may be noted that in each these methods the unknown parameters are determined in different ways depending onthe method used.

As long as if one has to begin with a trial velocity profile satisfying the boundary conditions on might determine the un-known parameters by minimizing the square of the total error in least square sense. The total error is defined as integral of e2

over the whole range of space becomes

E ¼Z l

0

Z 1

0e2ðx; y; mÞdy dx: ð82Þ

The functional E may be minimized with respect to the free parameters by employing the Euler–Lagrange differentialequation, which results in certain equations whose solution leads to determination of unknown parameters in the trialvelocity profile.

If terms of self-similar variables the trial velocity profile u/ U1 = F(f) and f = n/a becomes a simple function, satisfying theboundary conditions, involves some unknown parameter a, then the error of residual in the boundary layer Eq. (2) becomes

eðn;aÞ ¼ F 000 þ FF 00 þ bð1� F 02Þ: ð83Þ

The least square method for the minimization error by Euler–Lagrange differential equation is

ð @@a� d

dx@

@axÞZ 1

0e2ðn;aÞdn ¼ 0: ð84Þ

where ax ¼ da=dx. The postulated velocity profile for general b and k is

F 0ðfÞ ¼ 1� ð1� kÞ expð�afÞ; ð85Þ

which satisfies the boundary conditions (7b,c) provided constant a is a positive number, independent of x (i.e., ax = da/dx = 0). Certain cases where da/dx – 0, were considered by Afzal [33]. Thus use of Euler–Lagrange Eq. (84) for present trialprofile (85) corresponds to minimization of total integral of least square error for estimation of unknown constant in the trialvelocity profile. An integral of (85) subjected to the boundary condition (7a) may be expressed as

FðfÞ ¼ cþ 1a½nþ ð1� kÞðexpð�nÞ � 1Þ�; n ¼ af: ð86Þ

In the present work least square error is minimized for estimation of unknown constant a. The residual function e(n,a), fromboundary layer Eq. (83) for the trial profile (86) is given by the relation

eðn;aÞ ¼ ð1� kÞ½�a2 þ caþ ð1� kÞð2b� 1Þ þ n� expð�nÞ þ ð1� kÞ2ð1� bÞ expð�2nÞ: ð87Þ

Substituting (87) into Eq. (84) for least square method for the minimization error, and solving for a we get

a2 � ca� 12� 1

3ð1� kÞð4b� 1Þ

� �a� 1

2c

� �¼ 0: ð88Þ


The second factor value a = c/2 is not appropriate and we get

Fig. 7.wall su

Fig. 8.the wa

a2 � ca ¼ 12þ 1

3ð1� kÞð4b� 1Þ: ð89Þ

This solution of quadratic equation with positive sign as adopted for a > 0 to get

a ¼ 12

c�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 2þ 4

3ð1� kÞð4b� 1Þ

r !: ð90Þ

The final form of the solution (86) becomes

FðfÞ ¼ cþ fþ 1a½ð1� kÞðexpð�afÞ � 1Þ� ð91Þ

and skin friction and displacement thickness become

F 00ð0Þ ¼ ð1� kÞa; K ¼Z 1

0ðF 0 � 1Þdf ¼ c� 1

að1� kÞ: ð92a;bÞ

The solutions (92a,b) and (90) for no-slip condition (k = 0) become

F 00ð0Þ ¼ a; K ¼ c� 1a; a ¼ 1

2c�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 2þ 4

3ð4b� 1Þ

r !: ð93a;b; cÞ

Minimum error solution (85a) for no-slip condition k = 0, for wall shear stress F00(0) versus the stretching sheet parameter b, for various values thection/injection parameter c = 1, 0 and �1.

Minimum error solution (85a) for impermeable surface c = 0, for wall shear stress F00(0) versus the stretching sheet parameter b for various values kll slip velocity parameter k = 1, 0.5, 0 and �0.5.

Fig. 9. Minimum error solution (85a) for no-slip k = 0 on surface, for wall shear stress F00(0) versus suction/injection c for various values of stretchingparameter b = 2, 1 and 0.


Further, for k = 0 the asymptotic solution of Afzal and Luthra [40] for large b given below

F 00ð0Þ ¼ b1=2ð1:1547bþ 0:0746b�1 þ 0:00509b�2 � 0:00182b�3 þ � � �Þ ð94Þ

and the convergence of series (94) was accelerated by Euler transform Y = (1 + b)�1 to yield

F 00ð0Þ ¼ Y�1=2ð1:1547þ 0:0746Y þ 0:00509Y2 � 0:00182Y3 þ � � �Þ: ð95Þ

The prediction F00(0) vs b from the analytical solution (92a) for k = 0 is shown in Fig. 7 for various values of transpirationparameter c = 1, 0 and �1. The asymptotic solution (95) of Afzal and Luthra [36] for k = 0 = c also shown in Fig. 7 compareswell the present minmum error solution (92a). The analytical solution (92a) F00(0) vs b for c = 0 is shown in Fig. 8 for variousvalues of slip velocity parameter k=1, 0.5, 0 and �0.5. The asymptotic solution (95) for k = 0 = c also shown in Fig. 8 compareswell the present minimum error solution (92a). The effects of transpiration parameter on prediction F00(0) vs c from analyt-ical solution (92a) for k = 0 are shown in Fig. 9 for various values stretching parameter b = 0, 1 and 2. The exact numericalsolution b = 0 also shown in same figure compare very well with the prediction.

4. Summary and conclusion

The problem of self-similar boundary layer flow over a moving/stretching sheet is analytically solved in certain cases toexhibit the combined effects of wall transpiration and plate movement compatible with self-similar flow governed by Falk-ner–Skan equation. The exact solution for b = �1 and closed form asymptotic solutions for each case of b large, large c and fork ? 1 have been presented. For b = �1 the solution for each value of the suction parameter c shows the existence of dualsolutions, that are clearly displayed in the [k,F00(0)]-parameter space. For b large the dual solutions are clearly displayedin the [k,F00(0)]-parameter space, but suction parameter c plays no role if the prediction of skin friction, but plays an additiverole to the stream function. The zero transpiration also, the dual solutions exist for each nonzero value of k. Suction increasesthe range of stable solutions and blowing decreases this range The analysis of solution behaviors at the singular and regularfocal points and of the asymptotic behavior of the wall shear stress parameter F00(0) at large c are given. The critical values ofc, b and k are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solutionby integral method for a trial velocity profile compares well exact solutions. The present work on suction and blowing overmoving continuous sheet under pressure gradient for laminar flow reported here is of great interest in turbulent motion ofthe fluid. This is because under eddy viscosity closure model (Afzal et al. [41]) the analogous equations describe the outerlayer of turbulent boundary layer over a moving continuous subjected to external pressure gradient fluid stream, whichis of interest in cooling of the objects during industrial manufacture process.

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Appl. Math. Stat. (2011) (Special issue on Similarity solutions in nonlinear PDE’s).

falkner–skan equation for flow past a stretching surface with suction or blowing: analytical...

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