- Email: [email protected]
Falkner–Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions
Applied Mathematics and Computation 217 (2010) 2724–2736
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Falkner–Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions Noor Afzal Faculty of Engineering, Aligarh Muslim University, Aligarh 202 002, India
a r t i c l e
i n f o
Keywords: Suction Blowing Dual solutions Stretching surface Similarity solutions Falkner–Skan equation
a b s t r a c t The simultaneous effects of suction and injection on tangential movement of a nonlinear power-law stretching surface governed by laminar boundary layer flow of a viscous and incompressible fluid beneath a non-uniform free with stream pressure gradient is considered. The self-similar flow is governed by Falkner–Skan equation, with transpiration parameter 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, large suction c, and k ? 1 have also been presented. Dual solutions are found for b = 1 for each value of the transpiration parameter, including the non-permeable surface, for each prescribed value of the wall slip velocity k. The large b asymptotic solution also dual with respect to wall slip velocity k, but do not depend on suction and blowing. The critical values of c, b and k are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile is presented and results are compared with the exact solutions. Ó 2010 Elsevier Inc. All rights reserved.
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 engineering processes. The examples are the thermal processing of sheet-like materials is a necessary operation in the production of paper, linoleum, polymeric sheets, wire drawing, drawing of plastic films, metal spinning, roofing shingles, insulating materials, fine-fiber matts, cooling of films or sheets, conveyor belts, metallic plates and cylinders. In virtually all such processing operations, the sheet moves parallel to its own plane. The moving sheet may induce motion in the neighboring fluid or, alternatively, the fluid may have an independent forced-convection motion that is parallel to that of the sheet. Both the kinematics of stretching and the simultaneous heating or cooling during such processes have a decisive influence on the quality of the 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 ambient fluid speed Ue. For Uw > Ue or Uw < Ue these two problems are physically different and cannot be mathematically transformed into 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 is the basic velocity (for Uw < Ue) the first set of boundary conditions was studied by Klemp and Acrivos [2] and Hussaini et al. [3]. Afzal [4,5] considered a reference velocity Ur as Ur = Uw + Ue, and proposed a single set of boundary layer equation along the boundary conditions, irrespective of whether Uw > Ue or Uw < Ue. Abraham and Sparrow [6] considered the reference E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.07.080
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 the drag formula for the case in which only one of the participating media is in motion. They found that the results of exact solutions demonstrate that this model is flawed and under predicts the drag force, and thus the use of the relative-velocity model can lead to gross errors in the drag force. The extent of the error increases as the two participating velocities approach each other in magnitude. The solution depends not only on the velocity difference jUw Uej but also on the velocity ratio Uw/Ue. In view of these applications, Sakiadis [7] initiated the study of boundary layer flow over a continuous solid surface moving with a constant speed in an otherwise quiescent fluid medium. Due to entrainment of ambient fluid, this boundary layer flow is quite different from that over a semi-infinite flat plate (Blasius [8] problem). An important class of similarity solutions corresponding to the boundary layer on nonlinear stretching impermeable wall was first presented by Afzal [9–11], and work [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 with similarity solutions arising during free convection in porous media. The effects of suction and injection on momentum and thermal boundary layers over a two-dimensional or axisymmetric nonlinear stretching surface in a stationary fluid has 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 (see Schlichting [16], Rosenhead [17]. The boundary layer problem of a semi-infinite flat plate moving in a free stream with mass transfer (suction or injection) has been recently discussed by Ahmad [18] and Weidman, Kubitschek and Davis [19] where as mass transfer on a stationary plate was studied by Watanabe [20] and Yih [21]. Yang and Chien [22] presented analytic solution 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. The numerical solutions of the Falkner–Skan equation with suction and injection were presented by Koh and Hartnett [24]. Riley and Weidman [25] have studied multiple solutions of the Falkner–Skan equation for flow past a stretching boundary when the 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 McLeod and Rajagopal [29]. The effects of non-Newtonian fluids past a porous plate with suction or injection has been studied by Mansutti 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 boundary layer flow driven by far field velocity has been considered here. The analytical solutions including mass injection as well as suction on the walls are considered for Falkner–Skan equation for flow past a stretching boundary when the external velocity and the boundary velocity are each proportional to the same power-law of the downstream distance. An approximate minimum error solution for a trial velocity profile is also presented by minimizing the error in the square of the integral of 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, in standard notation are
@u @ v þ ¼ 0; @x @y
ð1Þ 2
u
@u @u dU e @ u þv ¼ Ue þm 2: @x @y @y dx
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
U w ðxÞ ¼ U w xm ;
U e ðxÞ ¼ U e xm ;
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 U w ðxÞ;
v ¼ V w ðxÞ;
y=d ! 1;
u ¼ U e ðxÞ:
ð4Þ
The similarity transformation is given by 0
u ¼ U e ðxÞF ðgÞ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ mÞU e ðxÞ f¼y 2mx
ð5Þ
and the laminar boundary layer equations reduce to
F 000 þ FF 00 þ bð1 F 02 Þ ¼ 0; Fð0Þ ¼ c;
F 0 ð0Þ ¼ k;
F 0 ð1Þ ¼ 1:
ð6Þ ð7a; b; cÞ
2726
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
Fig. 1. Physical model and co-ordinate system in the boundary layer with pressure gradient over a stretching wedge surface with suction or blowing.
The flow parameters b and k are constants, and c is also a constant (positive for suction and negative for blowing) given by relation
b¼
2m ; mþ1
k¼
Uw Ue
c ¼ V w ðxÞ ðm þ 1Þ
mU e ðxÞ 2x
1=2 :
ð8Þ
The boundary layer displacement thickness d is given by relation
d ¼ K ¼
Z
1
1 F 0 df:
ð9Þ
0
Under the transformation
pffiffiffi FðfÞ ¼ gðzÞ k;
z f ¼ pffiffiffi ; k
ð10Þ
the Eqs. (6) and (7) may be expressed aa
g 000 þ gg 00 bg 02 ¼ gð0Þ ¼ g w ;
b k2
g 0 ð0Þ ¼ 1;
ð11Þ
; g 0 ð1Þ ¼
1 ; k
ð12Þ
pffiffiffi where g w ¼ c= k. For large wall velocity k ? 1 the Eqs. (11) and (12) yield
g 000 þ gg 00 bg 02 ¼ 0; gð0Þ ¼ g w ;
g 0 ð0Þ ¼ 1;
ð13Þ g 0 ð1Þ 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 Afzal and 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
1 1 1 F 0 þ F 2 ¼ f2 þ Kf þ k þ c2 2 2 2
ð15Þ
and F00 (0) = K kc. Under the following transformation
FðfÞ ¼ f þ K þ 2
K 0 ðfÞ ; KðfÞ
ð16Þ
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2727
the Eq. (15) yields a Riccati type equation as K00 + (f + K)K0 = 0 whose solution becomes
K¼A
p1=2 2
erf
fþK pffiffiffi þ B: 2
ð17Þ
The relations (16) and (17) provided the solution
h i 2 exp 12 ðf þ KÞ2 FðfÞ ¼ f þ K pffiffiffi ; perf fþ pffiffiK þ B 2 A 2
ð18Þ
The boundary condition (7a) yields
B 2 K2 ¼ exp A Kc 2
!
p1=2 2
K erf pffiffiffi : 2
ð19Þ
The relations (18) and (10) yield the solution
FðfÞ ¼ f þ K
ðK cÞ expðKf 12 f2 Þ 2 h i : pffiffiffi pffiffiK erf pKffiffi 1 ðK cÞ p8 exp K2 erf fþ 2 2
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
1=2 : F 00 ð0Þ ¼ kc þ K ¼ kc 2ðk 1Þ þ c2
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 is shown 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 are shown in Fig. 3(a) and (b). The solution of Eqs. (13) and (14) for b = 1 yield
gðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 þ g 2w þ hc ; 2 þ g 2w tanh 2
g 00 ð0Þ ¼ g w ;
hc ¼ tanh
1
! gw pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; 2 þ g 2w
gð1Þ ¼ ð2 þ g 2w Þ1=2 :
ð23Þ ð24Þ
2.2. Large b asymptote The asymptotic solution for large b is analysed in terms of following variables
FðfÞ ¼ b1=2 HðnÞ;
f ¼ b1=2 n;
ð25Þ
and the boundary layer Eqs. (4) and (5) become
H000 H02 þ 1 ¼ b1 HH00 :
ð26Þ
Fig. 2. 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, for various values of the wall suction velocity parameter c = 0.5, 0 and 1.
2728
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
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) + b1H2(n) + . The leading order equation and boundary conditions 02 H000 1 H 1 þ 1 ¼ 0;
H1 ð0Þ ¼ Hw ;
ð29Þ
H01 ð0Þ
¼ k;
H01 ð1Þ
¼ 1:
ð30Þ
An integral of relation (29) subject to the boundary conditions (30) yields
2 03 4 H002 1 H 1 þ 2H 1 ¼ 3 3
ð31Þ
and shear stress on the surface become
H001 ð0Þ ¼ ðk 1Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðk þ 2Þ: 3
ð32Þ
A further integral of (29) yields
n pffiffiffi þ / ; 2
pffiffiffi n H1 ðnÞ ¼ Hw þ n þ 3 2 tanh / tanh pffiffiffi þ / ; 2 2
H01 ðnÞ ¼ 2 þ 3tanh
ð33Þ ð34Þ
where
/ ¼ tan1
rffiffiffiffiffiffiffiffiffiffiffiffi kþ2 : 3
ð35Þ
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2729
(i) Case 1: In terms of original variables the solution becomes
rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffi b kþ2 ; þ tan1 2 3 sffiffiffi"rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi!# rffiffiffi 2 kþ2 b kþ2 1 FðfÞ ¼ f þ c þ 3 tanh f þ tan : b 3 2 3 2
F 0 ðfÞ ¼ 2 þ 3tanh
f
ð36Þ
ð37Þ
The velocity gradient at the wall is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b ðk þ 2Þ; F ð0Þ ¼ ðk 1Þ 3 sffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi ! 2 kþ2 K¼cþ3 1 : b 3 00
ð38Þ ð39Þ
(ii) Case 2: In terms of original variables the solution becomes
rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffi b kþ2 1 tan ; F ðfÞ ¼ 2 þ 3tanh f 2 3 sffiffiffi" rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi!# rffiffiffi 2 kþ2 b kþ2 tanh f : FðfÞ ¼ f þ c þ 3 tan1 b 3 2 3 2
0
ð40Þ
ð41Þ
The velocity gradient at the wall is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b ðk þ 2Þ; F 00 ð0Þ ¼ ðk 1Þ 3 sffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffi 2 kþ2 1 : K¼cþ3 b 3
ð42Þ ð43Þ
pffiffiffi The square root sign indicates that the solution for large b exists for k P 2, that to is dual. The shear stress f 00 ð0Þ= b vs k based on relations (40) and (42) have been shown in Fig. 4. The solutions are dual for 2 < k < 1 describing the aiding and opposing flows regimes. The dual solutions coincide for k = 2 and k = 1. The solution for k > 1 on lower branch, shown by solid line, is regarded as unique as the solution of corresponding upper branch, shown by dotted line, is physically unrealistic. The solution are shown in Fig. 5. The solution of Eqs. (13) and (14) large slip velocity (b ? 1) becomes
gðzÞ ¼ g w þ 00
zb1 ; z þ b1
g ð0Þ ¼ ð2b=3Þ
1=2
;
pffiffiffiffiffiffiffiffi b1 ¼ 6=b;
ð44Þ
pffiffiffiffiffiffiffiffi gð1Þ ¼ g w 6=b:
ð45Þ
pffiffiffi Fig. 4. Asymptotic solution for b ? 1: the wall shear stress F 00 ð0Þ= b versus non-dimensional slip velocity of sheet parameter k, does not depend on suction/injection parameter c.
2730
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
pffiffiffi Fig. 5. Velocity distribution F0 (f) vs f b 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 around this exact solution by expanding the stream function in the power of parameter k 1. Introducing the following change of variables
FðfÞ ¼ f þ c þ ðk 1ÞGðfÞ;
ð46Þ
the boundary layer Eqs. (6) and (7) yield
G000 þ ðf þ cÞG00 bG0 ¼ ðk 1ÞðGG00 þ bG02 Þ; Gð0Þ ¼ 0;
0
ð47Þ
0
G ð0Þ ¼ 1 G ð1Þ ¼ 0:
ð48Þ
The asymptotic expansion for G is
GðfÞ ¼ G1 ðfÞ þ ðk 1ÞG2 ðfÞ þ ðk 1Þ2 G3 ðfÞ þ
ð49Þ
The first order equations are 00 0 G000 1 þ ðf þ cÞG1 bG1 ¼ 0;
G1 ð0Þ ¼ 0;
G01 ð0Þ ¼ 1;
ð50Þ G01 ð1Þ ¼ 0:
ð51Þ
The solution to first order Eqs. (50) and (51) may be expressed in closed form
1 1 1 2b 3 1 ; ; ðf þ cÞ2 ; G01 ðfÞ ¼ AU b; ; ðf þ cÞ2 þ Bðf þ cÞU 2 2 2 2 2
ð52Þ
where U(a, b, x) is the confluent hypergeometric function. It is well-known that U(a b, 0) = 1 and as x ? 1 we have (Abromwitch and Stugen [32])
Uða; b; xÞ ¼
CðbÞ a x þ Cðb aÞ
ð53Þ
where C(a) is the gamma function. The constants A and B estimated from the boundary conditions (51) yield
1 1 1 1 2b 3 1 A ¼ U b; ; c2 W cU ; ; ; c2 2 2 2 2 2
B ¼ AW;
ð54Þ
where
W¼
pffiffiffi Cð1 þ bÞ : 2 Cð1þ2b Þ 2
ð55Þ
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2731
The solution (52) based on relations (54) and (55) yields the velocity distribution
G01 ðfÞ
¼
U b; 12 ; 12 ðf þ cÞ2 Wðf þ cÞU
12b 3 ; 2 ; 12 ðf 2
12b
U b; 12 ; 12 c2 W cU
2
; 32 ; 12 c2
þ cÞ2
ð56Þ
:
The solution in terms of F(f) yields
F 0 ðfÞ ¼ 1 þ ðk 1Þ
U b; 12 ; 12 ðf þ cÞ2 Wðf þ cÞU
12b 3 ; 2 ; 12 ðf 2
12b
U b; 12 ; 12 c2 W cU
2
; 32 ; 12 c2
þ cÞ2
ð57Þ
and friction parameter becomes
1 1 1 1 2b 3 1 ; ; c2 F 00 ð0Þ ¼ ðk 1ÞW U b; ; c2 W cU : 2 2 2 2 2
ð58Þ
(i) Special case c = 0: The solution (57) for no transpiration from the surface (c = 0) becomes
1 1 1 2b 3 1 ; ; f2 : F 0 ðfÞ ¼ 1 þ ðk 1Þ U b; ; f2 WfU 2 2 2 2 2
ð59Þ
The velocity gradient at the wall yields
F 00 ð0Þ 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 0 ðfÞ ¼ 1 þ ðk 1Þ
1
qffiffiffi 2 2 1 3 1 pðf þ cÞU 2 ; 2 ; 2 ðf þ cÞ qffiffiffi : 1 p2 cU 12 ; 32 ; 12 c2
ð61Þ
Using the relation (Abromwitch and Stugen [28])
pffiffiffiffi 1 3 p erfðzÞ ; ; z2 ¼ 2z 2 2
U
ð62Þ
and the solution Eq. (57) is simplified as
F 0 ðfÞ ¼ 1 þ ðk 1Þ
erfc
fþc pffiffi 2
erfc
; pcffiffi
ð63Þ
2
where erfc(x) = 1 erf(x) is the complimentary error function. The velocity gradient at the wall yields
2 rffiffiffiffi c 2 exp 2 : F ð0Þ ¼ ðk 1Þ p erfc pcffiffi 2 00
ð64Þ
2.4. Large suction For large suction, introducing the variables
FðfÞ ¼ c þ
hðgÞ
c
;
f¼
g ; c
ð65Þ
into (4) and (6) the boundary layer equations yield 000
00
00
02
h þ h ¼ c2 ½hh þ bð1 h Þ; hð0Þ ¼ 0;
0
h ð0Þ ¼ k;
0
h ð1Þ ¼ 1:
ð66Þ ð67Þ
The asymptotic expansion for h is
hðgÞ ¼ h0 ðgÞ þ c2 h2 ðgÞ þ c4 h4 ðgÞ þ
ð68Þ
2732
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
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 various values 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 000
00
000 h2 000 h4
00 h2 00 h4
h0 þ h0 ¼ 0; þ þ
¼ ¼
ð69Þ
00 h0 h0 00 h0 h2
þ
02 bð1 h0 Þ; 0 0 00 2bh0 h2 h0 h2 :
ð70Þ ð71Þ
The solution to first and second order problems are
h0 ¼ g þ ð1 kÞðeg 1Þ;
1 1 h2 ¼ ð1 kÞ ðk þ 2b þ 1Þfðg þ 1Þeg 1g þ g2 eg ð1 kÞð1 bÞðe2g 2eg þ 1Þ 2 4 00
The wall shear stresses h ð0Þ and boundary layer thickness dh ¼
1 2b þ 1 ð1 kÞð1 þ bÞ þ 2 c
1k 1 dh ¼ 1 k 2 2b þ 2 ð1 kÞðb þ 3Þ þ 4 c 00
h ð0Þ ¼ 1 k þ
1k
R1 0
ð72Þ ð73Þ
0
ð1 h Þdg are given by relations
2
ð74Þ ð75Þ
The solution in terms of original variables f becomes
FðfÞ ¼ c þ f þ
1k
c
ðecf 1Þ
ð1 kÞ
c3
1 1 ½ðk þ 2b þ 1Þfðcf þ 1Þecf 1g þ c2 f2 ecf þ ð1 kÞð1 bÞðe2cf 2ecf þ 1Þ 2 4 ð76Þ
The wall shear stress and displacement thickness becomes
1k 1 F 00 ð0Þ ¼ ð1 kÞc þ 2b þ 1 ð1 kÞð1 þ bÞ þ 2 c
Z 1 1 k ð1 kÞ 1 0 ðk 1Þðb þ 3Þ þ K¼ ðF ðfÞ 1Þdf ¼ c þ 2b þ 2 4 c c3 0
ð77Þ ð78Þ
The large suction asymptotic prediction (77) for friction factor F00 (0) versus transpiration parameter c (for typical no slip condition k = 0) is shown in Fig. 6 for various values of stretching parameter b = 2, 1 and 0. A comparison with minimum error solution of Section 3, and exact numerical solution for b = 1 are also shown in same figure. The leading term in Eq. (77) due to Weidman 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 all the boundary conditions. The Karman momentum integal coupled with Karman–Pohlhausen method has been commonly used [15,16]. Afzal [33] proposed alternate approach of minimum error solutions obtained by minimizing the integral of
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2733
the square of the error by Euler–Lagrange differential equation (fundamental equation of calculus of variations) used for estimation 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 compared with traditional Karman–Pohlhausen method [29]. If u/ U1 = f(g), g = y/d is the approximating functions satisfying the boundary conditions, involves some unknown parameters, then the error of residual in the boundary layer Eq. (2) becomes
eðx; y; mÞ ¼ m
@2u dU 1 @u @u v : U1 u @y2 @x @y dx
ð79Þ
If the integral of the error is taken as zero as
Z
d
eðx; y; mÞdy ¼ 0;
ð80Þ
0
then we get traditional Karman momentum integral, from which unknown parameter in the trial velocity profile is estimated. It is shown in Afzal [33] that various local potential methods, used by Venkateswarlu and Deshpande [34], Prigogine and Glansdroff [35], Lebon and Lambermont [36] and Doty and Blick [37], correspond to
Z
d
eðx; y; mÞ/dy ¼ 0;
/¼
0
@u ; @d
ð81Þ
which is equivalent to a moment of function / of the error in the boundary layer equation, from which unknown parameter in the trial velocity profile is estimated. It is also possible to choose another moment function but then the method becomes arbitrary. Hsu [38] employed the Galerkin method to study a class of two-dimensional boundary layer problems. It can be shown that the local potential or the Galerkin method for a given problem, will yield same result when the approximating function are the members of a complete set. When the solutions are not based on complete set, the functional has to satisfy certain 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 on the method used. As long as if one has to begin with a trial velocity profile satisfying the boundary conditions on might determine the unknown 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
E¼
Z
l 0
Z
1
e2 ðx; y; mÞdy dx:
ð82Þ
0
The functional E may be minimized with respect to the free parameters by employing the Euler–Lagrange differential equation, which results in certain equations whose solution leads to determination of unknown parameters in the trial velocity profile. If terms of self-similar variables the trial velocity profile u/ U1 = F(f) and f = n/a becomes a simple function, satisfying the boundary 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 02 Þ:
ð83Þ
The least square method for the minimization error by Euler–Lagrange differential equation is
@ d @ ð Þ @ a dx @ ax
Z
1
e2 ðn; aÞdn ¼ 0:
ð84Þ
0
where ax ¼ da=dx. The postulated velocity profile for general b and k is
F 0 ðfÞ ¼ 1 ð1 kÞ expðafÞ;
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 trial profile (85) corresponds to minimization of total integral of least square error for estimation of unknown constant in the trial velocity profile. An integral of (85) subjected to the boundary condition (7a) may be expressed as
1 FðfÞ ¼ c þ ½n þ ð1 kÞðexpðnÞ 1Þ;
n ¼ af:
a
ð86Þ
In the present work least square error is minimized for estimation of unknown constant a. The residual function e(n, a), from boundary layer Eq. (83) for the trial profile (86) is given by the relation
eðn; aÞ ¼ ð1 kÞ½a2 þ ca þ ð1 kÞð2b 1Þ þ n expðnÞ þ ð1 kÞ2 ð1 bÞ expð2nÞ:
ð87Þ
Substituting (87) into Eq. (84) for least square method for the minimization error, and solving for a we get
1 2
1 3
a2 ca ð1 kÞð4b 1Þ
1 2
a c ¼ 0:
ð88Þ
2734
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
The second factor value a = c/2 is not appropriate and we get
1 2
1 3
a2 ca ¼ þ ð1 kÞð4b 1Þ:
ð89Þ
This solution of quadratic equation with positive sign as adopted for a > 0 to get
a¼
1 c 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 c2 þ 2 þ ð1 kÞð4b 1Þ : 3
ð90Þ
The final form of the solution (86) becomes
1 FðfÞ ¼ c þ f þ ½ð1 kÞðexpðafÞ 1Þ
ð91Þ
a
and skin friction and displacement thickness become
F 00 ð0Þ ¼ ð1 kÞa;
K¼
Z
1
1 ðF 0 1Þdf ¼ c ð1 kÞ:
0
a
ð92a; bÞ
The solutions (92a,b) and (90) for no-slip condition (k = 0) become 00
F ð0Þ ¼ a;
1
K¼c ;
a
1 a¼ c 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 c2 þ 2 þ ð4b 1Þ : 3
ð93a; b; cÞ
Fig. 7. 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 the wall suction/injection parameter c = 1, 0 and 1.
Fig. 8. Minimum error solution (85a) for impermeable surface c = 0, for wall shear stress F00 (0) versus the stretching sheet parameter b for various values k the wall slip velocity parameter k = 1, 0.5, 0 and 0.5.
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2735
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 stretching parameter b = 2, 1 and 0.
Further, for k = 0 the asymptotic solution of Afzal and Luthra [40] for large b given below
F 00 ð0Þ ¼ b1=2 ð1:1547b þ 0:0746b1 þ 0:00509b2 0:00182b3 þ Þ
ð94Þ 1
and the convergence of series (94) was accelerated by Euler transform Y = (1 + b)
F 00 ð0Þ ¼ Y 1=2 ð1:1547 þ 0:0746Y þ 0:00509Y 2 0:00182Y 3 þ Þ:
to yield
ð95Þ
00
The prediction F (0) vs b from the analytical solution (92a) for k = 0 is shown in Fig. 7 for various values of transpiration parameter c = 1, 0 and 1. The asymptotic solution (95) of Afzal and Luthra [36] for k = 0 = c also shown in Fig. 7 compares well the present minmum error solution (92a). The analytical solution (92a) F00 (0) vs b for c = 0 is shown in Fig. 8 for various values 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 compares well the present minimum error solution (92a). The effects of transpiration parameter on prediction F00 (0) vs c from analytical solution (92a) for k = 0 are shown in Fig. 9 for various values stretching parameter b = 0, 1 and 2. The exact numerical solution 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 to exhibit the combined effects of wall transpiration and plate movement compatible with self-similar flow governed by Falkner–Skan equation. The exact solution for b = 1 and closed form asymptotic solutions for each case of b large, large c and for k ? 1 have been presented. For b = 1 the solution for each value of the suction parameter c shows the existence of dual solutions, that are clearly displayed in the [k, F00 (0)]-parameter space. For b large the dual solutions are clearly displayed in the [k, F00 (0)]-parameter space, but suction parameter c plays no role if the prediction of skin friction, but plays an additive role to the stream function. The zero transpiration also, the dual solutions exist for each nonzero value of k. Suction increases the range of stable solutions and blowing decreases this range The analysis of solution behaviors at the singular and regular focal points and of the asymptotic behavior of the wall shear stress parameter F00 (0) at large c are given. The critical values of c, b and k are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile compares well exact solutions. The present work on suction and blowing over moving continuous sheet under pressure gradient for laminar flow reported here is of great interest in turbulent motion of the fluid. This is because under eddy viscosity closure model (Afzal et al. [41]) the analogous equations describe the outer layer of turbulent boundary layer over a moving continuous subjected to external pressure gradient fluid stream, which is of interest in cooling of the objects during industrial manufacture process. References [1] T.A. Abdelhafez, Skin friction and heat transfer on a continuous flat surface moving in a parallel free stream, Int. J. Heat Mass Trans. 28 (1985) 1234– 1237. [2] J.B. Klemp, A.A. Acrivos, A moving-wall boundary layer with reverse flow, J. Fluid Mech. 76 (1976) 363–381. [3] M.Y. Hussaini, W.D. Lakin, N. Nachman, On similarity solutions of a boundary layer problem with an upstream moving wall, SIAM J. Appl. Math. 47 (1987) 699–709. [4] N. Afzal, Momentum transfer on power law stretching plate with free stream pressure gradient, Int. J. Eng. Sci. 41 (2003) 1197–1207. [5] N. Afzal, A. Badaruddin, A.A. Elgarvi, Momentum and transport on a continuous flat surface moving in a parallel stream, Int. J. Heat Mass Trans. 36 (1993) 3399–3403.
2736
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
[6] J.P. Abraham, E.M. Sparrow, Friction drag resulting from the simultaneous imposed motions of a freestream and its bounding surface, Int. J. Heat Fluid Flow 26 (2005) 289–295. [7] B.C. Sakiadis, Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow, AIChE J. 7 (1961) 26–28. [8] H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung, Z. Math. Phys. 56 (1908) 137. [9] N. Afzal, I.S. Varshney, The cooling of a low heat resistance stretching sheet moving through a fluid, Heat Mass Trans. 14 (1980) 289–293. [10] N. Afzal, The cooling of a low heat resistance stretching sheet moving through a fluid: a rejoinder, Heat Mass Trans. 17 (1983) 217–219. [11] N. Afzal, Heat transfer from a stretching surface, Int. J. Heat Mass Trans. 36 (1993) 1128–1131. [12] A. Aziz, T.Y. Na, Perturbation Methods in Heat Transfer, Hemisphere, New York, 1984. p. 182. [13] B. Brighi, J.D. Hoernel, Recent advances on similarity solutions arising during free convection, Progress in Nonlinear Differential Equations Their Applications, vol. 63, Birkhauser Verlag, Basel/Switzerland, 2005. pp. 83–92. [14] M. Guedda, Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous media, C.R. Mecanique 330 (2002) 279– 283. [15] N. Afzal, Momentum and thermal boundary layers over a two-dimensional or axisymmetric non-linear stretching surface in a stationary fluid, Int. J. Heat Mass Trans. 53 (2010) 540–547. [16] H. Schlichting, Boundary Layer Theory, Mc-Graw Hill Book Co., New York, 1968. [17] L. Rosenhead, Laminar Boundary Layers, Oxford University Press, Oxford, 1963. p. 232. [18] F. Ahmad, Degeneracy in the Blasius problem, Electron. J. Differ. Equ. 79 (2007) 18. [19] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on self-similar boundary layer flow over moving surfaces, Int. J. Eng. Sci. 44 (2006) 730–737. [20] T. Watanabe, Thermal boundary layers over a wedge with uniform suction or injection in forced flow, Acta Mech. 83 (1990) 119–126. [21] K.A. Yih, Uniform suction/blowing effect on forced convection about a wedge: uniform heat flux, Acta Mech. 128 (1998) 173–181. [22] H.T. Yang, L.C. Chien, Analytic solution of the Falkner–Skan equation when b = 1 and c = 0, SIAM J. Appl. Math. 29 (1975) 558–569. [23] J.C.Y. Koh, J.P. Hartnett, Skin-friction and heat transfer for incompressible laminar flow over porous wedges with suction and variable wall temperature, Int. J. Heat Mass Trans. 2 (1961) 185–198. [24] R.B. Kudenatti, V.B. Awati, Solution of pressure gradient stretching plate with suction, Appl. Math. Comput. 210 (2009) 151–157. [25] N. Riley, P.D. Weidman, Multiple solutions of the Falkner–Skan equation for flow past a stretching boundary, SIAM J. Appl. Math. 49 (1989) 1350. [26] V.M. Falkner, S.W. Skan, Some approximate solutions of the boundary-layer equations, Phylos. Mag. 12 (1931) 865–896. [27] S.P. Hastings, Reversed flow solutions of the Falkner–Skan equation, SIAM J. Appl. Math. 22 (1972) 329–334. [28] S.N. Brown, K. Stewartson, On the reversed flow solutions of the Falkner–Skan equation, Mathematika 13 (1966) 1–6. [29] J.B. McLeod, K.R. Rajagopal, On the uniqueness of flow of a Navier–Stokes fluid due to a stretching boundary, Arch. Ration. Mech. Anal. 98 (1987) 385– 393. [30] D. Mansutti, G. Pontrelli, K.R. Rajagopal, Steady flows of non-Newtonian fluids past a porous plate with suction or injection, Int. J. Numer. Meth. Fluids 17 (1993) 927–941. [31] M. Massoudi, M. Ramezan, Effect of injection or suction on the Falkner–Skan flows of second grade fluids, Int. J. Non Linear Mech. 24 (1989) 221–227. [32] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover publications, Inc., New York, 1970. [33] N. Afzal, Minimum error solutions of boundary layer equations, Indian Academy of Sciences, (Math. Sci.: Series A) 91 (1982) 183–193. [34] P. Venkateswarlu, S.M. Deshpande, A critical study of classical and non-classical variational principles in fluid mechanics, J. Non-Equilib. Thermodyn. 7 (1982) 105–128. [35] I. Prigogine, P. Glansdroff, Variational properties and fluctuation theory, Physica 31 (1965) 1243–1256. [36] G. Lebon, J. Lambermont, Generalization of the principle of continuous dissipative systems, J. Chem. Phys. 69 (1973) 2929–2936. [37] R.T. Doty, E.P. Blick, Local potential variational method applied to Himenz flow, AIAA J. 11 (1973) 880. [38] C.C. Hsu, A Galerkin method for a class of steady, two-dimensional, incompressible, laminar boundary-layer flows, J. Fluid Mech. 69 (1975) 783–802. [39] D.A. MacDonald, On the method of the local potential as applied to the solution of the equations of diffusion, Int. J. Heat Mass Trans. 17 (1974) 393– 400. [40] N. Afzal, V.K. Luthra, Highly accelerated laminar flow at moderately large Reynolds number, AIAA J. 12 (1974) 529–532. [41] N. Afzal, A. Seena, A. Bushra, Thermo-fluid-dynamics of turbulent boundary layer over a moving continuous flat sheet in a parallel free stream, Int. J. Appl. Math. Stat. (2011) (Special issue on Similarity solutions in nonlinear PDE’s).
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Falkner–Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions Noor Afzal Faculty of Engineering, Aligarh Muslim University, Aligarh 202 002, India
a r t i c l e
i n f o
Keywords: Suction Blowing Dual solutions Stretching surface Similarity solutions Falkner–Skan equation
a b s t r a c t The simultaneous effects of suction and injection on tangential movement of a nonlinear power-law stretching surface governed by laminar boundary layer flow of a viscous and incompressible fluid beneath a non-uniform free with stream pressure gradient is considered. The self-similar flow is governed by Falkner–Skan equation, with transpiration parameter 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, large suction c, and k ? 1 have also been presented. Dual solutions are found for b = 1 for each value of the transpiration parameter, including the non-permeable surface, for each prescribed value of the wall slip velocity k. The large b asymptotic solution also dual with respect to wall slip velocity k, but do not depend on suction and blowing. The critical values of c, b and k are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile is presented and results are compared with the exact solutions. Ó 2010 Elsevier Inc. All rights reserved.
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 engineering processes. The examples are the thermal processing of sheet-like materials is a necessary operation in the production of paper, linoleum, polymeric sheets, wire drawing, drawing of plastic films, metal spinning, roofing shingles, insulating materials, fine-fiber matts, cooling of films or sheets, conveyor belts, metallic plates and cylinders. In virtually all such processing operations, the sheet moves parallel to its own plane. The moving sheet may induce motion in the neighboring fluid or, alternatively, the fluid may have an independent forced-convection motion that is parallel to that of the sheet. Both the kinematics of stretching and the simultaneous heating or cooling during such processes have a decisive influence on the quality of the 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 ambient fluid speed Ue. For Uw > Ue or Uw < Ue these two problems are physically different and cannot be mathematically transformed into 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 is the basic velocity (for Uw < Ue) the first set of boundary conditions was studied by Klemp and Acrivos [2] and Hussaini et al. [3]. Afzal [4,5] considered a reference velocity Ur as Ur = Uw + Ue, and proposed a single set of boundary layer equation along the boundary conditions, irrespective of whether Uw > Ue or Uw < Ue. Abraham and Sparrow [6] considered the reference E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.07.080
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 the drag formula for the case in which only one of the participating media is in motion. They found that the results of exact solutions demonstrate that this model is flawed and under predicts the drag force, and thus the use of the relative-velocity model can lead to gross errors in the drag force. The extent of the error increases as the two participating velocities approach each other in magnitude. The solution depends not only on the velocity difference jUw Uej but also on the velocity ratio Uw/Ue. In view of these applications, Sakiadis [7] initiated the study of boundary layer flow over a continuous solid surface moving with a constant speed in an otherwise quiescent fluid medium. Due to entrainment of ambient fluid, this boundary layer flow is quite different from that over a semi-infinite flat plate (Blasius [8] problem). An important class of similarity solutions corresponding to the boundary layer on nonlinear stretching impermeable wall was first presented by Afzal [9–11], and work [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 with similarity solutions arising during free convection in porous media. The effects of suction and injection on momentum and thermal boundary layers over a two-dimensional or axisymmetric nonlinear stretching surface in a stationary fluid has 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 (see Schlichting [16], Rosenhead [17]. The boundary layer problem of a semi-infinite flat plate moving in a free stream with mass transfer (suction or injection) has been recently discussed by Ahmad [18] and Weidman, Kubitschek and Davis [19] where as mass transfer on a stationary plate was studied by Watanabe [20] and Yih [21]. Yang and Chien [22] presented analytic solution 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. The numerical solutions of the Falkner–Skan equation with suction and injection were presented by Koh and Hartnett [24]. Riley and Weidman [25] have studied multiple solutions of the Falkner–Skan equation for flow past a stretching boundary when the 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 McLeod and Rajagopal [29]. The effects of non-Newtonian fluids past a porous plate with suction or injection has been studied by Mansutti 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 boundary layer flow driven by far field velocity has been considered here. The analytical solutions including mass injection as well as suction on the walls are considered for Falkner–Skan equation for flow past a stretching boundary when the external velocity and the boundary velocity are each proportional to the same power-law of the downstream distance. An approximate minimum error solution for a trial velocity profile is also presented by minimizing the error in the square of the integral of 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, in standard notation are
@u @ v þ ¼ 0; @x @y
ð1Þ 2
u
@u @u dU e @ u þv ¼ Ue þm 2: @x @y @y dx
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
U w ðxÞ ¼ U w xm ;
U e ðxÞ ¼ U e xm ;
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 U w ðxÞ;
v ¼ V w ðxÞ;
y=d ! 1;
u ¼ U e ðxÞ:
ð4Þ
The similarity transformation is given by 0
u ¼ U e ðxÞF ðgÞ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ mÞU e ðxÞ f¼y 2mx
ð5Þ
and the laminar boundary layer equations reduce to
F 000 þ FF 00 þ bð1 F 02 Þ ¼ 0; Fð0Þ ¼ c;
F 0 ð0Þ ¼ k;
F 0 ð1Þ ¼ 1:
ð6Þ ð7a; b; cÞ
2726
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
Fig. 1. Physical model and co-ordinate system in the boundary layer with pressure gradient over a stretching wedge surface with suction or blowing.
The flow parameters b and k are constants, and c is also a constant (positive for suction and negative for blowing) given by relation
b¼
2m ; mþ1
k¼
Uw Ue
c ¼ V w ðxÞ ðm þ 1Þ
mU e ðxÞ 2x
1=2 :
ð8Þ
The boundary layer displacement thickness d is given by relation
d ¼ K ¼
Z
1
1 F 0 df:
ð9Þ
0
Under the transformation
pffiffiffi FðfÞ ¼ gðzÞ k;
z f ¼ pffiffiffi ; k
ð10Þ
the Eqs. (6) and (7) may be expressed aa
g 000 þ gg 00 bg 02 ¼ gð0Þ ¼ g w ;
b k2
g 0 ð0Þ ¼ 1;
ð11Þ
; g 0 ð1Þ ¼
1 ; k
ð12Þ
pffiffiffi where g w ¼ c= k. For large wall velocity k ? 1 the Eqs. (11) and (12) yield
g 000 þ gg 00 bg 02 ¼ 0; gð0Þ ¼ g w ;
g 0 ð0Þ ¼ 1;
ð13Þ g 0 ð1Þ 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 Afzal and 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
1 1 1 F 0 þ F 2 ¼ f2 þ Kf þ k þ c2 2 2 2
ð15Þ
and F00 (0) = K kc. Under the following transformation
FðfÞ ¼ f þ K þ 2
K 0 ðfÞ ; KðfÞ
ð16Þ
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2727
the Eq. (15) yields a Riccati type equation as K00 + (f + K)K0 = 0 whose solution becomes
K¼A
p1=2 2
erf
fþK pffiffiffi þ B: 2
ð17Þ
The relations (16) and (17) provided the solution
h i 2 exp 12 ðf þ KÞ2 FðfÞ ¼ f þ K pffiffiffi ; perf fþ pffiffiK þ B 2 A 2
ð18Þ
The boundary condition (7a) yields
B 2 K2 ¼ exp A Kc 2
!
p1=2 2
K erf pffiffiffi : 2
ð19Þ
The relations (18) and (10) yield the solution
FðfÞ ¼ f þ K
ðK cÞ expðKf 12 f2 Þ 2 h i : pffiffiffi pffiffiK erf pKffiffi 1 ðK cÞ p8 exp K2 erf fþ 2 2
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
1=2 : F 00 ð0Þ ¼ kc þ K ¼ kc 2ðk 1Þ þ c2
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 is shown 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 are shown in Fig. 3(a) and (b). The solution of Eqs. (13) and (14) for b = 1 yield
gðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 þ g 2w þ hc ; 2 þ g 2w tanh 2
g 00 ð0Þ ¼ g w ;
hc ¼ tanh
1
! gw pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; 2 þ g 2w
gð1Þ ¼ ð2 þ g 2w Þ1=2 :
ð23Þ ð24Þ
2.2. Large b asymptote The asymptotic solution for large b is analysed in terms of following variables
FðfÞ ¼ b1=2 HðnÞ;
f ¼ b1=2 n;
ð25Þ
and the boundary layer Eqs. (4) and (5) become
H000 H02 þ 1 ¼ b1 HH00 :
ð26Þ
Fig. 2. 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, for various values of the wall suction velocity parameter c = 0.5, 0 and 1.
2728
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
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) + b1H2(n) + . The leading order equation and boundary conditions 02 H000 1 H 1 þ 1 ¼ 0;
H1 ð0Þ ¼ Hw ;
ð29Þ
H01 ð0Þ
¼ k;
H01 ð1Þ
¼ 1:
ð30Þ
An integral of relation (29) subject to the boundary conditions (30) yields
2 03 4 H002 1 H 1 þ 2H 1 ¼ 3 3
ð31Þ
and shear stress on the surface become
H001 ð0Þ ¼ ðk 1Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðk þ 2Þ: 3
ð32Þ
A further integral of (29) yields
n pffiffiffi þ / ; 2
pffiffiffi n H1 ðnÞ ¼ Hw þ n þ 3 2 tanh / tanh pffiffiffi þ / ; 2 2
H01 ðnÞ ¼ 2 þ 3tanh
ð33Þ ð34Þ
where
/ ¼ tan1
rffiffiffiffiffiffiffiffiffiffiffiffi kþ2 : 3
ð35Þ
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2729
(i) Case 1: In terms of original variables the solution becomes
rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffi b kþ2 ; þ tan1 2 3 sffiffiffi"rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi!# rffiffiffi 2 kþ2 b kþ2 1 FðfÞ ¼ f þ c þ 3 tanh f þ tan : b 3 2 3 2
F 0 ðfÞ ¼ 2 þ 3tanh
f
ð36Þ
ð37Þ
The velocity gradient at the wall is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b ðk þ 2Þ; F ð0Þ ¼ ðk 1Þ 3 sffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi ! 2 kþ2 K¼cþ3 1 : b 3 00
ð38Þ ð39Þ
(ii) Case 2: In terms of original variables the solution becomes
rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffi b kþ2 1 tan ; F ðfÞ ¼ 2 þ 3tanh f 2 3 sffiffiffi" rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi!# rffiffiffi 2 kþ2 b kþ2 tanh f : FðfÞ ¼ f þ c þ 3 tan1 b 3 2 3 2
0
ð40Þ
ð41Þ
The velocity gradient at the wall is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b ðk þ 2Þ; F 00 ð0Þ ¼ ðk 1Þ 3 sffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffi 2 kþ2 1 : K¼cþ3 b 3
ð42Þ ð43Þ
pffiffiffi The square root sign indicates that the solution for large b exists for k P 2, that to is dual. The shear stress f 00 ð0Þ= b vs k based on relations (40) and (42) have been shown in Fig. 4. The solutions are dual for 2 < k < 1 describing the aiding and opposing flows regimes. The dual solutions coincide for k = 2 and k = 1. The solution for k > 1 on lower branch, shown by solid line, is regarded as unique as the solution of corresponding upper branch, shown by dotted line, is physically unrealistic. The solution are shown in Fig. 5. The solution of Eqs. (13) and (14) large slip velocity (b ? 1) becomes
gðzÞ ¼ g w þ 00
zb1 ; z þ b1
g ð0Þ ¼ ð2b=3Þ
1=2
;
pffiffiffiffiffiffiffiffi b1 ¼ 6=b;
ð44Þ
pffiffiffiffiffiffiffiffi gð1Þ ¼ g w 6=b:
ð45Þ
pffiffiffi Fig. 4. Asymptotic solution for b ? 1: the wall shear stress F 00 ð0Þ= b versus non-dimensional slip velocity of sheet parameter k, does not depend on suction/injection parameter c.
2730
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
pffiffiffi Fig. 5. Velocity distribution F0 (f) vs f b 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 around this exact solution by expanding the stream function in the power of parameter k 1. Introducing the following change of variables
FðfÞ ¼ f þ c þ ðk 1ÞGðfÞ;
ð46Þ
the boundary layer Eqs. (6) and (7) yield
G000 þ ðf þ cÞG00 bG0 ¼ ðk 1ÞðGG00 þ bG02 Þ; Gð0Þ ¼ 0;
0
ð47Þ
0
G ð0Þ ¼ 1 G ð1Þ ¼ 0:
ð48Þ
The asymptotic expansion for G is
GðfÞ ¼ G1 ðfÞ þ ðk 1ÞG2 ðfÞ þ ðk 1Þ2 G3 ðfÞ þ
ð49Þ
The first order equations are 00 0 G000 1 þ ðf þ cÞG1 bG1 ¼ 0;
G1 ð0Þ ¼ 0;
G01 ð0Þ ¼ 1;
ð50Þ G01 ð1Þ ¼ 0:
ð51Þ
The solution to first order Eqs. (50) and (51) may be expressed in closed form
1 1 1 2b 3 1 ; ; ðf þ cÞ2 ; G01 ðfÞ ¼ AU b; ; ðf þ cÞ2 þ Bðf þ cÞU 2 2 2 2 2
ð52Þ
where U(a, b, x) is the confluent hypergeometric function. It is well-known that U(a b, 0) = 1 and as x ? 1 we have (Abromwitch and Stugen [32])
Uða; b; xÞ ¼
CðbÞ a x þ Cðb aÞ
ð53Þ
where C(a) is the gamma function. The constants A and B estimated from the boundary conditions (51) yield
1 1 1 1 2b 3 1 A ¼ U b; ; c2 W cU ; ; ; c2 2 2 2 2 2
B ¼ AW;
ð54Þ
where
W¼
pffiffiffi Cð1 þ bÞ : 2 Cð1þ2b Þ 2
ð55Þ
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2731
The solution (52) based on relations (54) and (55) yields the velocity distribution
G01 ðfÞ
¼
U b; 12 ; 12 ðf þ cÞ2 Wðf þ cÞU
12b 3 ; 2 ; 12 ðf 2
12b
U b; 12 ; 12 c2 W cU
2
; 32 ; 12 c2
þ cÞ2
ð56Þ
:
The solution in terms of F(f) yields
F 0 ðfÞ ¼ 1 þ ðk 1Þ
U b; 12 ; 12 ðf þ cÞ2 Wðf þ cÞU
12b 3 ; 2 ; 12 ðf 2
12b
U b; 12 ; 12 c2 W cU
2
; 32 ; 12 c2
þ cÞ2
ð57Þ
and friction parameter becomes
1 1 1 1 2b 3 1 ; ; c2 F 00 ð0Þ ¼ ðk 1ÞW U b; ; c2 W cU : 2 2 2 2 2
ð58Þ
(i) Special case c = 0: The solution (57) for no transpiration from the surface (c = 0) becomes
1 1 1 2b 3 1 ; ; f2 : F 0 ðfÞ ¼ 1 þ ðk 1Þ U b; ; f2 WfU 2 2 2 2 2
ð59Þ
The velocity gradient at the wall yields
F 00 ð0Þ 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 0 ðfÞ ¼ 1 þ ðk 1Þ
1
qffiffiffi 2 2 1 3 1 pðf þ cÞU 2 ; 2 ; 2 ðf þ cÞ qffiffiffi : 1 p2 cU 12 ; 32 ; 12 c2
ð61Þ
Using the relation (Abromwitch and Stugen [28])
pffiffiffiffi 1 3 p erfðzÞ ; ; z2 ¼ 2z 2 2
U
ð62Þ
and the solution Eq. (57) is simplified as
F 0 ðfÞ ¼ 1 þ ðk 1Þ
erfc
fþc pffiffi 2
erfc
; pcffiffi
ð63Þ
2
where erfc(x) = 1 erf(x) is the complimentary error function. The velocity gradient at the wall yields
2 rffiffiffiffi c 2 exp 2 : F ð0Þ ¼ ðk 1Þ p erfc pcffiffi 2 00
ð64Þ
2.4. Large suction For large suction, introducing the variables
FðfÞ ¼ c þ
hðgÞ
c
;
f¼
g ; c
ð65Þ
into (4) and (6) the boundary layer equations yield 000
00
00
02
h þ h ¼ c2 ½hh þ bð1 h Þ; hð0Þ ¼ 0;
0
h ð0Þ ¼ k;
0
h ð1Þ ¼ 1:
ð66Þ ð67Þ
The asymptotic expansion for h is
hðgÞ ¼ h0 ðgÞ þ c2 h2 ðgÞ þ c4 h4 ðgÞ þ
ð68Þ
2732
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
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 various values 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 000
00
000 h2 000 h4
00 h2 00 h4
h0 þ h0 ¼ 0; þ þ
¼ ¼
ð69Þ
00 h0 h0 00 h0 h2
þ
02 bð1 h0 Þ; 0 0 00 2bh0 h2 h0 h2 :
ð70Þ ð71Þ
The solution to first and second order problems are
h0 ¼ g þ ð1 kÞðeg 1Þ;
1 1 h2 ¼ ð1 kÞ ðk þ 2b þ 1Þfðg þ 1Þeg 1g þ g2 eg ð1 kÞð1 bÞðe2g 2eg þ 1Þ 2 4 00
The wall shear stresses h ð0Þ and boundary layer thickness dh ¼
1 2b þ 1 ð1 kÞð1 þ bÞ þ 2 c
1k 1 dh ¼ 1 k 2 2b þ 2 ð1 kÞðb þ 3Þ þ 4 c 00
h ð0Þ ¼ 1 k þ
1k
R1 0
ð72Þ ð73Þ
0
ð1 h Þdg are given by relations
2
ð74Þ ð75Þ
The solution in terms of original variables f becomes
FðfÞ ¼ c þ f þ
1k
c
ðecf 1Þ
ð1 kÞ
c3
1 1 ½ðk þ 2b þ 1Þfðcf þ 1Þecf 1g þ c2 f2 ecf þ ð1 kÞð1 bÞðe2cf 2ecf þ 1Þ 2 4 ð76Þ
The wall shear stress and displacement thickness becomes
1k 1 F 00 ð0Þ ¼ ð1 kÞc þ 2b þ 1 ð1 kÞð1 þ bÞ þ 2 c
Z 1 1 k ð1 kÞ 1 0 ðk 1Þðb þ 3Þ þ K¼ ðF ðfÞ 1Þdf ¼ c þ 2b þ 2 4 c c3 0
ð77Þ ð78Þ
The large suction asymptotic prediction (77) for friction factor F00 (0) versus transpiration parameter c (for typical no slip condition k = 0) is shown in Fig. 6 for various values of stretching parameter b = 2, 1 and 0. A comparison with minimum error solution of Section 3, and exact numerical solution for b = 1 are also shown in same figure. The leading term in Eq. (77) due to Weidman 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 all the boundary conditions. The Karman momentum integal coupled with Karman–Pohlhausen method has been commonly used [15,16]. Afzal [33] proposed alternate approach of minimum error solutions obtained by minimizing the integral of
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2733
the square of the error by Euler–Lagrange differential equation (fundamental equation of calculus of variations) used for estimation 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 compared with traditional Karman–Pohlhausen method [29]. If u/ U1 = f(g), g = y/d is the approximating functions satisfying the boundary conditions, involves some unknown parameters, then the error of residual in the boundary layer Eq. (2) becomes
eðx; y; mÞ ¼ m
@2u dU 1 @u @u v : U1 u @y2 @x @y dx
ð79Þ
If the integral of the error is taken as zero as
Z
d
eðx; y; mÞdy ¼ 0;
ð80Þ
0
then we get traditional Karman momentum integral, from which unknown parameter in the trial velocity profile is estimated. It is shown in Afzal [33] that various local potential methods, used by Venkateswarlu and Deshpande [34], Prigogine and Glansdroff [35], Lebon and Lambermont [36] and Doty and Blick [37], correspond to
Z
d
eðx; y; mÞ/dy ¼ 0;
/¼
0
@u ; @d
ð81Þ
which is equivalent to a moment of function / of the error in the boundary layer equation, from which unknown parameter in the trial velocity profile is estimated. It is also possible to choose another moment function but then the method becomes arbitrary. Hsu [38] employed the Galerkin method to study a class of two-dimensional boundary layer problems. It can be shown that the local potential or the Galerkin method for a given problem, will yield same result when the approximating function are the members of a complete set. When the solutions are not based on complete set, the functional has to satisfy certain 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 on the method used. As long as if one has to begin with a trial velocity profile satisfying the boundary conditions on might determine the unknown 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
E¼
Z
l 0
Z
1
e2 ðx; y; mÞdy dx:
ð82Þ
0
The functional E may be minimized with respect to the free parameters by employing the Euler–Lagrange differential equation, which results in certain equations whose solution leads to determination of unknown parameters in the trial velocity profile. If terms of self-similar variables the trial velocity profile u/ U1 = F(f) and f = n/a becomes a simple function, satisfying the boundary 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 02 Þ:
ð83Þ
The least square method for the minimization error by Euler–Lagrange differential equation is
@ d @ ð Þ @ a dx @ ax
Z
1
e2 ðn; aÞdn ¼ 0:
ð84Þ
0
where ax ¼ da=dx. The postulated velocity profile for general b and k is
F 0 ðfÞ ¼ 1 ð1 kÞ expðafÞ;
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 trial profile (85) corresponds to minimization of total integral of least square error for estimation of unknown constant in the trial velocity profile. An integral of (85) subjected to the boundary condition (7a) may be expressed as
1 FðfÞ ¼ c þ ½n þ ð1 kÞðexpðnÞ 1Þ;
n ¼ af:
a
ð86Þ
In the present work least square error is minimized for estimation of unknown constant a. The residual function e(n, a), from boundary layer Eq. (83) for the trial profile (86) is given by the relation
eðn; aÞ ¼ ð1 kÞ½a2 þ ca þ ð1 kÞð2b 1Þ þ n expðnÞ þ ð1 kÞ2 ð1 bÞ expð2nÞ:
ð87Þ
Substituting (87) into Eq. (84) for least square method for the minimization error, and solving for a we get
1 2
1 3
a2 ca ð1 kÞð4b 1Þ
1 2
a c ¼ 0:
ð88Þ
2734
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
The second factor value a = c/2 is not appropriate and we get
1 2
1 3
a2 ca ¼ þ ð1 kÞð4b 1Þ:
ð89Þ
This solution of quadratic equation with positive sign as adopted for a > 0 to get
a¼
1 c 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 c2 þ 2 þ ð1 kÞð4b 1Þ : 3
ð90Þ
The final form of the solution (86) becomes
1 FðfÞ ¼ c þ f þ ½ð1 kÞðexpðafÞ 1Þ
ð91Þ
a
and skin friction and displacement thickness become
F 00 ð0Þ ¼ ð1 kÞa;
K¼
Z
1
1 ðF 0 1Þdf ¼ c ð1 kÞ:
0
a
ð92a; bÞ
The solutions (92a,b) and (90) for no-slip condition (k = 0) become 00
F ð0Þ ¼ a;
1
K¼c ;
a
1 a¼ c 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 c2 þ 2 þ ð4b 1Þ : 3
ð93a; b; cÞ
Fig. 7. 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 the wall suction/injection parameter c = 1, 0 and 1.
Fig. 8. Minimum error solution (85a) for impermeable surface c = 0, for wall shear stress F00 (0) versus the stretching sheet parameter b for various values k the wall slip velocity parameter k = 1, 0.5, 0 and 0.5.
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
2735
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 stretching parameter b = 2, 1 and 0.
Further, for k = 0 the asymptotic solution of Afzal and Luthra [40] for large b given below
F 00 ð0Þ ¼ b1=2 ð1:1547b þ 0:0746b1 þ 0:00509b2 0:00182b3 þ Þ
ð94Þ 1
and the convergence of series (94) was accelerated by Euler transform Y = (1 + b)
F 00 ð0Þ ¼ Y 1=2 ð1:1547 þ 0:0746Y þ 0:00509Y 2 0:00182Y 3 þ Þ:
to yield
ð95Þ
00
The prediction F (0) vs b from the analytical solution (92a) for k = 0 is shown in Fig. 7 for various values of transpiration parameter c = 1, 0 and 1. The asymptotic solution (95) of Afzal and Luthra [36] for k = 0 = c also shown in Fig. 7 compares well the present minmum error solution (92a). The analytical solution (92a) F00 (0) vs b for c = 0 is shown in Fig. 8 for various values 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 compares well the present minimum error solution (92a). The effects of transpiration parameter on prediction F00 (0) vs c from analytical solution (92a) for k = 0 are shown in Fig. 9 for various values stretching parameter b = 0, 1 and 2. The exact numerical solution 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 to exhibit the combined effects of wall transpiration and plate movement compatible with self-similar flow governed by Falkner–Skan equation. The exact solution for b = 1 and closed form asymptotic solutions for each case of b large, large c and for k ? 1 have been presented. For b = 1 the solution for each value of the suction parameter c shows the existence of dual solutions, that are clearly displayed in the [k, F00 (0)]-parameter space. For b large the dual solutions are clearly displayed in the [k, F00 (0)]-parameter space, but suction parameter c plays no role if the prediction of skin friction, but plays an additive role to the stream function. The zero transpiration also, the dual solutions exist for each nonzero value of k. Suction increases the range of stable solutions and blowing decreases this range The analysis of solution behaviors at the singular and regular focal points and of the asymptotic behavior of the wall shear stress parameter F00 (0) at large c are given. The critical values of c, b and k are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile compares well exact solutions. The present work on suction and blowing over moving continuous sheet under pressure gradient for laminar flow reported here is of great interest in turbulent motion of the fluid. This is because under eddy viscosity closure model (Afzal et al. [41]) the analogous equations describe the outer layer of turbulent boundary layer over a moving continuous subjected to external pressure gradient fluid stream, which is of interest in cooling of the objects during industrial manufacture process. References [1] T.A. Abdelhafez, Skin friction and heat transfer on a continuous flat surface moving in a parallel free stream, Int. J. Heat Mass Trans. 28 (1985) 1234– 1237. [2] J.B. Klemp, A.A. Acrivos, A moving-wall boundary layer with reverse flow, J. Fluid Mech. 76 (1976) 363–381. [3] M.Y. Hussaini, W.D. Lakin, N. Nachman, On similarity solutions of a boundary layer problem with an upstream moving wall, SIAM J. Appl. Math. 47 (1987) 699–709. [4] N. Afzal, Momentum transfer on power law stretching plate with free stream pressure gradient, Int. J. Eng. Sci. 41 (2003) 1197–1207. [5] N. Afzal, A. Badaruddin, A.A. Elgarvi, Momentum and transport on a continuous flat surface moving in a parallel stream, Int. J. Heat Mass Trans. 36 (1993) 3399–3403.
2736
N. Afzal / Applied Mathematics and Computation 217 (2010) 2724–2736
[6] J.P. Abraham, E.M. Sparrow, Friction drag resulting from the simultaneous imposed motions of a freestream and its bounding surface, Int. J. Heat Fluid Flow 26 (2005) 289–295. [7] B.C. Sakiadis, Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow, AIChE J. 7 (1961) 26–28. [8] H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung, Z. Math. Phys. 56 (1908) 137. [9] N. Afzal, I.S. Varshney, The cooling of a low heat resistance stretching sheet moving through a fluid, Heat Mass Trans. 14 (1980) 289–293. [10] N. Afzal, The cooling of a low heat resistance stretching sheet moving through a fluid: a rejoinder, Heat Mass Trans. 17 (1983) 217–219. [11] N. Afzal, Heat transfer from a stretching surface, Int. J. Heat Mass Trans. 36 (1993) 1128–1131. [12] A. Aziz, T.Y. Na, Perturbation Methods in Heat Transfer, Hemisphere, New York, 1984. p. 182. [13] B. Brighi, J.D. Hoernel, Recent advances on similarity solutions arising during free convection, Progress in Nonlinear Differential Equations Their Applications, vol. 63, Birkhauser Verlag, Basel/Switzerland, 2005. pp. 83–92. [14] M. Guedda, Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous media, C.R. Mecanique 330 (2002) 279– 283. [15] N. Afzal, Momentum and thermal boundary layers over a two-dimensional or axisymmetric non-linear stretching surface in a stationary fluid, Int. J. Heat Mass Trans. 53 (2010) 540–547. [16] H. Schlichting, Boundary Layer Theory, Mc-Graw Hill Book Co., New York, 1968. [17] L. Rosenhead, Laminar Boundary Layers, Oxford University Press, Oxford, 1963. p. 232. [18] F. Ahmad, Degeneracy in the Blasius problem, Electron. J. Differ. Equ. 79 (2007) 18. [19] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on self-similar boundary layer flow over moving surfaces, Int. J. Eng. Sci. 44 (2006) 730–737. [20] T. Watanabe, Thermal boundary layers over a wedge with uniform suction or injection in forced flow, Acta Mech. 83 (1990) 119–126. [21] K.A. Yih, Uniform suction/blowing effect on forced convection about a wedge: uniform heat flux, Acta Mech. 128 (1998) 173–181. [22] H.T. Yang, L.C. Chien, Analytic solution of the Falkner–Skan equation when b = 1 and c = 0, SIAM J. Appl. Math. 29 (1975) 558–569. [23] J.C.Y. Koh, J.P. Hartnett, Skin-friction and heat transfer for incompressible laminar flow over porous wedges with suction and variable wall temperature, Int. J. Heat Mass Trans. 2 (1961) 185–198. [24] R.B. Kudenatti, V.B. Awati, Solution of pressure gradient stretching plate with suction, Appl. Math. Comput. 210 (2009) 151–157. [25] N. Riley, P.D. Weidman, Multiple solutions of the Falkner–Skan equation for flow past a stretching boundary, SIAM J. Appl. Math. 49 (1989) 1350. [26] V.M. Falkner, S.W. Skan, Some approximate solutions of the boundary-layer equations, Phylos. Mag. 12 (1931) 865–896. [27] S.P. Hastings, Reversed flow solutions of the Falkner–Skan equation, SIAM J. Appl. Math. 22 (1972) 329–334. [28] S.N. Brown, K. Stewartson, On the reversed flow solutions of the Falkner–Skan equation, Mathematika 13 (1966) 1–6. [29] J.B. McLeod, K.R. Rajagopal, On the uniqueness of flow of a Navier–Stokes fluid due to a stretching boundary, Arch. Ration. Mech. Anal. 98 (1987) 385– 393. [30] D. Mansutti, G. Pontrelli, K.R. Rajagopal, Steady flows of non-Newtonian fluids past a porous plate with suction or injection, Int. J. Numer. Meth. Fluids 17 (1993) 927–941. [31] M. Massoudi, M. Ramezan, Effect of injection or suction on the Falkner–Skan flows of second grade fluids, Int. J. Non Linear Mech. 24 (1989) 221–227. [32] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover publications, Inc., New York, 1970. [33] N. Afzal, Minimum error solutions of boundary layer equations, Indian Academy of Sciences, (Math. Sci.: Series A) 91 (1982) 183–193. [34] P. Venkateswarlu, S.M. Deshpande, A critical study of classical and non-classical variational principles in fluid mechanics, J. Non-Equilib. Thermodyn. 7 (1982) 105–128. [35] I. Prigogine, P. Glansdroff, Variational properties and fluctuation theory, Physica 31 (1965) 1243–1256. [36] G. Lebon, J. Lambermont, Generalization of the principle of continuous dissipative systems, J. Chem. Phys. 69 (1973) 2929–2936. [37] R.T. Doty, E.P. Blick, Local potential variational method applied to Himenz flow, AIAA J. 11 (1973) 880. [38] C.C. Hsu, A Galerkin method for a class of steady, two-dimensional, incompressible, laminar boundary-layer flows, J. Fluid Mech. 69 (1975) 783–802. [39] D.A. MacDonald, On the method of the local potential as applied to the solution of the equations of diffusion, Int. J. Heat Mass Trans. 17 (1974) 393– 400. [40] N. Afzal, V.K. Luthra, Highly accelerated laminar flow at moderately large Reynolds number, AIAA J. 12 (1974) 529–532. [41] N. Afzal, A. Seena, A. Bushra, Thermo-fluid-dynamics of turbulent boundary layer over a moving continuous flat sheet in a parallel free stream, Int. J. Appl. Math. Stat. (2011) (Special issue on Similarity solutions in nonlinear PDE’s).