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Magnetohydrodynamic viscoelastic boundary layer flow past a stretching plate and heat transfer
Applied Mathematics and Computation 155 (2004) 165–177 www.elsevier.com/locate/amc
Magnetohydrodynamic viscoelastic boundary layer flow past a stretching plate and heat transfer M. Zakaria Department of Mathematics, Faculty of Education, El-Shatby 21526, Alexandria University, Alexandria, Egypt
Abstract The heat transfer from a non-isothermal stretching sheet in the presence of a transverse magnetic field is analyzed using the theory of viscoelastic fluid formulated by Walters [Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, 1964] and Beard and Walters [Proc. Cambridge Philol. Soc. 60 (1964)]. By means of the successive approximation, method the governing equations for momentum and energy have been solved. The effects of the coefficient of elastic velocity of fluid K0 , surface mass transfer fx , Alfven velocity a, Prandtl number P , and relaxation time parameter s0 on the velocity, and temperature have discussed. Numerical results are given and illustrated graphically for the problem considered. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Viscoelastic fluid; Suction; Injection; Magnetic field
1. Introduction The study of viscoelastic fluid has become of increasing importance in the last few years. This is mainly due to its many applications in petroleum in drilling, manufacturing of foods and paper and many other similar activities. The boundary layer concept of such fluids is of special importance due to its applications to many engineering problems among which we cite the possibility of reducing frictional drag on the hulls of ships and submarines. E-mail address: [email protected] (M. Zakaria). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00769-0
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
Walters [1] and Beard and Walters [2] deduced the governing equations for the boundary layer flow for a prototype viscoelastic fluid, which they have designated as liquid B when this liquid has a very short memory. Sherief and Ezzat [3] have discussed a problem of a viscoelastic magnetohydrodynamic fluctuating boundary layer flow past an infinite porous plate. The flow of a viscoelastic fluid past a porous plate are studied by Ariel [4]. Continuous surface are surfaces such as those of polymer sheets or filaments continuously drawn from a die. Boundary layer flow on continuous surfaces is an important type of flow occurring in a number of technical processes. Sakiadis [5] introduced the continuous surface concept. Soundalgekar and Takhar [6] studied the flow and heat transfer past a continuously moving flat plate in a micropolar fluid Hiremath and Patial [7]. Recently analyzed Ahmad et al. [8] have studied viscoelastic boundary layer flow past a stretching plate and heat transfer. Recently, Zakaria [9–11] studied the effects of induced magnetic field of a polar fluid through a porous medium. Crane [12] considered a moving strip the velocity of which is proportional to the local distance. The heat and mass transfer on a stretching sheet with suction or blowing was investigated by Gupta [13]. Dutta et al. [14] studied the temperature distribution on the flow over a stretching sheet. The Newtonian fluid flow behavior was assumed by these authors [12–14]. Our aim in this paper to study the heat transfer to a viscoelastic fluid from a non-isothermal stretching sheet with suction or injection in the presence of a transverse magnetic field, which also includes the relaxation time of heat conduction. The inclusion of the relaxation time modifies the governing thermal equation, changing from the parabolic to a hyperbolic type, and thereby eliminating the unrealistic result that thermal disturbance is realized instantaneously everywhere within a fluid. A constant magnetic field is applied transversely to the direction of the flow. This produces an induced magnetic field and an induced electric field.
2. Formulation of the problem Let us consider an unsteady, incompressible, viscoelastic fluid in the presence of a transverse magnetic field flowing past a horizontal stretching sheet, which issues from a thin slit as found on polymer processing applications. Fig. 1 shows the flow model and coordinate system. Initially we assume that the velocity of a point on a sheet is proportional to its distance from the slit. The x and y axes are taken along and perpendicular to the surface, respectively. Let a constant magnetic field of strength H0 acts in the direction of the yaxis. This produces an induced magnetic field h and induced electric field E which satisfy the linearzed equations of electromagnetic, valid for slowly moving media of a perfect conductor [15].
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
167
Boundary Layer
Slot y,v,H0 ,h 2 V0 x,u,h1
Stretching Surface u = Cx
Fig. 1. Coordinate system for the physical model of the stretching sheet.
curl h ¼ J þ e0 curl E ¼ l0
oE ; ot
oh ; ot
ð1Þ ð2Þ
E ¼ l0 ðV H0 Þ;
ð3Þ
div h ¼ 0;
ð4Þ
where J is the electric current density, t is the time, l0 and e0 are the magnetic and electric permeabilities, and V ¼ ðu; v; 0Þ is the velocity vector of the fluid. As mentioned above the applied magnetic field H0 has components H0 ¼ ð0; H0 ; 0Þ. It can be easily seen from the above equations the induced magnetic field has the components h ¼ ðh1 ; h2 ; 0Þ. The vector E and J will have non-vanishing components only in the z-direction, i.e. E ¼ ð0; 0; EÞ;
J ¼ ð0; 0; J Þ:
The flow, temperature, and magnetic fields are now governing by the following equations [1] ou ov þ ¼ 0; ox oy
ð5Þ
ou ou ou o2 u a2 oh1 oh2 ou þu þv ¼m 2þ l0 e 0 H 0 ot ox oy oy ot H0 oy ox 3 3 3 2 2 ou ou o u ou o u ou o u K0 þv 3 þ ; þu ot oy 2 ox oy 2 oy ox oy 2 oy ox oy 2 oT oT oT k o2 T m ou o oT oT oT þ u ; þ s þu þv ¼ þ v 0 ot ox oy qCp oy 2 Cp oy ot ot ox oy
ð6Þ
ð7Þ
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
oh1 ou ¼ H0 ; oy ot
ð8Þ
oh2 ou ¼ H0 ; ox ot
ð9Þ
where q is the density of the fluid, m is the kinematics viscosity, a is the Alfven velocity given by a2 ¼ l0 H02 =q, K0 is the coefficient of elastic velocity of fluid, T is the temperature of the fluid in the boundary layer, s0 is the relaxation time, Cp is the specific heat at constant pressure and k is the thermal conductivity. The boundary and initial conditions imposed on Eqs. (5)–(7), are y ¼ 0:
u ¼ Cx;
v ¼ V0 ;
t ¼ 0;
m
y ¼ 0: y ¼ 0:
T T1 ¼ T0 x ; t ¼ 0; u ¼ Cxext ; v ¼ V0 ext ; t > 0;
y ¼ 0:
T T1 ¼ T0 xm ext ;
y!1
u ! 0;
ð10Þ
t > 0;
T ! T1 ;
where C, and x are constants, V0 is the velocity condition at the surface, T0 is the mean temperature of the surface, T1 is the temperature condition far away from the surface, and m is the power law exponent. Eliminating h1 and h2 between Eqs. (7)–(9), and taking into account the boundary layer approximations. Eq. (7) yields o2 u o2 u ou ou o2 u ov ou þ þv þ ð1 þ a2 l0 e0 Þ 2 þ u ot ot ox ot ox ot oy ot oy 2 4 o ou ou o4 u ou o3 u o4 u ¼ m þ a2 K þ þ v þ u 0 ot oy 2 ot2 oy 2 ot ox oy 2 ot ox oy 2 ot oy 3 3 3 2 2 3 2 2 ov o u ou o u o u o u ou o u ou ou þ þ þ : ot oy 3 ox ot oy 2 ot ox oy 2 oy ot ox oy ot oy ox oy ð11Þ We introduce the following non-dimensional quantities rffiffiffiffi rffiffiffiffi C C h1 x ¼ ; x; y ¼ y; t ¼ Ct; h 1 ¼ m m H0 h2 u v qCp m ; ; u ¼ pffiffiffiffiffiffi ; v ¼ pffiffiffiffiffiffi ; P ¼ h 2 ¼ k H0 Cm Cm pffiffiffiffiffiffi Cm T T1 a T ¼ ; s 0 ¼ Cs0 ; a ¼ pffiffiffiffiffiffi ; Ec ¼ ; T0 Cp T0 Cm V0 x C 1 pffiffiffiffiffiffi E; fx ¼ pffiffiffiffiffiffi ; x ¼ ; K0 ¼ ; E ¼ C m H0 l0 Cm Cm
ð12Þ
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
169
where fx is the mass transfer, P is the Prandtl number Ec is the Eckert number. The mass transfer parameter fx is positive for injection and negative for suction. Invoking the non-dimensional quantities above, Eqs. (5), (7), and (11), are reduced to the non-dimensional equations, dropping the asterisks for convenience, ou ov þ ¼ 0; ox oy a1
ð13Þ
o2 u o2 u ou ou o2 u ov ou þ þ v þ þ u ot2 ot ox ot ox ot oy ot oy 2 4 o ou ou o4 u ou o3 u o4 u ¼ K þ þ v þ a2 þ u 0 ot oy 2 ot2 oy 2 ot ox oy 2 ot ox oy 2 ot oy 3 ov o3 u ou o3 u o2 u o2 u ou o3 u o2 u o2 u þ þ þ ; ot oy 3 ox ot oy 2 ot ox oy 2 oy ot ox oy ot oy ox oy ð14Þ
2 oT oT oT 1 o2 T ou o oT oT oT þu þv ¼ þu þv ; þ Ec s0 ot ox oy P oy 2 oy ot ot ox oy
ð15Þ
where a1 ¼ 1 þ a2 =c2 and c is the speed of light given by c2 ¼ 1=e0 l0 . From Eq. (10) the reduced boundary conditions are y ¼ 0:
u ¼ x;
v ¼ fx ;
m
y ¼ 0:
T ¼x ;
y ¼ 0:
xt
u ¼ xe ;
t ¼ 0;
t ¼ 0; v ¼ fx ext ; t > 0;
ð16Þ
m xt
y ¼ 0:
T ¼ x e ; t > 0;
y!1
u ! 0;
T ! 0:
3. The method of successive approximations A process of successive approximations [14] will integrate the unsteady boundary layer equations (13)–(15). Selecting a system of coordinates which is at rest with respect to the plate and that the magnetohydrodynamic flow of a perfectly conducting fluid moves with respect to the plane surface, we can assume that the velocities u, v, angular velocity x, and the temperature T , possess a series solutions of the form
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
uðx; y; tÞ ¼
1 X
ui ðx; y; tÞ;
vðx; y; tÞ ¼
i¼0
T ðx; y; tÞ ¼
1 X
1 X
vi ðx; y; tÞ;
i¼0
Ti ðx; y; tÞ;
ð17Þ
i¼1
where ui ¼ 0ðei Þ i––integer and e is a small number. Each term in the series (17) must satisfy the continuity equation (13) oui ovi þ ¼0 ox oy
ði ¼ 0; 1; 2; . . .Þ:
ð18Þ
Substituting the series (17) into Eqs. (14) and (15) and setting equal to zero terms of the same order, one obtains equations for finding components of the series (17)
2 o o u0 o4 u0 o2 u0 K0 2 2 a1 2 ¼ 0; þ a2 2 ot oy ot oy ot 2 o o u1 o4 u1 o2 u1 a þ a2 1 ot oy 2 ot2 oy 2 ot2
ð19Þ
o2 u0 ou0 ou0 o2 u0 ov0 ou0 þ þ v0 þ ot ox ot ox ot oy ot oy o4 u0 ou0 o3 u0 o4 u0 ov0 o3 u0 ou0 o3 u0 þ v0 þ þ þ K0 u0 þ 2 2 3 ot ox oy ot ox oy ot oy ot oy 3 ox ot oy 2 o2 u0 o2 u0 ou0 o3 u0 o2 u0 o2 u 0 þ ; ð20Þ ot ox oy 2 oy ot ox oy ot oy ox oy o2 T0 o oT0 ¼ 0; ð21Þ P 1 þ s 0 ot ot oy 2 ¼ u0
2 o2 T1 o oT1 o oT0 oT0 ou0 ¼P 1 þs P 1 þs : u þv Ec 0 0 0 0 2 ot ot ot oy ox oy oy
ð22Þ
Combining Eq. (17) and Eq. (16), we have the corresponding boundary conditions y¼0: y¼0:
u0 ¼ xeat ;
ui ¼ 0;
i ¼ 1; 2; . . . ;
t > 0;
xt
vi ¼ 0;
i ¼ 1; 2; . . . ;
t > 0;
m xt
t > 0;
v0 ¼ fx e ;
y¼0:
T0 ¼ x e ;
Ti ¼ 0;
i ¼ 1; 2; . . . ;
y!1
ui ! 0;
Ti ! 0;
i ¼ 0; 1; 2; . . .
ð23Þ
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
171
In the following analysis, the first two terms in the series solution (17) will be retained. It is known fact [14] that such solution is satisfactory in the phases of the non-periodic motion after it has been started from rest (till the moment when the first separation of boundary layer occurs) and in the case of periodic motion when the amplitude of oscillation is small. Higher-order approximations u3 , can be obtained easy in principle. However, the complexity of the method of successive approximations increases rapidly as higher approximations are considered. It is also known that third and higher terms series solutions give small changes in the results compared with the two terms series solutions.
4. Solution of the problem Let us suppose that the exact solutions of the differential equation (19), is of the form u0 ðx; y; tÞ ¼ xext f10 ðyÞ;
ð24Þ
T0 ðx; y; tÞ ¼ xm ext w1 ðyÞ;
ð25Þ
using Eq. (18) v0 ðx; y; tÞ ¼ eat f1 ðyÞ;
ð26Þ
from Eqs. (19) and (21) and using Eqs. (24)–(26) one obtains the differential equations of the unknown functions f1 ðyÞ, w1 ðyÞ and the corresponding boundary conditions f1000 k12 f10 ¼ 0;
ð27Þ
w001
ð28Þ
P1 w1 ¼ 0;
y ¼ 0: y ¼ 0:
f1 ¼ fx ; w1 ¼ 1;
y!1
f10 ! 0;
f10 ¼ 1; ð29Þ w1 ! 0;
2
a1 x where k12 ¼ a2 þxK 2 , P1 ¼ xP ð1 þ xs0 Þ. 0x The solutions of the system (27)–(29) are of the form
1 ð1 ek1 y Þ fx ; k1 pffiffiffiffi w1 ðyÞ ¼ e P1 y :
f1 ðyÞ ¼
ð30Þ ð31Þ
Assuming the solutions of the differential equation (20), is of the form u1 ðx; y; tÞ ¼ xe2xt f20 ðyÞ;
ð32Þ
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
we can obtain an exact solution of Eq. (22) if we consider the case m ¼ 2 T1 ðx; y; tÞ ¼ x2 e2xt w2 ðyÞ;
ð33Þ
and using Eqs. (24)–(26), (32), and (33) one obtains from Eqs. (20), (22), and (23) the differential equations for f2 ðyÞ, and w2 ðyÞ and the corresponding boundary conditions i k2 h f2000 k22 f20 ¼ 2 f102 f1 f100 þ K0 2f10 f0000 f0 f 0000 ðf000 Þ2 ; ð34Þ 2xa1 w002 P2 w2 ¼ y ¼ 0:
P2 2 ½2w1 f10 w01 f1 Ecf100 ; 2a
f2 ¼ 0;
y ¼ 0 : w2 ¼ 0; y ! 1; w2 ! 0;
ð35Þ
f20 ¼ 0; ð36Þ f20
! 0;
x2
4a1 where k22 ¼ ½2xð12xK 2 , P2 ¼ 2xP ð1 þ 2xs0 Þ. 0 Þþa Using Eqs. (30) and (31) one obtains the solutions of the system (34)–(36)
f2 ðyÞ ¼ A1 þ A2 ek1 y þ A3 ek2 y ; pffiffiffiffi pffiffiffiffi pffiffiffiffi w2 ðyÞ ¼ B1 e P1 y þ B2 e2k1 y þ B3 eðk1 þ P1 Þy þ B4 e P2 y ;
ð37Þ ð38Þ
where k1 k1 ½1 k1 fx þ K0 k12 K0 k13 fx ; A1 ¼ A2 A3 ; A3 ¼ A2 ; A2 ¼ 2a1 xðk22 k12 Þ k2 pffiffiffiffiffi P2 P1 ð1 k1 fx Þ Eck12 B1 ¼ ; B2 ¼ ; 2ak1 ðP1 P2 Þ P1 P2 pffiffiffiffiffi 2k1 P1 P2 ; B4 ¼ ðB1 þ B2 þ B3 Þ: B3 ¼ 2ak1 ðP1 P2 Þ From Eqs. (8) and (9) by virtue of transform equation (12) we get oh1 ou ¼ ; oy ot
ð39Þ
oh2 ou ¼ : ox ot
ð40Þ
Now from Eqs. (24), (32), (39), and (40) the components of the induced magnetic field are given by h1 ðx; y; tÞ ¼
xext 00 2f1 ðyÞ þ eext f200 ðyÞ ; 2x
ð41Þ
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
h2 ðx; y; tÞ ¼
ext 0 2f1 ðyÞ þ eeat f20 ðyÞ : 2x
173
ð42Þ
From Eqs. (1), (3), (24), and (32), by virtue of transform equation (12) the electric field and the electric current density are given by Eðx; y; tÞ ¼ xext f10 ðyÞ þ eext f20 ðyÞ ; ð43Þ J ðx; y; tÞ ¼
ext 0 2 f1 xf1000 þ eeat f20 xf2000 : 2a
ð44Þ
From the velocity filed, we can study the wall shear stress s, as given by ou sðx; tÞ ¼ l : ð45Þ oy y¼0 From Eqs. (17), (24), (30), (32), and (37) we obtain ou s ¼ lC ¼ lCxext ½k1 þ eext ðk12 A2 þ k22 A3 Þ : oy y¼0
ð46Þ
The local friction coefficient Cf is then given by s ; Cf ¼ 1 lC 2 it follows from Eq. (47) Cf ¼ 2xext ½k1 þ eext ðk12 A2 þ k22 A3 Þ : The local heat flux q may be written by FourierÕs law as oT q ¼ k : oy y¼0
ð47Þ
ð48Þ
From Eqs. (26), (37), (45), (48) and (55) we obtain U0 oT q ¼ k ðT0 T1 Þ oy y¼0 m pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi U0 ¼ kx2 ext ðT0 T1 Þ P1 B1 þ 2k1 B2 þ k1 þ P1 B3 þ P2 B4 : m ð49Þ The local heat transfer coefficient is given by qðx; tÞ ðT0 T1 Þ pffiffiffiffiffi pffiffiffiffiffi U0 pffiffiffiffiffi ¼ kx2 ext P1 B1 þ 2k1 B2 þ k1 þ P1 B3 þ P2 B4 : m
hðx; tÞ ¼
ð50Þ
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
The local Nusselt number is given by N ðx; tÞ ¼
hðx; tÞ k
¼ x2 ext
pffiffiffiffiffi pffiffiffiffiffi U0 pffiffiffiffiffi P1 B1 þ 2k1 B2 þ k1 þ P1 B3 þ P2 B4 : m
ð51Þ
5. Results and discussion The velocity profiles for K0 ¼ 0:2, a ¼ 0:2, and for different values of fx are shown in Fig. 2. As might be expected, suction (fx < 0) broadens the velocity distribution and thickens the boundary layer thickness, while injection (fx > 0) thins it. Also the wall shear stress would be increased with the application of suction whereas injection tends to decrease the wall shear stress. This can be readily understood from the fact that the wall velocity gradient is increased with increasing the value of fx . The effects of Alfven velocity a on the velocity profiles are presented in Fig. 3 for fx ¼ 2, K0 ¼ 0:2 and xt ¼ 1. In these figures the dotes lines represent the solution of this flow when x ¼ 0:2, and the solid lines represent the solution of this flow obtained when, x ¼ 0:5. It can be seen from these figure that the velocity field increases with increasing values of the Alfven velocity parameter a and an increases in the value of x leads to a decrease in the velocity. In Fig. 4 the transient velocity is plotted for a ¼ 0:2, fx ¼ 2 and for various values of the coefficient of elastic velocity of fluid K0 . It is seen that, an increases in the value of K0 leads to a decreases in the velocity. u 1.2
fω = 3.0 fω = 1.0 fω = 0.0 fω = - 1 fω = - 2 fω = - 3
0.8
0.4
y
0 0
1
2
3
4
5
6
Fig. 2. Effect of surface mass transfer on velocity distribution.
7
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
175
u 1.2
0.8
α = 0.6 α = 0.3 α = 0.0
ω = 0.2 ω = 0.5
0.4
y
0 0
4
8
12
16
20
Fig. 3. Effect of Alfven velocity a on velocity distribution.
Results for typical temperature profile are illustrated in Fig. 5 for various values of Prandtl number and relaxation time. The thermal boundary layer thickness is more reduced, together with a larger wall temperature gradient, when the relaxation time s0 ¼ 0:02. Also it is observed that an increases in the value of P leads to a decrease in the temperature field. The skin friction coefficient Cf is plotted against x in Fig. 6 for different value of a, and two value of x. The effects of Alfven velocity a are observed from Fig. 6. An increase in the value of a leads to a decrease in the skin fraction
1.2
u
0.8
K0 = 0.0 K0 = 0.2 K0 = 0.4
0.4
ω = 0.2 ω = 0.5
y
0 0
1.5
3
4.5
6
7.5
Fig. 4. Effect of elastic coefficient K0 on velocity distribution.
9
176
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177 T 5
4 P 0.7 1.0 2.0 3.0 4.0 5.0
3
2
τ 0 = 0.00 τ 0 = 0.02
1
y
0 0
1
2
3
4
5
Fig. 5. Temperature distribution for various Prandtl number P .
coefficient. Also the skin fraction coefficient is found to increase when x ¼ 0:5 as compared to x ¼ 0:2. The effects of Prandtl number is observed from Fig. 7. An increase in the Prandtl number leads to increase in the Local Nusselt number. Also it can be seen from these figure the local Nusselt number increases slowly when s0 increases.
Cf 4
3
ω = 0.2 ω = 0.5
2
α = 0.0 α = 0.2 α = 0.4
1
x
0 0
0.5
1
1.5
Fig. 6. Local friction factor versus x.
2
2.5
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
1.2
177
N
τ0 = 0.00 τ0 = 0.02
0.8 P 0.9 0.7 0.5
0.4
x
0 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 7. Local Nusselt number versus x.
References [1] K. Walters, Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, 1964. [2] D. Beard, K. Walters, Elastic-viscous boundary layer flows. Part I. Two dimensional flow near a stagnation point, Proc. Cambridge Philol. Soc. 60 (1964). [3] H. Sherief, M.A. Ezzat, A problem of a viscoelastic magneto-hydrodynamic fluctuating boundary layer flow past an infinite porous plate, Can. J. Phys. 71 (1994). [4] P.D. Ariel, The flow of a viscoelastic fluid past a porous plate, Acta Mech. 107 (1994). [5] B.C. Sakiadis, Boundary layer behavior on continuous solid surface; the boundary layer on a continuous flat surface, Am. Int. Chem. Engrs. J. 7 (1961). [6] V.M. Soundalgekar, H.S. Takhar, Boundary layer flow of a micropolar fluid on a continuous moving plate, Int. J. Eng. Sci. 21 (1983). [7] P.S. Hiremath, P.M. Patial, Free convection effects on the oscillatory flow of a couple stress fluid through a porous medium, Acta Mech. 98 (1993). [8] N. Ahmad, G.S. Patel, B. Siddappa, Visco-elastic boundary layer flow past a stretching plate and heat transfer, J. ZAMP 41 (1990). [9] M. Zakaria, Hydromagnetic fluctuating flow of a couple stress fluid through a porous medium, Appl. Math. Comput. 10 (2002). [10] M. Zakaria, An electromagnetic free convection flow of a micropolar fluid with relaxation time, Korean J. Comput. Appl. Math. 8 (2) (2001) 447–458. [11] M. Zakaria, Magnetohydrodynamic unsteady free convection flow of a couple stress fluid with one relaxation time through a porous medium, Appl. Math. Comput., in press. [12] L.J. Crane, Heat transfer on continuous solid surface, J. Appl. Math. Phys. 21 (1970). [13] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction and blowing, Can. J. Chem. Eng. 55 (1977). [14] B.K. Dutta, P. Roy, A.S. Gupta, Heat transfer on a stretching sheet, Int. Commun. Heat Mass Transfer 12 (1985). [15] M. Zakaria, Solution of the boundary layer equation for a magnetohydrodynamic flow of a perfectly conducting fluid, J. KSIAM 6 (2) (2002).
Magnetohydrodynamic viscoelastic boundary layer flow past a stretching plate and heat transfer M. Zakaria Department of Mathematics, Faculty of Education, El-Shatby 21526, Alexandria University, Alexandria, Egypt
Abstract The heat transfer from a non-isothermal stretching sheet in the presence of a transverse magnetic field is analyzed using the theory of viscoelastic fluid formulated by Walters [Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, 1964] and Beard and Walters [Proc. Cambridge Philol. Soc. 60 (1964)]. By means of the successive approximation, method the governing equations for momentum and energy have been solved. The effects of the coefficient of elastic velocity of fluid K0 , surface mass transfer fx , Alfven velocity a, Prandtl number P , and relaxation time parameter s0 on the velocity, and temperature have discussed. Numerical results are given and illustrated graphically for the problem considered. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Viscoelastic fluid; Suction; Injection; Magnetic field
1. Introduction The study of viscoelastic fluid has become of increasing importance in the last few years. This is mainly due to its many applications in petroleum in drilling, manufacturing of foods and paper and many other similar activities. The boundary layer concept of such fluids is of special importance due to its applications to many engineering problems among which we cite the possibility of reducing frictional drag on the hulls of ships and submarines. E-mail address: [email protected] (M. Zakaria). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00769-0
166
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
Walters [1] and Beard and Walters [2] deduced the governing equations for the boundary layer flow for a prototype viscoelastic fluid, which they have designated as liquid B when this liquid has a very short memory. Sherief and Ezzat [3] have discussed a problem of a viscoelastic magnetohydrodynamic fluctuating boundary layer flow past an infinite porous plate. The flow of a viscoelastic fluid past a porous plate are studied by Ariel [4]. Continuous surface are surfaces such as those of polymer sheets or filaments continuously drawn from a die. Boundary layer flow on continuous surfaces is an important type of flow occurring in a number of technical processes. Sakiadis [5] introduced the continuous surface concept. Soundalgekar and Takhar [6] studied the flow and heat transfer past a continuously moving flat plate in a micropolar fluid Hiremath and Patial [7]. Recently analyzed Ahmad et al. [8] have studied viscoelastic boundary layer flow past a stretching plate and heat transfer. Recently, Zakaria [9–11] studied the effects of induced magnetic field of a polar fluid through a porous medium. Crane [12] considered a moving strip the velocity of which is proportional to the local distance. The heat and mass transfer on a stretching sheet with suction or blowing was investigated by Gupta [13]. Dutta et al. [14] studied the temperature distribution on the flow over a stretching sheet. The Newtonian fluid flow behavior was assumed by these authors [12–14]. Our aim in this paper to study the heat transfer to a viscoelastic fluid from a non-isothermal stretching sheet with suction or injection in the presence of a transverse magnetic field, which also includes the relaxation time of heat conduction. The inclusion of the relaxation time modifies the governing thermal equation, changing from the parabolic to a hyperbolic type, and thereby eliminating the unrealistic result that thermal disturbance is realized instantaneously everywhere within a fluid. A constant magnetic field is applied transversely to the direction of the flow. This produces an induced magnetic field and an induced electric field.
2. Formulation of the problem Let us consider an unsteady, incompressible, viscoelastic fluid in the presence of a transverse magnetic field flowing past a horizontal stretching sheet, which issues from a thin slit as found on polymer processing applications. Fig. 1 shows the flow model and coordinate system. Initially we assume that the velocity of a point on a sheet is proportional to its distance from the slit. The x and y axes are taken along and perpendicular to the surface, respectively. Let a constant magnetic field of strength H0 acts in the direction of the yaxis. This produces an induced magnetic field h and induced electric field E which satisfy the linearzed equations of electromagnetic, valid for slowly moving media of a perfect conductor [15].
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
167
Boundary Layer
Slot y,v,H0 ,h 2 V0 x,u,h1
Stretching Surface u = Cx
Fig. 1. Coordinate system for the physical model of the stretching sheet.
curl h ¼ J þ e0 curl E ¼ l0
oE ; ot
oh ; ot
ð1Þ ð2Þ
E ¼ l0 ðV H0 Þ;
ð3Þ
div h ¼ 0;
ð4Þ
where J is the electric current density, t is the time, l0 and e0 are the magnetic and electric permeabilities, and V ¼ ðu; v; 0Þ is the velocity vector of the fluid. As mentioned above the applied magnetic field H0 has components H0 ¼ ð0; H0 ; 0Þ. It can be easily seen from the above equations the induced magnetic field has the components h ¼ ðh1 ; h2 ; 0Þ. The vector E and J will have non-vanishing components only in the z-direction, i.e. E ¼ ð0; 0; EÞ;
J ¼ ð0; 0; J Þ:
The flow, temperature, and magnetic fields are now governing by the following equations [1] ou ov þ ¼ 0; ox oy
ð5Þ
ou ou ou o2 u a2 oh1 oh2 ou þu þv ¼m 2þ l0 e 0 H 0 ot ox oy oy ot H0 oy ox 3 3 3 2 2 ou ou o u ou o u ou o u K0 þv 3 þ ; þu ot oy 2 ox oy 2 oy ox oy 2 oy ox oy 2 oT oT oT k o2 T m ou o oT oT oT þ u ; þ s þu þv ¼ þ v 0 ot ox oy qCp oy 2 Cp oy ot ot ox oy
ð6Þ
ð7Þ
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oh1 ou ¼ H0 ; oy ot
ð8Þ
oh2 ou ¼ H0 ; ox ot
ð9Þ
where q is the density of the fluid, m is the kinematics viscosity, a is the Alfven velocity given by a2 ¼ l0 H02 =q, K0 is the coefficient of elastic velocity of fluid, T is the temperature of the fluid in the boundary layer, s0 is the relaxation time, Cp is the specific heat at constant pressure and k is the thermal conductivity. The boundary and initial conditions imposed on Eqs. (5)–(7), are y ¼ 0:
u ¼ Cx;
v ¼ V0 ;
t ¼ 0;
m
y ¼ 0: y ¼ 0:
T T1 ¼ T0 x ; t ¼ 0; u ¼ Cxext ; v ¼ V0 ext ; t > 0;
y ¼ 0:
T T1 ¼ T0 xm ext ;
y!1
u ! 0;
ð10Þ
t > 0;
T ! T1 ;
where C, and x are constants, V0 is the velocity condition at the surface, T0 is the mean temperature of the surface, T1 is the temperature condition far away from the surface, and m is the power law exponent. Eliminating h1 and h2 between Eqs. (7)–(9), and taking into account the boundary layer approximations. Eq. (7) yields o2 u o2 u ou ou o2 u ov ou þ þv þ ð1 þ a2 l0 e0 Þ 2 þ u ot ot ox ot ox ot oy ot oy 2 4 o ou ou o4 u ou o3 u o4 u ¼ m þ a2 K þ þ v þ u 0 ot oy 2 ot2 oy 2 ot ox oy 2 ot ox oy 2 ot oy 3 3 3 2 2 3 2 2 ov o u ou o u o u o u ou o u ou ou þ þ þ : ot oy 3 ox ot oy 2 ot ox oy 2 oy ot ox oy ot oy ox oy ð11Þ We introduce the following non-dimensional quantities rffiffiffiffi rffiffiffiffi C C h1 x ¼ ; x; y ¼ y; t ¼ Ct; h 1 ¼ m m H0 h2 u v qCp m ; ; u ¼ pffiffiffiffiffiffi ; v ¼ pffiffiffiffiffiffi ; P ¼ h 2 ¼ k H0 Cm Cm pffiffiffiffiffiffi Cm T T1 a T ¼ ; s 0 ¼ Cs0 ; a ¼ pffiffiffiffiffiffi ; Ec ¼ ; T0 Cp T0 Cm V0 x C 1 pffiffiffiffiffiffi E; fx ¼ pffiffiffiffiffiffi ; x ¼ ; K0 ¼ ; E ¼ C m H0 l0 Cm Cm
ð12Þ
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
169
where fx is the mass transfer, P is the Prandtl number Ec is the Eckert number. The mass transfer parameter fx is positive for injection and negative for suction. Invoking the non-dimensional quantities above, Eqs. (5), (7), and (11), are reduced to the non-dimensional equations, dropping the asterisks for convenience, ou ov þ ¼ 0; ox oy a1
ð13Þ
o2 u o2 u ou ou o2 u ov ou þ þ v þ þ u ot2 ot ox ot ox ot oy ot oy 2 4 o ou ou o4 u ou o3 u o4 u ¼ K þ þ v þ a2 þ u 0 ot oy 2 ot2 oy 2 ot ox oy 2 ot ox oy 2 ot oy 3 ov o3 u ou o3 u o2 u o2 u ou o3 u o2 u o2 u þ þ þ ; ot oy 3 ox ot oy 2 ot ox oy 2 oy ot ox oy ot oy ox oy ð14Þ
2 oT oT oT 1 o2 T ou o oT oT oT þu þv ¼ þu þv ; þ Ec s0 ot ox oy P oy 2 oy ot ot ox oy
ð15Þ
where a1 ¼ 1 þ a2 =c2 and c is the speed of light given by c2 ¼ 1=e0 l0 . From Eq. (10) the reduced boundary conditions are y ¼ 0:
u ¼ x;
v ¼ fx ;
m
y ¼ 0:
T ¼x ;
y ¼ 0:
xt
u ¼ xe ;
t ¼ 0;
t ¼ 0; v ¼ fx ext ; t > 0;
ð16Þ
m xt
y ¼ 0:
T ¼ x e ; t > 0;
y!1
u ! 0;
T ! 0:
3. The method of successive approximations A process of successive approximations [14] will integrate the unsteady boundary layer equations (13)–(15). Selecting a system of coordinates which is at rest with respect to the plate and that the magnetohydrodynamic flow of a perfectly conducting fluid moves with respect to the plane surface, we can assume that the velocities u, v, angular velocity x, and the temperature T , possess a series solutions of the form
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
uðx; y; tÞ ¼
1 X
ui ðx; y; tÞ;
vðx; y; tÞ ¼
i¼0
T ðx; y; tÞ ¼
1 X
1 X
vi ðx; y; tÞ;
i¼0
Ti ðx; y; tÞ;
ð17Þ
i¼1
where ui ¼ 0ðei Þ i––integer and e is a small number. Each term in the series (17) must satisfy the continuity equation (13) oui ovi þ ¼0 ox oy
ði ¼ 0; 1; 2; . . .Þ:
ð18Þ
Substituting the series (17) into Eqs. (14) and (15) and setting equal to zero terms of the same order, one obtains equations for finding components of the series (17)
2 o o u0 o4 u0 o2 u0 K0 2 2 a1 2 ¼ 0; þ a2 2 ot oy ot oy ot 2 o o u1 o4 u1 o2 u1 a þ a2 1 ot oy 2 ot2 oy 2 ot2
ð19Þ
o2 u0 ou0 ou0 o2 u0 ov0 ou0 þ þ v0 þ ot ox ot ox ot oy ot oy o4 u0 ou0 o3 u0 o4 u0 ov0 o3 u0 ou0 o3 u0 þ v0 þ þ þ K0 u0 þ 2 2 3 ot ox oy ot ox oy ot oy ot oy 3 ox ot oy 2 o2 u0 o2 u0 ou0 o3 u0 o2 u0 o2 u 0 þ ; ð20Þ ot ox oy 2 oy ot ox oy ot oy ox oy o2 T0 o oT0 ¼ 0; ð21Þ P 1 þ s 0 ot ot oy 2 ¼ u0
2 o2 T1 o oT1 o oT0 oT0 ou0 ¼P 1 þs P 1 þs : u þv Ec 0 0 0 0 2 ot ot ot oy ox oy oy
ð22Þ
Combining Eq. (17) and Eq. (16), we have the corresponding boundary conditions y¼0: y¼0:
u0 ¼ xeat ;
ui ¼ 0;
i ¼ 1; 2; . . . ;
t > 0;
xt
vi ¼ 0;
i ¼ 1; 2; . . . ;
t > 0;
m xt
t > 0;
v0 ¼ fx e ;
y¼0:
T0 ¼ x e ;
Ti ¼ 0;
i ¼ 1; 2; . . . ;
y!1
ui ! 0;
Ti ! 0;
i ¼ 0; 1; 2; . . .
ð23Þ
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
171
In the following analysis, the first two terms in the series solution (17) will be retained. It is known fact [14] that such solution is satisfactory in the phases of the non-periodic motion after it has been started from rest (till the moment when the first separation of boundary layer occurs) and in the case of periodic motion when the amplitude of oscillation is small. Higher-order approximations u3 , can be obtained easy in principle. However, the complexity of the method of successive approximations increases rapidly as higher approximations are considered. It is also known that third and higher terms series solutions give small changes in the results compared with the two terms series solutions.
4. Solution of the problem Let us suppose that the exact solutions of the differential equation (19), is of the form u0 ðx; y; tÞ ¼ xext f10 ðyÞ;
ð24Þ
T0 ðx; y; tÞ ¼ xm ext w1 ðyÞ;
ð25Þ
using Eq. (18) v0 ðx; y; tÞ ¼ eat f1 ðyÞ;
ð26Þ
from Eqs. (19) and (21) and using Eqs. (24)–(26) one obtains the differential equations of the unknown functions f1 ðyÞ, w1 ðyÞ and the corresponding boundary conditions f1000 k12 f10 ¼ 0;
ð27Þ
w001
ð28Þ
P1 w1 ¼ 0;
y ¼ 0: y ¼ 0:
f1 ¼ fx ; w1 ¼ 1;
y!1
f10 ! 0;
f10 ¼ 1; ð29Þ w1 ! 0;
2
a1 x where k12 ¼ a2 þxK 2 , P1 ¼ xP ð1 þ xs0 Þ. 0x The solutions of the system (27)–(29) are of the form
1 ð1 ek1 y Þ fx ; k1 pffiffiffiffi w1 ðyÞ ¼ e P1 y :
f1 ðyÞ ¼
ð30Þ ð31Þ
Assuming the solutions of the differential equation (20), is of the form u1 ðx; y; tÞ ¼ xe2xt f20 ðyÞ;
ð32Þ
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
we can obtain an exact solution of Eq. (22) if we consider the case m ¼ 2 T1 ðx; y; tÞ ¼ x2 e2xt w2 ðyÞ;
ð33Þ
and using Eqs. (24)–(26), (32), and (33) one obtains from Eqs. (20), (22), and (23) the differential equations for f2 ðyÞ, and w2 ðyÞ and the corresponding boundary conditions i k2 h f2000 k22 f20 ¼ 2 f102 f1 f100 þ K0 2f10 f0000 f0 f 0000 ðf000 Þ2 ; ð34Þ 2xa1 w002 P2 w2 ¼ y ¼ 0:
P2 2 ½2w1 f10 w01 f1 Ecf100 ; 2a
f2 ¼ 0;
y ¼ 0 : w2 ¼ 0; y ! 1; w2 ! 0;
ð35Þ
f20 ¼ 0; ð36Þ f20
! 0;
x2
4a1 where k22 ¼ ½2xð12xK 2 , P2 ¼ 2xP ð1 þ 2xs0 Þ. 0 Þþa Using Eqs. (30) and (31) one obtains the solutions of the system (34)–(36)
f2 ðyÞ ¼ A1 þ A2 ek1 y þ A3 ek2 y ; pffiffiffiffi pffiffiffiffi pffiffiffiffi w2 ðyÞ ¼ B1 e P1 y þ B2 e2k1 y þ B3 eðk1 þ P1 Þy þ B4 e P2 y ;
ð37Þ ð38Þ
where k1 k1 ½1 k1 fx þ K0 k12 K0 k13 fx ; A1 ¼ A2 A3 ; A3 ¼ A2 ; A2 ¼ 2a1 xðk22 k12 Þ k2 pffiffiffiffiffi P2 P1 ð1 k1 fx Þ Eck12 B1 ¼ ; B2 ¼ ; 2ak1 ðP1 P2 Þ P1 P2 pffiffiffiffiffi 2k1 P1 P2 ; B4 ¼ ðB1 þ B2 þ B3 Þ: B3 ¼ 2ak1 ðP1 P2 Þ From Eqs. (8) and (9) by virtue of transform equation (12) we get oh1 ou ¼ ; oy ot
ð39Þ
oh2 ou ¼ : ox ot
ð40Þ
Now from Eqs. (24), (32), (39), and (40) the components of the induced magnetic field are given by h1 ðx; y; tÞ ¼
xext 00 2f1 ðyÞ þ eext f200 ðyÞ ; 2x
ð41Þ
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
h2 ðx; y; tÞ ¼
ext 0 2f1 ðyÞ þ eeat f20 ðyÞ : 2x
173
ð42Þ
From Eqs. (1), (3), (24), and (32), by virtue of transform equation (12) the electric field and the electric current density are given by Eðx; y; tÞ ¼ xext f10 ðyÞ þ eext f20 ðyÞ ; ð43Þ J ðx; y; tÞ ¼
ext 0 2 f1 xf1000 þ eeat f20 xf2000 : 2a
ð44Þ
From the velocity filed, we can study the wall shear stress s, as given by ou sðx; tÞ ¼ l : ð45Þ oy y¼0 From Eqs. (17), (24), (30), (32), and (37) we obtain ou s ¼ lC ¼ lCxext ½k1 þ eext ðk12 A2 þ k22 A3 Þ : oy y¼0
ð46Þ
The local friction coefficient Cf is then given by s ; Cf ¼ 1 lC 2 it follows from Eq. (47) Cf ¼ 2xext ½k1 þ eext ðk12 A2 þ k22 A3 Þ : The local heat flux q may be written by FourierÕs law as oT q ¼ k : oy y¼0
ð47Þ
ð48Þ
From Eqs. (26), (37), (45), (48) and (55) we obtain U0 oT q ¼ k ðT0 T1 Þ oy y¼0 m pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi U0 ¼ kx2 ext ðT0 T1 Þ P1 B1 þ 2k1 B2 þ k1 þ P1 B3 þ P2 B4 : m ð49Þ The local heat transfer coefficient is given by qðx; tÞ ðT0 T1 Þ pffiffiffiffiffi pffiffiffiffiffi U0 pffiffiffiffiffi ¼ kx2 ext P1 B1 þ 2k1 B2 þ k1 þ P1 B3 þ P2 B4 : m
hðx; tÞ ¼
ð50Þ
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M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
The local Nusselt number is given by N ðx; tÞ ¼
hðx; tÞ k
¼ x2 ext
pffiffiffiffiffi pffiffiffiffiffi U0 pffiffiffiffiffi P1 B1 þ 2k1 B2 þ k1 þ P1 B3 þ P2 B4 : m
ð51Þ
5. Results and discussion The velocity profiles for K0 ¼ 0:2, a ¼ 0:2, and for different values of fx are shown in Fig. 2. As might be expected, suction (fx < 0) broadens the velocity distribution and thickens the boundary layer thickness, while injection (fx > 0) thins it. Also the wall shear stress would be increased with the application of suction whereas injection tends to decrease the wall shear stress. This can be readily understood from the fact that the wall velocity gradient is increased with increasing the value of fx . The effects of Alfven velocity a on the velocity profiles are presented in Fig. 3 for fx ¼ 2, K0 ¼ 0:2 and xt ¼ 1. In these figures the dotes lines represent the solution of this flow when x ¼ 0:2, and the solid lines represent the solution of this flow obtained when, x ¼ 0:5. It can be seen from these figure that the velocity field increases with increasing values of the Alfven velocity parameter a and an increases in the value of x leads to a decrease in the velocity. In Fig. 4 the transient velocity is plotted for a ¼ 0:2, fx ¼ 2 and for various values of the coefficient of elastic velocity of fluid K0 . It is seen that, an increases in the value of K0 leads to a decreases in the velocity. u 1.2
fω = 3.0 fω = 1.0 fω = 0.0 fω = - 1 fω = - 2 fω = - 3
0.8
0.4
y
0 0
1
2
3
4
5
6
Fig. 2. Effect of surface mass transfer on velocity distribution.
7
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
175
u 1.2
0.8
α = 0.6 α = 0.3 α = 0.0
ω = 0.2 ω = 0.5
0.4
y
0 0
4
8
12
16
20
Fig. 3. Effect of Alfven velocity a on velocity distribution.
Results for typical temperature profile are illustrated in Fig. 5 for various values of Prandtl number and relaxation time. The thermal boundary layer thickness is more reduced, together with a larger wall temperature gradient, when the relaxation time s0 ¼ 0:02. Also it is observed that an increases in the value of P leads to a decrease in the temperature field. The skin friction coefficient Cf is plotted against x in Fig. 6 for different value of a, and two value of x. The effects of Alfven velocity a are observed from Fig. 6. An increase in the value of a leads to a decrease in the skin fraction
1.2
u
0.8
K0 = 0.0 K0 = 0.2 K0 = 0.4
0.4
ω = 0.2 ω = 0.5
y
0 0
1.5
3
4.5
6
7.5
Fig. 4. Effect of elastic coefficient K0 on velocity distribution.
9
176
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177 T 5
4 P 0.7 1.0 2.0 3.0 4.0 5.0
3
2
τ 0 = 0.00 τ 0 = 0.02
1
y
0 0
1
2
3
4
5
Fig. 5. Temperature distribution for various Prandtl number P .
coefficient. Also the skin fraction coefficient is found to increase when x ¼ 0:5 as compared to x ¼ 0:2. The effects of Prandtl number is observed from Fig. 7. An increase in the Prandtl number leads to increase in the Local Nusselt number. Also it can be seen from these figure the local Nusselt number increases slowly when s0 increases.
Cf 4
3
ω = 0.2 ω = 0.5
2
α = 0.0 α = 0.2 α = 0.4
1
x
0 0
0.5
1
1.5
Fig. 6. Local friction factor versus x.
2
2.5
M. Zakaria / Appl. Math. Comput. 155 (2004) 165–177
1.2
177
N
τ0 = 0.00 τ0 = 0.02
0.8 P 0.9 0.7 0.5
0.4
x
0 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 7. Local Nusselt number versus x.
References [1] K. Walters, Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, 1964. [2] D. Beard, K. Walters, Elastic-viscous boundary layer flows. Part I. Two dimensional flow near a stagnation point, Proc. Cambridge Philol. Soc. 60 (1964). [3] H. Sherief, M.A. Ezzat, A problem of a viscoelastic magneto-hydrodynamic fluctuating boundary layer flow past an infinite porous plate, Can. J. Phys. 71 (1994). [4] P.D. Ariel, The flow of a viscoelastic fluid past a porous plate, Acta Mech. 107 (1994). [5] B.C. Sakiadis, Boundary layer behavior on continuous solid surface; the boundary layer on a continuous flat surface, Am. Int. Chem. Engrs. J. 7 (1961). [6] V.M. Soundalgekar, H.S. Takhar, Boundary layer flow of a micropolar fluid on a continuous moving plate, Int. J. Eng. Sci. 21 (1983). [7] P.S. Hiremath, P.M. Patial, Free convection effects on the oscillatory flow of a couple stress fluid through a porous medium, Acta Mech. 98 (1993). [8] N. Ahmad, G.S. Patel, B. Siddappa, Visco-elastic boundary layer flow past a stretching plate and heat transfer, J. ZAMP 41 (1990). [9] M. Zakaria, Hydromagnetic fluctuating flow of a couple stress fluid through a porous medium, Appl. Math. Comput. 10 (2002). [10] M. Zakaria, An electromagnetic free convection flow of a micropolar fluid with relaxation time, Korean J. Comput. Appl. Math. 8 (2) (2001) 447–458. [11] M. Zakaria, Magnetohydrodynamic unsteady free convection flow of a couple stress fluid with one relaxation time through a porous medium, Appl. Math. Comput., in press. [12] L.J. Crane, Heat transfer on continuous solid surface, J. Appl. Math. Phys. 21 (1970). [13] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction and blowing, Can. J. Chem. Eng. 55 (1977). [14] B.K. Dutta, P. Roy, A.S. Gupta, Heat transfer on a stretching sheet, Int. Commun. Heat Mass Transfer 12 (1985). [15] M. Zakaria, Solution of the boundary layer equation for a magnetohydrodynamic flow of a perfectly conducting fluid, J. KSIAM 6 (2) (2002).