Originals

Heat and Mass Transfer 36 (2000) 439±447 Ó Springer-Verlag 2000

MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate M. K. Chowdhury, M. N. Islam

Abstract This work provides a comprehensive theoretical analysis of a two-dimensional unsteady free convection ¯ow of an incompressible, visco-elastic ¯uid past an in®nite vertical porous plate. Solutions for the zero order perturbation velocity pro®le, the ®rst order perturbation velocity and temperature pro®les in closed form are obtained with the help of Laplace transform technique. The numerical solutions are carried out for the Prandtl number 0.1, 0.72, 1.0, 1.5 and 2.0 which are appropriate for different types of liquid metals and for different values of magnetic ®eld parameter, M. List of a Cp Gr M n Pr S T T1 a b m k r0 d h q l j j0 sw g u; v; w x; y

symbols transpiration parameter speci®c heat at constant pressure Grashof number magnetic ®eld parameter power/exponent Prandtl number visco-elasticity parameter temperature of ¯uid temperature of ambient ¯uid constant coef®cient of thermal expansion kinematic viscosity suction parameter electric conductivity boundary layer thickness dimensionless temperature function density of the ¯uid coef®cient of viscosity the coef®cient of thermal diffusivity the rotational viscosity coef®cient non-dimensional skin friction scaled coordinate de®ned in equations velocity components in the boundary layer co-ordinates along and normal to the plate, respectively

Received on 1 September 1999

M. K. Chowdhury (&), M. N. Islam Department of Mathematics University of Engineering & Technology (B.U.E.T) Dhaka-1000, Bangladesh

1 Introduction The natural convection boundary layer ¯ow of an electrically conducting ¯uid up a hot vertical wall in the presence of strong magnetic ®eld has been studied by Sing and Cowling [1], Sparrow and Cess [2], Riley [3] and Kuiken [4] because of its application in the nuclear engineering in connection with the cooling of reactors. Sometimes it becomes necessary to control the convective boundary layer ¯ows by injecting or withdrawing ¯uid through a porous heated boundary wall. The study of heat transfer has, over the past several years, been related to a wide variety of problems, each with its own demands of precision and elaboration in the understanding of the particular processes of interest. Atmospheric, geophysical and environmental problems in connection with heat rejection, space research and manufacturing system require such type of studies. Elliot [5] generalized Illingworth's [6] problem by assuming a time-dependent velocity and temperature for the plate but neglected the viscous dissipation. However, in both papers, only mathematical results were derived and no physical situation was discussed. From the engineering point of view the physical aspects of such types of problems are important. In these two papers the ¯ow past an impulsively started semi-in®nite horizontal plate has been considered. Stewartson [7] studied it by analytical methods, whereas Hall [8] discussed the same problem by ®nite-difference method. The boundary layer treatment for an idealized viscoelastic ¯uid was introduced by Beard and Walters [9]. There has been a continued interest in the investigation of natural convection heat transfer of non-Newtonian ¯uid, which exhibit visco-elasticity. Recently Rajagopal [10] investigated the heat transfer in the forced convection ¯ow of a visco-elastic ¯uid of Walters model. Most recently Dandpath and Gupta [11] have investigated the ¯ow and heat transfer in an incompressible second order ¯uid caused by a stretching sheet with a view to examining the in¯uence of visco-elasticity on the ¯ow and heart transfer characteristics. The above work were con®ned to the study of steady forced convection ¯ow. Less interest was shown in the problem of transient forced and or free convection ¯ow of a visco-elastic ¯uid. Teipel [12] ®rst studied the transient ¯ow of non-Newtonian visco-elastic ¯uid for an impulsive motion of a ¯at plate. The ¯ow-along a harmonically oscillating ¯at plate of the visco-elastic ¯uid has been studied by Rajagopal [10] and Panda and Roy [13]. It is now a well-known fact that magnetic ¯uid has an sta-

439

In order to non-dimensionalize the governing bilizing effect on the boundary layer growth. With this understanding Singh [14] has investigated the effect of a equations, we introduce the following non-dimensional variables: transverse magnetic ®eld of an electrically conducting visco-elastic ¯uid past an accelerated ¯at plate of in®nite u u0 u2 T T1 extension. W ˆ ; g ˆ y; s ˆ 0 t; h ˆ

440

2 Mathematical formulation An unsteady free convection ¯ow of an electrically conducting viscous incompressible ¯uid past an in®nite vertical porous plate with time dependent suction has been considered. A magnetic ®eld of uniform strength B0 is applied transversely to the plate. The induced magnetic ®eld is neglected as the magnetic Reynolds number of the ¯ow is taken to be very small. We assume that all the ¯uid properties are constant and the in¯uence of density variation with temperature is considered only in the body force term. The ¯ow is assumed to be in the x-direction which is along the vertical plate in the upward direction and y-axis is taken to be normal to the plate. Initially the temperature of the plate and the ¯uid are same. The governing equations are: Continuity equation ov ˆ0 oy Momentum equation

ou ot

k

ou ˆ bg…T oy

…2:1†

T1 † ‡ m

o2 u oy2

r0 b20 u q

j0

o3 u oy2 ot

u0

m

m

Tw

T1

…2:5† With the help of these non-dimensional variables the momentum equation (2.2) takes the form:

ow os

k ow o2 w ˆ 2 u0 og og

S

o3 w og2 os

Mw ‡ Grh

…2:6†

Energy equation (2.3) takes the form

oh os

k oh 1 o2 h ˆ u0 og Pr og2

…2:7†

where Gr ˆ gbDTx3 =m2 is the Grashof number, Pr ˆ lCp =j is the Prandtl number, S ˆ j0 …u0 =m†2 is the visco-elasticity parameter, M ˆ mr0 b20 =u20 q is the magnetic ®eld parameter. The boundary conditions (2.4) become

s  0; s > 0;

w…g; s† ˆ h…g; s† ˆ 0 w…0; s† ˆ sn ; h…0; s† ˆ 1 w…1; s† ˆ 0; h…1; s† ˆ 0

…2:8†

where n is an exponent.

3 Solution for the temperature distribution The method of solutions applied is the Laplace transform Energy equation technique. After using Laplace transform and solving oT oT j o2 T m ˆ …2:3† Eq. (2.7) subject to boundary condition, ot oy qCp oy2   s† ˆ 1 ; h…1; s† ˆ 0 …3:1† h…0; q Here u and v are the velocity components associated with the direction of increasing x and y co-ordinates. T is the we get temperature of the ¯uid in the boundary layer, g is the p
p acceleration due to gravity, b is the volumetric coef®cient Pr a‡ a2 ‡q g e of expansion, j is the thermal conductivity, q is the density  …3:2† hˆ q of the ¯uid, r0 is the electric conductivity, m is the kinematic viscosity, Cp is the speci®c heat at constant pressure, where q is the Laplace transformation parameter and p T1 is the temperature of the ambient ¯uid, j0 is the a Pr ˆ a. The temperature distribution h is now obtained rotational viscosity coef®cient and k is the suction by taking the inverse Laplace transformation of Eq. (3.2) to parameter. obtain. The associated initial and boundary conditions are  p    p 1 ap g Pr p Prg a Prg p ‡ at h…g; t† ˆ e e erfc t  0; u…y; t† ˆ 0; T…y; t† ˆ T1 ; 2 2 t p    t > 0; u…0; t† ˆ u0 f …t†; T…0; t† ˆ Tw ; …2:4† p g Pr p a Prg p  ‡ e at …3:3† erfc t > 0; u…1; t† ˆ 0; T…1; t† ˆ T1 : 2 t Initially the temperature of the plate is the same as that of For a ˆ 0 i.e. k ˆ 0 the solution in (3.3) is exactly the same the ¯uid (no slip condition). At t > 0, the plate starts given by Menold and Yang [15]. moving in its own plane with a velocity u ˆ u0 f …t† where For Pr ˆ 1, Eq. (3.3) turns into u0 is constant and f …t† is a function of time. The plate    p 1 ag ag g temperature is instantaneously raised or lowered to Tw , h…g; t† ˆ e e erfc p ‡ at which is thereafter maintained constant in order to pro2 2 t   duce buoyancy effect. The heat due to viscous and joule p g ag dissipation are neglected in the energy equation because of p  ‡ e erfc at …3:4† 2 t small velocity usually encounters in free convection ¯ow. …2:2†

From Eq. (3.4) we get the rate of heat transfer de®ned by and the boundary conditions (4.5) take the form

p Pr ku0 DT p m …pT†

qw ˆ

…3:5†

1  0 …0; q† ˆ ; w q

 0 …1; q† ˆ 0 : w

…4:7†

Solution of the differential equation (4.6) subject to (4.7) is where DT ˆ Tw T1 . It follows from Eq. (3.5) that the p p a‡ a2 ‡4…q‡M† a‡ a2 ‡4…q‡M† rate of heat transfer is directly proportional to the square g g 2 2  0 ˆ Ae ‡ Be w root of the Prandtl number and inversely proportional to p 
p 2 the square root of the time variable t. Gr e Pr a‡ a ‡q g p
p
p q Pr a ‡ a2 ‡ q 2 a Pr a ‡ a2 ‡ q …M ‡ q†

4 Solution of momentum equation Momentum equation (2.6) can be written as ow os

2

ow o w a ˆ 2 og og

3

ow S 2 og os

Mw ‡ Grh

…4:8†

…4:1†

w…g; s† ˆ 0;

s > 0; s > 0;

w…0; s† ˆ sn ; w…1; s† ˆ 0 :

1. Using inverse Laplace

Gr Gr I1 I2 Pr 1 Pr 1 for Pr 6ˆ 1, where p a‡ a2 ‡4…q‡M† g 2 e I0 ˆ L 1 q p a‡ a2 ‡4…q‡M† g 2 e i I1 ˆ L 1 h p
2 q a ‡ a2 ‡ q Q2r p
p Pr a‡ a2 ‡q g e i I2 ˆ L 1 h p
2 q a ‡ a2 ‡ q Q2r w0 ˆ I0 ‡

subject to the boundary conditions,

s  0;

for Pr 6ˆ 1, where Q2r ˆ M=Pr transform on (4.8) we obtain

…4:2†

…4:9†

…4:10†

Equation (4.1) is a third order partial differential equation. For S ˆ 0, it reduces to equation governing the Newtonian …4:11† ¯uid. Hence, the presence of elastic parameter increases the order of the governing equation from two to three. There are prescribed only two boundary conditions (4.2). Therefore it needs one boundary condition more for a …4:12† unique solution. Thus to overcome the dif®culty we adopt the perturbation technique in which the elastic parameter S can be regarded as a small quantity. We therefore, follow Similarly application of Laplace transform on Eq. (4.4), gives the technique of Beard and Walters [9] and assume the 1 dw solution of (4.1) in the form w ˆ w0 ‡ Sw1 . Thus, Eq. (4.1)  1 w1 …0; g† a qw reduces to dg

o…w0 ‡ Sw1 † o…w0 ‡ Sw1 † a os og 2 o …w0 ‡ Sw1 † M…w0 ‡ Sw1 † ˆ og2 o3 …w0 ‡ Sw1 † ‡ Grh S og2 os

ˆ

1 d2 w dg2

1 Mw

d2 ‰qw0 dg2

w0 …0; g†Š

or

1 1 d2 w dw ‡a 2 dg dg

1 ˆ q …M ‡ q†w

0 d2 w dg2

subject to the boundary conditions Equating the coef®cient of free S and those of S, we obtain 1  1 …0; q† ˆ w the following equations:

ow0 os ow1 os

a a

ow0 o2 w0 ˆ og og2 2

ow1 o w1 ˆ og og2

q

Mw0 ‡ Grh ; Mw1

3

o w0 : og2 os

…4:3† …4:4†

with the boundary conditions (4.2) becoming

s  0; s > 0;

w0 …g; s† ˆ 0; w1 …g; s† ˆ 0; w0 …0; s† ˆ sn ; w1 …0; s† ˆ 0; w0 …1; s† ˆ w1 …1; s† ˆ 0 :

…4:5†

Applying Laplace transform on (4.3), we get

0 0 o2 w ow ‡a 2 og og

0 ˆ …q ‡ M†w

Grh :

…4:6†

…4:13†

…4:14†

Substituting w0 from (4.9) in (4.13) we get

1 1 d2 w dw 1 …M ‡ q†w ‡a 2 dg dg 2 p 
1 2 d2 4 1 Gr e 2 a‡ a ‡4…q‡M† g h i ‡ ˆq 2
2 dg q q…Pr 1† a ‡ p a2 ‡ q Q2r 3 p
p Pr a‡ a2 ‡q g Gr e h i5 p q…Pr 1† a ‡ a2 ‡ q
2 Q2 r 2 3 Gr h i5 ˆ 41 ‡ p
2 Q2r …Pr 1† a ‡ a2 ‡ q

441

  e

 po 2 1n a ‡ a2 ‡ 4…q ‡ M† 2  p 1 2 2

a‡

J1 ˆ 0

a ‡4…q‡M† g

p
2 Pr a ‡ a2 ‡ q Gr h i
2 …Pr 1† a ‡ p a2 ‡ q Q2r p 
p 2  e Pr a‡ a ‡q g 442

Zs 

p 2Qr … 2aQr ‡Q2r †v e erfc……a Qr † v† 4Qr p i a ‡ Qr h …2aQr ‡Q2r †v e 2…a ‡ Qr †v erfc……a ‡ Qr † v† ‡ 4 p i a Qr h … 2aQr ‡Q2r †v e 2…a Qr †v erfc……a Qr † v† 4 r  v 1 ag a2 ‡4M…s v† a2 v e 2 4 ae p 2 g2 g q e 4…s v† dv …4:19† p p …s v†3 a

…4:15†

with the boundary condition (4.14). Solving Eq. (4.15) subject to (4.14) gives

p
2 p a‡ a2 ‡4…q‡M† Pr a ‡ a2 ‡ q g 2 1 ˆ w n o2 e
2 …Pr 1†2 a ‡ p 2 2 a ‡q Qr p2  g a ‡ a2 ‡ 4…q ‡ M† p 4 a2 ‡ 4…q ‡ M† 2 3 Gr n o5  41 ‡ p
2 2 2 Qr …Pr 1† a ‡ a ‡ q p a‡ a2 ‡4…q‡M† g 2 e p
2 Pr a ‡ a2 ‡ q Gr n o2
2 …Pr 1†2 a ‡ p a2 ‡ q Q2r p 
p 2  e Pr a‡ a ‡q g …4:16† Gr

Using inverse Laplace transform on (4.16) we obtain

p
2 a ‡ a2 ‡ q n o2 p
2 a ‡ a2 ‡ q Q2r p2  p a‡ a2 ‡4…q‡M† g 1 a ‡ a2 ‡ 4…q ‡ M† g 2 p L e 4 a2 ‡ 4…q ‡ M† 2 3 Gr n o5  41 ‡ p
2 …Pr 1† a ‡ a2 ‡ q Q2r p a‡ a2 ‡4…q‡M† g 2 e p
2 a ‡ a2 ‡ q Gr Pr 1 L n o2 p
2 …Pr 1†2 a ‡ a2 ‡ q Q2r p 
p 2  e Pr a‡ a ‡q g …4:17†

Gr Pr w1 ˆ L …Pr 1†2

1

This solution can be written as

w1 ˆ

Gr Pr J1 …Pr 1†2

for Pr 6ˆ 1, where

g J2 4

Gr Pr J3 …Pr 1†2

…4:18†

p 2Qr ‡ a …2aQr ‡Q2r †v e erfc……a ‡ Qr † v† 4Qr

p 2  p a‡ a2 ‡4…q‡M† a ‡ a2 ‡ 4…q ‡ M† g 2 p J2 ˆ L e a2 ‡ 4…q ‡ M† p 2  a2 ‡ 4…q ‡ M† Gr 1 a‡ p L ‡ Pr 1 a2 ‡ 4…q ‡ M† p a‡ a2 ‡4…q‡M† 1 g 2  e p
2 Q2r a ‡ a2 ‡ q   p a ‡ 2Qr a ‡ Qr …2as ‡ 2Qr s ‡ g Pr† J3 ˆ ‡ 4Qr 4   p p p g 2asQr ‡Qr g Pr‡Q2r s erfc p ‡ a s ‡ Qr s e 2 s   p a 2Qr a Qr …2as 2Qr s g Pr† 4Qr 4   p 2 p p g  e 2asQr Qr g Pr‡Qr s erfc p ‡ a s Qr s 2 s r  2 s a2 s p Pr ag g4s e …4:20† a p 1

Substituting I0 , I1 and I2 , in Eq. (4.9) we get

p! …a2 ‡ 4M†s g e w0 ˆ erfc p ‡ 2 2 2 s !# p  p a2 ‡4M …a2 ‡ 4M†s g g 2 ‡e erfc p 2 2 s p! ag Z s " p a2 ‡4M …a2 ‡ 4M†v Ge 2 g g ‡ e 2 erfc p ‡ 2 4…Pr 1†Qr 2 v 0 p !# p a2 ‡4M …a2 ‡ 4M† g g 2 g ‡e erfc p 2 2 s h p i  …a ‡ Qr †e2aQr …s v† erfcf…a ‡ Qr † s vg h  p i …a Qr †e 2aQr …s v† erfc …a Qr † s v p Ge Prag Q2r …s v† e dv 2Qr …Pr 1†…4a2 Q2r † e

ag 2

"

p a2 ‡4M g 2

  Q2r e …4a2

p  p Prg erfc p a s 2 s p  p p Prg 2 Prag Qr †e erfc p ‡ a s 2 s p Prag

…4:21†

for Pr 6ˆ 1. On the other hand, substituting J1 , J2 and J3 in Eq. (4.18) one gets ag

Gr Prge 2 w1 ˆ p 2…Pr 1†2 p

Z s "( 0

2Qr ‡ a …a ‡ Qr †2 ‡ v 2 4Qr

p p erfc…a v ‡ Qr v† ) 2Qr † …a Qr †2 2 v e… 2aQr ‡Qr †v 2 4Qr r # p p v 2 erfc…a v Qr v† ae a v p …2aQr ‡Q2r †v

e ( …a

)

…a2 ‡4M† …s 4

v†

g2 4…s v†

ag a2 ‡4M g2 g p e 2 4 s 4s 8 ps    ag 2ag 2 g2 gGre 2 2 p  a ‡ 1 ‡ s s 2s 2:8…Pr 1† pQr   Z s " a2 ‡4Mv g2  4v e 4 2ag 2 g2 p 1 ‡  a2 ‡ v v 2v v 0 n p
p  …a ‡ Qr †e2aQr …s v† erfc a s v ‡ Qr s v p
o p …a Qr †e 2aQr …s v† erfc a s v Qr s v # GrPr 2  eQr …s v† dv …Pr 1†2 "( p
) a ‡ 2Qr …a ‡ Qr † 2as ‡ 2Qr s ‡ Prg ‡ 4 4Qr     p p p g 2asQr ‡Q2r s‡Qr g Pr e erfc p ‡ a s ‡ Qr s 2 s



e

q …s v†3

dv

Fig. 1. a Zero order perturbation velocity pro®les against g for different values of Pr while Gr ˆ 2:0, M ˆ 1:0, a ˆ 0:5 and S ˆ 0:4 from Eq. (4.21). b First order perturbation velocity pro®les for different values of Pr against g while Gr ˆ 2:0, M ˆ 1:0, a ˆ 0:5 and S ˆ 0:4 from Eq. (4.22). c Temperature pro®le for different values of Pr against g while Gr ˆ 2:0, M ˆ 1:0, a ˆ 0:5 and S ˆ 0:4 from Eq. (3.3)

443

(

444

p
) Prg

parameter a ˆ 0:5 and visco-elasticity parameters S ˆ 0:4. From Fig. 1a it can be concluded that the zero-order 4 perturbation velocity pro®le decreases as the values of the   p Prandtl number Pr increases in the region g 2 …0:0; 25†. p   p   g 2  e 2asQr ‡Qr s Qr g Pr erfc p ‡ a s Qr s Near the surface of the plate velocity pro®le increases, 2 s becomes maximum and then decreases and ®nally takes # r  g2 asymptotic value. From Fig. 1b it is observed that near the s a2 s ap Prg 4s e …4:22† surface of the plate the ®rst order perturbation velocity a p pro®le decreases and becomes minimum and then increases. We also observe that the velocity pro®le decreases for Pr 6ˆ 1. as prandtl number Pr increases. Finally here we also see that point of separation takes place for different values of 5 prandtl number. The ®rst order perturbation velocity Results and discussions pro®le for the largest prandtl number is distinguished. In the present investigation, free convection boundary layer ¯ow of a visco-elastic ¯uid past an in®nite vertical From Fig. 1c we see that the temperature pro®le is large near the surface of the plate and decreases away from the porous plate in the presence of a variable transverse plate and ®nally takes asymptotic value. Here we also see magneti®c ®eld is solved analytically. Figure 1a±c represents the zero-order perturbation ve- that temperature pro®le decreases with the increases of the locity pro®le and ®rst order perturbation velocity pro®le Prandtl number. Figure 2a±c represents the zero-order perturbation veand temperature pro®le against g for Prandtl number locity pro®le, ®rst order perturbation velocity pro®le and Pr ˆ 0:1; 0:3; 0:5; 0:72 and 1.0 while Grashof number Gr ˆ 2:0, magnetic ®eld parameter M ˆ 1:0, transpiration temperature pro®le against g for magnetic ®eld parameter

a

2Qr 4Qr

…a

Qr † 2as

2Qr s

Fig. 2. a Zero order perturbation velocity pro®les against g for different values of M while Pr ˆ 0:72, Gr ˆ 2:0, a ˆ 0:5 and S ˆ 0:4 from Eq. (4.21). b First order perturbation velocity pro®les for different values of M against g while Pr ˆ 0:72, Gr ˆ 2:0, a ˆ 0:5 and S ˆ 0:4 from Eq. (4.22). c Temperature pro®le for different values of M against g while Pr ˆ 0:72, Gr ˆ 2:0, a ˆ 0:5 and S ˆ 0:4 from Eq. (3.3)

M ˆ 0:1; 0:2; 0:3; 0:4 and 0.5 with Grashof number Gr ˆ 2:0, Prandtl number Pr ˆ 0:72, transpiration parameter a ˆ 0:5 and visco-elasticity parameters S ˆ 0:4. From Fig. 2a we observe that the zero-order perturbation velocity pro®le decreases owing to increase in the value of the magnetic ®eld parameter M. Near the surface of the plate velocity pro®le increases and becomes maximum and then decreases and ®nally takes asymptotic values. From the same ®gure we may also conclude that an increase in the magnetic ®eld parameter decreases the velocity pro®le more rapidly. From Fig. 2b we see that when the magnetic ®eld parameter M increases the ®rst order perturbation velocity pro®le decreases. We also observe that near the surface of the plate the velocity pro®le decreases and becomes minimum and then increases again. Here we also see that point of separation takes place for different values of the magnetic ®eld parameter. For vertical porous plate the ®rst order perturbation velocity pro®le is markedly affected by magnetic ®eld parameter. From Fig. 2c we see that the temperature pro®le remains unchanged for different values of the magnetic ®eld

parameter. For 0:1  M  0:5 the temperature pro®le becomes maximum at the surface of the plate then decreases away from the plate and ®nally takes asymptotic values at g ˆ 10:0. Figure 3a±c represents respectively the zero-order perturbation velocity pro®le, ®rst order perturbation velocity pro®le and temperature pro®le against g for Grashof number Gr ˆ 5:0; 4:0; 3:0; 2:0 and 1.0 while Prandtl number Pr ˆ 1:0, the magnetic ®eld parameter M ˆ 4:0, transpiration parameter a ˆ 0:8 and visco-elasticity parameters S ˆ 0:4. From Fig. 3a we see that the zero-order perturbation velocity pro®le increases with the increase of the Grashof number Gr. We also observe that near the surface of the plate the velocity pro®le increases becomes maximum and then decreases and ®nally takes the asymptotic value. It is also observed that the velocity pro®le moves away from the plate as the Grashof number Gr increases. From Fig. 3b it is observed that the ®rst order perturbation velocity pro®le increases as Grashof number increase. We also observe that for different values of Grashof number Gr near the surface of the plate velocity

Fig. 3. a Zero order perturbation velocity pro®les against g for different values of Gr while Pr ˆ 1:0, M ˆ 4:0, a ˆ 0:8 and S ˆ 0:4 from Eq. (4.21). b First order perturbation velocity pro®les for different values of Gr against g while Pr ˆ 1:0, M ˆ 4:0, a ˆ 0:8 and S ˆ 0:4 from Eq. (4.22). c Temperature pro®le for different values of Gr against g while Pr ˆ 1:0, M ˆ 4:0, a ˆ 0:8 and S ˆ 0:4 from Eq. (3.3)

445

446

pro®le is maximum then decreases and ®nally takes asymptotic value. Finally here we also see that point of separation takes place for different values of Grashof number. The ®rst order perturbation velocity pro®le for the largest Grashof number is distinguished. From Fig. 3c we may observe that the effect of Grashof number on temperature distribution is constant. Figure 4a±c represents respectively the zero-order perturbation velocity pro®le, ®rst order perturbation velocity pro®le and temperature pro®le against g for transpiration parameter a ˆ 1:5; 2:0; 2:5; 3:0 and 3.5, while Prandtl number Pr ˆ 0:1, Grashof number Gr ˆ 2:0, magnetic ®eld parameter M ˆ 1:0 and visco-elasticity parameters S ˆ 0:4. From Fig. 4a it can be concluded that the zero-order perturbation velocity pro®le decreases as the values of the transpiration parameter a increases in the region g 2 …0:0; 8:0†. Near the surface of the plate velocity pro®le increases, becomes maximum and then decreases and ®nally takes asymptotic value. From Fig. 4b we see that the transpiration parameter a increases as the ®rst order perturbation velocity pro®le decreases. Here we also

see that point of separation takes place for different values of transpiration parameter. For vertical porous plate the ®rst order perturbation velocity pro®le is markedly affected by transpiration parameter. From Fig. 4c we see that the temperature pro®le is large near the surface of the plate and decreases away from the plate and ®nally takes asymptotic value. Here we also see that temperature pro®le decreases with the increases of the transpiration parameter a. Figure 5a and b represents respectively the zero-order perturbation velocity pro®le and ®rst order perturbation velocity pro®le against g for visco-elasticity parameters S ˆ 0:0; 0:1; 0:5; 1:0 and 2.0 while Prandtl number Pr ˆ 0:72, Grashof number Gr ˆ 2:0, magnetic ®eld parameter M ˆ 1:0 and transpiration parameter a ˆ 0:5. From Fig. 5a it is observed that, the zero-order perturbation velocity pro®le decreases as the visco-elasticity parameters S increases. Near the surface of the plate velocity pro®le increases, becomes maximum and then decreases and ®nally takes asymptotic value. From Fig. 5b it is observed that near the surface of the plate the ®rst order

Fig. 4. a Zero order perturbation velocity pro®les against g for different values of a while Pr ˆ 0:1, M ˆ 1:0, Gr ˆ 2:0 and S ˆ 0:4 from Eq. (4.21). b First order perturbation velocity pro®les for different values of a against g while Pr ˆ 0:1, M ˆ 1:0, Gr ˆ 2:0 and S ˆ 0:4 from Eq. (4.22). c Temperature pro®le for different values of a against g while Pr ˆ 0:1, M ˆ 1:0, Gr ˆ 2:0 and S ˆ 0:4 from Eq. (3.3)

447 Fig. 5. a Zero order perturbation velocity pro®les against g for different values of S while Pr ˆ 0:1, M ˆ 1:0, Gr ˆ 2:0, and a ˆ 0:4 from Eq. (4.21). b First order perturbation velocity pro®les for different values of S against g while Pr ˆ 0:1, M ˆ 1:0, Gr ˆ 2:0 and a ˆ 0:4 from Eq. (4.22)

perturbation velocity pro®le decreases becomes minimum References 1. Singh KR; Cowling TG (1963) Thermal convection in and then increases. We also observe that the velocity magnetogasdynamics. J Mech Appl Math 16: 1±5 pro®le decreases as visco-elasticity parameters S increases. Finally here we also see that point of separation takes place 2. Sparrow EM; Cess RD (1961) Effects of magnetic ®eld on free heat transfer. Int J Heat Mass Transfer 3: 267±274 for different values of visco-elasticity parameters. The ®rst 3. convection Riley N (1964) Magnetohydrodynamic free convection. order perturbation velocity pro®le for the largest J Fluid Mech 18: 577±586 visco-elasticity parameters is distinguished. 4. Kuiken HK (1970) Magnetohydrodynamics free convection in strong cross ¯ow ®eld. J Fluid Mech 40: 21±38 5. Elliot L (1969) Unsteady laminar ¯ow of gas near an in®nite 6 ¯at plate. Zeit, Angew, Math Mech 49: 647±652 Conclusions 6. Illingworth CR (1950) Unsteady laminar ¯ow of gas near an in®nite ¯at plate. Proc Camb Phil Soc 4: 603±613 (i) The zero-order perturbation velocity pro®le, ®rst 7. Stewartson K (1951) On the impulsive motion of a ¯at plate order perturbation velocity pro®le and temperature in a viscous ¯uid. Quart Appl Math and Mech 4: 182±198 pro®le decrease with the increase of the Prandtl 8. Hall MG (1969) Boundary layer over an impulsively started number Pr. ¯at plate. Proc Roy Soc (London) 310A: 401±414 (ii) Both the zero and ®rst order perturbation velocity 9. Beard DW; Walters K (1964) Elastico-viscous boundary layer ¯ow. I. Two dimensional ¯ow near a stagnation point. Proc pro®les decrease with the increase of the magnetic Camb Phil Soc 60: 667±674 ®eld parameter M. But the effect of magnetic ®eld 10. Rajagopal KR (1983) On Stokes problem for a non-Newtoparameter on the temperature pro®le is negligible. nian ¯uid. Acta Mechanica 48: 233±239 (iii) Both the zero and ®rst order perturbation velocity 11. Dandpath BS; Gupta AS (1989) Flow and heat transfer in a pro®les increase with the increase of the Grashof viscoelastic ¯uid over a stretching shell. Int J Non-Linear number Gr. But the Grashof number has no Mech 24(3): 215±219 signi®cant effect on the temperature pro®le. 12. Teipel L (1981) The impulsive motion of a ¯at plate in a visco-elastic ¯uid. Acta Mechanica 39: 277±279 (iv) The values of the zero-order perturbation velocity 13. Panda J; Roy JS (1979) Harmonically oscillating visco-elastic pro®le, ®rst order perturbation velocity pro®le and boundary layer ¯ows. Acta Mechanica 31: 213±220 temperature pro®le decrease with the increase of the 14. Singh AS (1984) MHD ¯ow of an elastico-viscous ¯uid past transpiration parameter. an impulsively started vertical plate. Ganit (J Bangladesh (v) The values of the zero-order perturbation velocity Math Soc) 4(1 and 2): 35±39 pro®le and ®rst order perturbation velocity pro®le for 15. Menold ER; Yang KT (1962) Asymptotic solution for unNewtonian ¯uid are greater than that for visco-elastic steady laminar free convection on a vertical plate. TRANS, ASME 5: 124±126 ¯uid.