Unsteady MHD convection flow of polar fluids past a vertical moving porous plate in a porous medium

Unsteady MHD convection flow of polar fluids past a vertical moving porous plate in a porous medium

International Journal of Heat and Mass Transfer 44 (2001) 2791±2799 www.elsevier.com/locate/ijhmt Unsteady MHD convection ¯ow of polar ¯uids past a ...

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International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

www.elsevier.com/locate/ijhmt

Unsteady MHD convection ¯ow of polar ¯uids past a vertical moving porous plate in a porous medium Youn J. Kim * School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-Dong, Suwon 440-746, South Korea Received 21 July 2000; received in revised form 18 October 2000

Abstract The objectives of the present study are to investigate the unsteady two-dimensional laminar ¯ow of a viscous incompressible electrically conducting polar ¯uid via a porous medium past a semi-in®nite vertical porous moving plate in the presence of a transverse magnetic ®eld. The plate moves with a constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially increasing or decreasing small perturbation law. A uniform magnetic ®eld acts perpendicularly to the porous surface which absorbs the polar ¯uid with a suction velocity varying with time. The e€ects of material parameters on the velocity and temperature ®elds across the boundary layer are investigated. The method of solution can be applied for small perturbation approximation. Numerical results of velocity distribution of polar ¯uids are compared with the corresponding ¯ow problems for a Newtonian ¯uid. For a constant plate moving velocity with the given magnetic and permeability parameters, and Prandtl and Grashof numbers, the e€ect of increasing values of suction velocity parameter results in an increasing surface skin friction. It is also observed that the surface skin friction decreases by increasing the plate moving velocity. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction The study of ¯ow and heat transfer for an electrically conducting polar ¯uid past a porous plate under the in¯uence of a magnetic ®eld has attracted the interest of many investigators in view of its applications in many engineering problems such as magnetohydrodynamic (MHD) generator, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions and the boundary layer control in the ®eld of aerodynamics [1]. Polar ¯uids are ¯uids with microstructure belonging to a class of ¯uids with non-symmetrical stress tensor. Physically, they represent ¯uids consisting of randomly oriented particles suspended in a viscous medium [2±4]. A great number of Darcian porous MHD studies have been carried out examining the e€ects of magnetic ®eld on hydrodynamic ¯ow without heat transfer in various con®gurations, e.g., in channels and past plates and wedges, etc. [5,6]. *

Tel.: +82-31-290-7448; fax: +82-31-290-5849. E-mail address: [email protected] (Y.J. Kim).

Gribben [7] considered the MHD boundary layer ¯ow over a semi-in®nite plate with an aligned magnetic ®eld in the presence of a pressure gradient. He has obtained solutions for large and small magnetic Prandtl numbers using the method of matched asymptotic expansion. Takhar and Ram [8] studied the e€ects of Hall currents on hydromagnetic free convection boundary layer ¯ow via a porous medium past a plate, using harmonic analysis. Takhar and Ram [9] also studied the MHD free porous convection heat transfer of water at 4°C through a porous medium. Soundalgekar [10] obtained approximate solutions for the two-dimensional ¯ow of an incompressible, viscous ¯uid past an in®nite porous vertical plate with constant suction velocity normal to the plate, the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents. Raptis and Kafousias [11] studied the in¯uence of a magnetic ®eld upon the steady free convection ¯ow through a porous medium bounded by an in®nite vertical plate with a constant suction velocity, and when the plate temperature is also constant. Raptis [12] studied

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Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

Nomenclature A B0 Cf Cp G g K k M N Nu n Pr T t U0 u; v V0 x; y

suction velocity parameter magnetic ¯ux density skin friction coecient speci®c heat at constant pressure Grashof number acceleration due to gravity permeability of the porous medium thermal conductivity magnetic ®eld parameter dimensionless material parameter Nusselt number dimensionless exponential index Prandtl number temperature dimensionless time scale of free stream velocity components of velocities along and perpendicular to the plate, respectively scale of suction velocity distances along and perpendicular to the plate, respectively

mathematically the case of time-varying two-dimensional natural convective heat transfer of an incompressible, electrically conducting viscous ¯uid via a highly porous medium bounded by an in®nite vertical porous plate. However, most of the previous works assume that the plate is at rest. In the present work, we consider the case of a semi-in®nite moving porous plate with a constant velocity in the longitudinal direction when the magnetic ®eld is imposed transversely to the plate. We also consider the free stream to consist of a mean velocity and temperature with a superimposed exponentially variation with time. In general, the study of Darcian porous MHD is very complicated. It is necessary to consider in detail the distribution of velocity and temperature distributions across the boundary layer, in addition to the surface skin friction. The present work is an attempt to shed some light on these issues. 2. Formulation We consider the two-dimensional unsteady ¯ow of a laminar, incompressible ¯uid past a semi-in®nite vertical porous moving plate embedded in a porous medium and subjected to a transverse magnetic ®eld in the presence of a pressure gradient. The physical model and geometrical coordinates are shown in Fig. 1. It is assumed

Greek a b bf c e r q K l m mr h x

symbols ¯uid thermal di€usivity dimensionless viscosity ratio coecient of volumetric expansion of the working ¯uid spin-gradient viscosity scalar constant (1)) electrical conductivity ¯uid density coecient of gyro-viscosity ¯uid dynamic viscosity ¯uid kinematic viscosity ¯uid kinematic rotational viscosity dimensionless temperature angular velocity vector

Superscripts 0 di€erentiation with respect to y  dimensional properties Subscripts p plate w wall condition 1 free stream condition

Fig. 1. Physical model and coordinate system of problem.

that there is no applied voltage which implies the absence of an electric ®eld. The transversely applied magnetic ®eld and magnetic Reynolds number are very small and hence the induced magnetic ®eld is negligible [13]. Viscous and Darcy's resistance terms are taken into account with constant permeability of the porous medium. The MHD term is derived from an order-ofmagnitude analysis of the full Navier±Stokes equations. It is assumed here that the hole size of the porous plate is signi®cantly larger than a characteristic microscopic length scale of the porous medium. We regard the porous medium as an assemblage of small identical

Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

spherical particles ®xed in space, following Yamamoto and Iwamura [14]. Due to the semi-in®nite plane surface assumption, furthermore, the ¯ow variables are functions of y  and t only. Under these conditions, the governing equations, i.e., the mass, momentum and energy conservation equations can be written in a Cartesian frame of reference, as: continuity: ov ˆ 0; oy 

…1†

linear momentum: ou ou ‡ v  ˆ  ot oy

1 op o2 u ‡ …m ‡ mr † 2 ‡ gbf …T  q ox oy u r 2  ox B0 u ‡ 2mr  ; m  q K oy

…3†

energy: oT oT o2 T ‡ v  ˆ a 2 ;  ot oy oy



T ˆ Tw ‡ e…Tw

T1 † e

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;

2 

ox ou ˆ at y  ˆ 0; oy  oy 2    u ! U1 ˆ U0 1 ‡ e en t ; T ! T1 ; x ! 0 as y  ! 1

…5† …6†

in which n is a scalar constant, and U0 is a scale of free stream velocity. From the continuity equation (1), it is clear that the suction velocity normal to the plate is a function of time only and we shall take it in the form:   v ˆ V0 1 ‡ eA en t ; …7† where A is a real positive constant e and eA small less than unity and V0 is a scale of suction velocity which has non-zero positive constant. Outside the boundary layer, Eq. (2) gives

T1 †

…2† angular momentum:     ox o2 x  ox qj ‡ v ˆ c 2 ; ot oy  oy

u ˆ up ;

n t 

…4†

where x and y  are the dimensional distances longitudinal and perpendicular to the plate, respectively, u ,v the components of dimensional velocities along the x and y  directions, respectively, q the density and m the kinematic viscosity, mr the kinematic rotational viscosity, g the acceleration of gravity, bf the coecient of volumetric thermal expansion of the ¯uid, K  the permeability of the porous medium, r the electrical conductivity of the ¯uid, B0 the magnetic induction, j the micro-inertia density, x the component of the angular velocity vector normal to the xy-plane, c the spin-gradient viscosity, T the temperature, and a is the e€ective ¯uid thermal di€usivity. The third term on the RHS of the momentum Eq. (2) denotes buoyancy e€ects, the fourth is the bulk matrix linear resistance, i.e., Darcy term, the ®fth is the MHD term. The heat due to viscous dissipation is neglected for small velocities in Eq. (4). Also, Darcy dissipation term is neglected because it is of the same order-of-magnitude as the viscous dissipation term. It is assumed that the porous plate moves with constant velocity …up † in the  longitudinal direction, and the free stream velocity …U1 † follows an exponentially increasing or decreasing small perturbation law. We also assume that the plate temperature (T ) and suction velocity …v † vary exponentially with time. Under these assumptions, the appropriate boundary conditions for the velocity and temperature ®elds are

 1 dp dU1 m  r 2  ˆ ‡  U1 ‡ B0 U1 :   q dx K q dt

…8†

We now introduce the dimensionless variables, as follows: up u v V0 y  U ; U1 ˆ 1 ; Up ˆ ; vˆ ; yˆ ; uˆ U0 V0 m U0 U0 m t V 2 T T1 n m xˆ x ; t ˆ 0 ; h ˆ ; nˆ 2; U0 V0 m Tw T1 V0 K  V02 V02  Kˆ 2 ; jˆ 2 j ; m m mqCp m is the Prandtl number; ˆ a k rB2 m M ˆ 02 is the magnetic field parameter; qV0 mbf g…Tw T1 † is the Grashof number: Gˆ U0 V02 Pr ˆ

…9†

Furthermore, the spin-gradient viscosity c which gives some relationship between the coecients of viscosity and micro-inertia, is de®ned as     K  1  cˆ l‡ j ˆ lj 1 ‡ b ; …10† 2 2 where b denotes the dimensionless viscosity ratio, de®ned as follows: bˆ

K l

…11†

in which K is the coecient of gyro-viscosity (or vortex viscosity). In view of Eqs. (7)±(11), the governing equations (2)±(4) reduce to the following non-dimensional form: ou ot

…1 ‡ eA ent †

ou dU1 o2 u ‡ …1 ‡ b† 2 ‡ Gh ˆ dt oy oy ox ; ‡ N …U1 u† ‡ 2b oy

…12†

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Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

ox 1 o2 x ; ˆ oy g oy 2

…13†

oh 1 o2 h ˆ ; oy Pr oy 2

…14†

ox ot

…1 ‡ eA ent †

oh ot

…1 ‡ eA ent †

where   1 ; Nˆ M‡ K



lj 2 : ˆ 2‡b c

The boundary conditions (5) and (6) are then given by the following dimensionless form: u ˆ Up ;

h ˆ 1 ‡ e ent ;

u ! U1 ;

h ! 0;

ox ˆ oy

x!0

2

ou oy 2

at y ˆ 0;

as y ! 1:

…15† …16†

3. Solution In order to reduce the above system of partial differential equations to a system of ordinary di€erential equations in dimensionless form, we may represent the linear and angular velocities and temperature as u ˆ u0 …y† ‡ e ent u1 …y† ‡ O…e2 † ‡   

…17†

x ˆ x0 …y† ‡ e ent x1 …y† ‡ O…e2 † ‡   

…18†

h ˆ h0 …y† ‡ e ent h1 …y† ‡ O…e2 † ‡   

…19†

Substituting Eqs. (17)±(19) in Eqs. (12)±(14) and equating the harmonic and non-harmonic terms, neglecting the coecient of O…e2 †, we get the following pairs of equations for …u0 ; x0 ; h0 † and …u1 ; x1 ; h1 †. …1 ‡ b†u000 ‡ u00

Nu0 ˆ

…1 ‡ b†u001 ‡ u01

…N ‡ n†u1

ˆ

…N ‡ n†

Au00

N

Gh1

2bx00 ;

Gh0

h000

‡

gx01

‡

Prh00

2bx01 ;

h001 ‡ Prh01

Agx00 ;

…23†

ˆ 0;

…24†

nPrh1 ˆ APrh00 :

…25†

Here primes denote di€erentiation with respect to y. The corresponding boundary conditions can be written as: u 0 ˆ Up ;

u1 ˆ 0;

x00 ˆ

where

s# " g 4n h1 ˆ 1‡ 1‡ ; 2 g h pi 1 1 ‡ 1 ‡ 4N …1 ‡ b† ; h2 ˆ 2…1 ‡ b† h pi 1 1 ‡ 1 ‡ 4…N ‡ n†…1 ‡ b† ; h3 ˆ 2…1 ‡ b† r! Pr 4n h4 ˆ 1‡ 1‡ ; 2 Pr

and a1 ˆ Up a2 a3 b1 b2

…21† …22†

ngx1 ˆ

u0 …y† ˆ 1 ‡ a1 e h2 y ‡ a2 e Pry ‡ a3 e gy ; u1 …y† ˆ 1 ‡ b1 e h1 y ‡ b2 e h2 y ‡ b3 e h3 y ‡ b4 e h4 y ‡ b5 e Pry ‡ b6 e gy ; x0 …y† ˆ c1 e gy ; Ag c1 e gy ; x1 …y† ˆ c2 e h1 y n h0 …y† ˆ e Pry ;  A h1 …y† ˆ e h4 y ‡ Pr e h4 y e Pry ; n

b3

x000 ‡ gx00 ˆ 0; x001

…20†

The solutions of Eqs. (20)±(25) with satisfying boundary conditions (26) and (27) are given by

u000 ;

h0 ˆ 1;

h1 ˆ 1 at y ˆ 0

u0 ˆ 1;

u1 ˆ 1;

x0 ! 0;

h0 ! 0;

h1 ! 0

as y ! 1:

x01 ˆ

x1 ! 0;

u001 ;

…26†

…27†

b4 b5

1

a2 a3 ; G ˆ ; …1 ‡ b†Pr2 Pr N 2bg c1 ; ˆ …1 ‡ b†g2 g N 2bh1 ˆ c2 ; …1 ‡ b†h21 h1 …N ‡ n† Ah2 a1 ; ˆ n 2bh1 ˆ 2 …1 ‡ b†h1 ‡ …2b 1†h1 2bh23 …N ‡ n†   …1 ‡ 3b†h21 h1 …N ‡ n† k1 ; k2 ‡  2bh h1  1  APr 1 ˆ G 1‡ ; n …1 ‡ b†h24 h4 …N ‡ n† APrG 1 ; ˆ n …1 ‡ b†Pr2 Pr N 2

Aga3 f2bAg gc1 n ; …1 ‡ b†g2 g …N ‡ n† …1 ‡ b†g 1 fN =gg ˆ …1 b†g2 g N ‡ 2bh22      Up 1 h22 ‡ Pr2 h22 a2 ;  1 ˆ k1 ‡ b1 h21 ‡ b3 h23 ; h1 Ag2 c1 ‡ b2 h22 ‡ b4 h24 ‡ b5 Pr2 ‡ b6 g2 ; ˆ n ˆ …1 ‡ b2 ‡ b4 ‡ b5 ‡ b6 †:

b6 ˆ c1

c2 k1 k2

…28† …29† …30† …31† …32† …33†

Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

By virtue of Eqs. (17)±(19), we obtain the streamwise and angular velocities and temperature, as follows: u…y; t† ˆ 1 ‡ a1 e h2 y ‡ a2 e Pry ‡ a3 e gy  ‡ e ent 1 ‡ b1 e h1 y ‡ b2 e h2 y ‡ b3 e  ‡ b4 e h4 y ‡ b5 e Pry ‡ b6 e gy ; x…y; t† ˆ c1 e

h…y; t† ˆ e

Pry

gy

 ‡ e ent c2 e

 ‡ e ent e

h4 y

h1 y

‡

Ag c1 e n

A Pr e n

h4 y

gy

…34†

 ;

e

Pry

…35† 

 :

…36†

Given the velocity ®eld in the boundary layer, we can now calculate the skin friction at the wall of the plate, which is given by sw ou sw ˆ ˆ oy yˆ0 qU0 V0 ˆ …1

We can also calculate the heat transfer coecient in terms of the Nusselt number, as follows: …oT =oy  †w ; Tw T1 oh ˆ oy yˆ0  A 2 Pr ˆ Pr ‡ e ent n

Nu ˆ x

h3 y

Up †h2 ‡ …h2 Pr†a2 ‡ …h2 g†a3 e ent ‰b1 h1 ‡ b2 h2 ‡ b3 h3 ‡ b4 h4 ‡ b5 Pr ‡ b6 gŠ: …37†

Fig. 2. Distributions of velocity and temperature pro®les of Newtonian ¯uid across the boundary layer for a stationary vertical porous plate in the absence of magnetic ®eld.

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Nu Rex 1

…38†

  A h4 1 ‡ Pr ; n

…39†

where Rex ˆ V0 x=m is the Reynolds number. 4. Results and discussion The formulation of the e€ect of magnetic ®elds and suction velocity varying exponentially with time about a non-zero constant mean value on the ¯ow and heat transfer of an incompressible polar conducting ¯uid along a semi-in®nite vertical porous moving plate has been carried out in the preceding sections. This enables us to carry out the numerical computations for the velocity and temperature for various values of the ¯ow and material parameters. In the present study the boundary condition for y ! 1 is replaced by identical ones at ymax which is a suciently large value of y where the velocity pro®le u approaches the relevant free stream velocity. We choose ymax ˆ 6 and a step size Dy ˆ 0:001. The general distributions of velocity and temperature pro®les of Newtonian ¯uids across the boundary layer for a stationary porous plate in the case of absence of magnetic ®eld are shown in Fig. 2. In Figs. 3±10 we have prepared some graphs of the velocity and angular velocity pro®les for polar ¯uids with the ®xed ¯ow and material parameters, n; t; A; ; M; K; G; Pr and Up which are listed in the ®gure captions. The e€ect of viscosity ratio b on the velocity and angular velocity for a stationary porous plate in the absence of magnetic ®eld is presented in Fig. 3. The numerical results show that the velocity distribution is lower for a

Fig. 3. Velocity and angular velocity pro®les against spanwise coordinate y for di€erent values of viscosity ratio b.

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Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

Fig. 4. Surface angular velocity versus viscosity ratio for di€erent values of magnetic parameter M.

Fig. 5. Velocity pro®les against spanwise coordinate y for di€erent values of dimensionless exponential index n .

Newtonian ¯uid …b ˆ 0† with the ®xed ¯ow and material parameters, as compared with a polar ¯uid when the viscosity ratio is less than 0.5. When b takes values larger than 0.5, however, the velocity distribution shows a decelerating nature near the porous plate. In addition,

the angular velocity distributions do not show consistent variations with increment of viscosity ratio parameter. In order to elucidate these behaviors, we calculate the surface angular velocity on the stationary porous plate versus viscosity ratio b for di€erent values of the magnetic parameter M in Fig. 4, where it is seen that the critical value of viscosity ratio exists. This value tends to increase as the magnetic ®eld parameter increases. For the case of a stationary porous plate, velocity pro®les against the spanwise coordinate y for di€erent values of dimensionless exponential index n are shown in Fig. 5. It is seen that when approaching n ! 0 , the magnitude of the velocity distribution across the boundary layer increases, and then decays to the relevant free stream velocity. However, the velocity distribution decreases as the exponential index n approaches the n ! 0‡ value. Fig. 6 illustrates the variation of velocity and angular velocity distribution across the boundary layer for various values of the plate velocity Up in the direction of ¯uid ¯ow. The peak value of velocity across the boundary layer decreases near the porous plate as the plate velocity increases. However, the values of angular velocity on the porous plate are increased as the plate velocity increases. For di€erent values of the magnetic ®eld parameter M, the velocity and angular velocity pro®les are plotted in Fig. 7. It is obvious that the e€ect of increasing values of magnetic ®eld parameter results in a decreasing velocity distribution across the boundary layer. Furthermore, the results show that the values of angular velocity on the porous plate are decreased as M increases. Fig. 8 shows the velocity pro®les for di€erent values of the permeability parameter K. Clearly as K increases the velocity boundary layer tends to decrease, and then decays to the relevant free stream velocity. The velocity and angular velocity pro®les against spanwise coordinatey for di€erent values of Grashof number G are described in Fig. 9. It is observed that an

Fig. 6. Velocity and angular velocity pro®les against spanwise coordinate y for di€erent values of plate moving velocity Up .

Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

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Fig. 7. Velocity and angular velocity pro®les against spanwise coordinate y for di€erent values of magnetic parameter M.

Fig. 8. Velocity pro®les against spanwise coordinate y for di€erent values of permeability K.

increase in G leads to a rise in the values of velocity, but decreases due to angular velocity. Here the positive value of G corresponds to a cooling of the surface by natural convection. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the porous plate as the Grashof number increases, and then decays to the free stream velocity.

Fig. 10(a) shows the velocity pro®les against spanwise coordinate y for di€erent values of Prandtl number Pr. The numerical results show that the e€ect of increasing values of Prandtl number results in a decreasing velocity, and then approaches a constant value of about 1.3 which is relevant to the free stream velocity at the edge of boundary layer. The results also reveal that the peak value of velocity decreases as Pr decreases. Typical variations of the temperature pro®les along the spanwise coordinate are shown in Fig. 10(b) for di€erent values of Prandtl number Pr . The numerical results show that an increase of Prandtl number results in a decreasing thermal boundary layer thickness and more uniform temperature distribution across the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to di€use away from the heated surface more rapidly than for higher values of Pr. Hence the boundary layer is thicker and the rate of heat transfer is reduced, for gradients have been reduced. As shown in Fig. 11, it has been observed that for a constant suction velocity parameter A with given ¯ow and material parameters, the e€ect of increasing values of plate moving velocity Up results in a decreasing

Fig. 9. Velocity and angular velocity pro®les against spanwise coordinate y for di€erent values of Grashof number G.

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Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799

Fig. 10. Velocity and temperature pro®les against spanwise coordinate y for di€erent values of Prandtl number Pr.

eral values of Prandtl number. Numerical results show that for given ¯ow and material parameters which are listed in the ®gure caption, the surface heat transfer from the porous plate tends to decrease slightly on increasing the magnitude of suction velocity.

5. Conclusions

Fig. 11. Variation of the surface skin friction with the plate moving velocity Up for various suction velocity parameters A.

Fig. 12. Variation of the surface heat transfer with the suction velocity parameter A for di€erent values of Prandtl number Pr.

surface skin friction on the porous plate. It is dicult to show clearly the corresponding pro®les of surface skin friction due to very little variation. It is also evident that for di€erent values of the suction velocity parameter A, the surface skin friction has zero value near Up ˆ 1:2: Fig. 12 illustrates the variation of surface heat transfer with the suction velocity parameter A for sev-

We have examined the governing equations for an unsteady, incompressible polar ¯uid past a semi-in®nite porous moving plate whose velocity is maintained at a constant value, and embedded in a porous medium and subjected to the presence of a transverse magnetic ®eld. The method of solution can be applied for small perturbation approximation. Numerical results are presented to illustrate the details of the ¯ow and heat transfer characteristics and their dependence on the material parameters. We observe that, when the magnetic parameter increases the velocity decreases, whereas when the permeability parameter or Grashof number increases the velocity increases. It is recognized that there are many other methods that could be considered in order to describe some reasonable solutions for this particular type of problem. For a better understanding of the thermal behavior of this work, however, it may be necessary to perform the experimental works. In the near future, we would be glad to compare these theoretical results with those obtained by anyone in the same ®eld.

References [1] V.M. Soundalgekar, H.S. Takhar, MHD forced and free convective ¯ow past a semi-in®nite plate, AIAA. J. 15 (1977) 457±458. [2] E.L. Aero, A.N. Bulygin, E.V. Kuvshinskii, Asymmetric hydromechanics, J. Appl. Math. Mech. 29 (2) (1965) 333± 346.

Y.J. Kim / International Journal of Heat and Mass Transfer 44 (2001) 2791±2799 [3] N.V. Dep, Equations of a ¯uid boundary layer with couple stresses, J. Appl. Math. Mech. 32 (4) (1968) 777±783. [4] G. Lukaszewicz, Micropolar Fluids ± Theory and Applications, Birkhauser, Boston, 1999. [5] H.S. Takhar, O.A. Beg, E€ects of transverse magnetic ®eld Prandtl number and Reynolds number on non-Darcy mixed convective ¯ow of an incompressible viscous ¯uid past a porous vertical ¯at plate in a saturated porous medium, Int. J. Eng. Res. 21 (1997) 87±100. [6] M. Kumari, MHD ¯ow over a wedge with large blowing rates, Int. J. Eng. Sci. 36 (3) (1998) 299±314. [7] R.J. Gribben, The magnetohydrodynamic boundary layer in the presence of a pressure gradient, Proc. R. Soc. London A 287 (1965) 123±141. [8] H.S. Takhar, P.C. Ram, Free convection in hydromagnetic ¯ows of a viscous heat-generating ¯uid with wall temperature and Hall currents, Astrophys. Space Sci. 183 (1991) 193±198.

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[9] H.S. Takhar, P.C. Ram, Magnetohydrodynamic free convection ¯ow of water at 4°C through a porous medium, Int. Comm. Heat Mass Transfer 21 (1994) 371±376. [10] V.M. Soundalgekar, Free convection e€ects on the oscillatory ¯ow past an in®nite, vertical, porous plate with constant suction, Proc. R. Soc. London A 333 (1973) 25±36. [11] A.A. Raptis, N. Kafousias, Heat transfer in ¯ow through a porous medium bounded by an in®nite vertical plate under the action of a magnetic ®eld, Int. J. Energy Res. 6 (1982) 241±245. [12] A.A. Raptis, Flow through a porous medium in the presence of a magnetic ®eld, Int. J. Energy Res. 10 (1986) 97±100. [13] T.G. Cowling, Magnetohydrodynamics, Interscience Publishers, New York, 1957. [14] K. Yamamoto, N. Iwamura, Flow with convective acceleration through a porous medium, J. Eng. Math. 10 (1976) 41±54.