Cmjv01i02p0163 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(2), 163-170 (2010).

Effects of variable viscosity and thermal conductivity on free convective heat and mass transfer flow with constant heat flux through a porous medium UTPAL SARMA and Dr. G.C.HAZARIKA Dibrugarh University, Dibrugarh (India)

ABSTRACT The MHD flow has been subjected to a porous vertical plate with Hall current and constant heat flux. A uniform magnetic field also applied which makes an angle ď Ą with the plane transverse to the plate. A similarity parameter ď ł has been introduced and the suction velocity is inversely proportional to this time dependent parameter. The non-linear partial differential equations are transformed in to ordinary differential equations with the help of similarity substitutions. Finally the equations are solved by applying Runga-Kutta shooting algorithm. The effects of various parameters i.e. viscosity parameter, thermal conductivity parameter and mass transfer parameter are displayed graphically. Key words: Variable viscosity, thermal conductivity, Hall current, constant heat flux. AMS N0.Fluid Mechanics-76D10

INTRODUCTION The hydrodynamic flow of a viscous incompressible fluid past an impulsively started infinite horizontal plate was studied by Stokes 15, and because of its practical importance this problem was extended to bodies of different shapes by various authors.

Soundalgekar1 studied free convection effects on the stokes problem for an infinite vertical plate, when it is cooled or heated by the free convection currents. Many of the researchers studied the effects of heat and mass transfer on magneto hydrodynamics (MHD) free convection flow: some of them are Raptis and Kafoussias 2 ,

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

164

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170 (2010).

Rahman and Sattar3, Yih4, In the above stated papers, the diffusion-thermo term and thermal-diffusion term were neglected from the energy and concentration equations respectively. Kafoussias and Williams7 studied thermaldiffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Recently, Takhar et al.8 studied unsteady free convection flow over an Infinite porous plate due to the combined effects of thermal and mass diffusion, magnetic field and Hall currents. Very recently, Postelnicu 9 studied numerically the influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. In the light of the applications of the flows arising from differences in concentration in geophysics, aeronautics and engineering many researchers studied the effects of magnetohydrodynamics (MHD) free convection flow : some of them are Aboeldahab and Elbarbary12, Megahead et al.13. Sattar and Hussain5 studied the effects of mass transfer as well as the effects of Hall currents on an unsteady MHD free convection flow past an accelerated porous plate with time dependent temperature and concentration. Sattar and Alam6 have also studied the effects of heat and mass transfer as well as the effects of Hall current on the unsteady MHD free convection flow past an accelerated porous plate with

tie dependent temperature and concentration through a porous medium. Following the works of Sattar and Alam 6 our aim is to study the effects of variable viscosity and thermal conductivity on various parameters like velocity, temperature and mass transfer on free convective heat and mass transfer flow through a porous medium with Hall current and constant heat flux. The aim of the present paper is to study the effects of variable viscosity and thermal conductivity on free convective heat and mass transfer flow and Lai and Kulacki 14 probably presented the expression for these two terms. Mathematical Analysis We consider an electrically conducting viscous incompressible fluid through a porous medium along an infinite vertical porous plate (y=0) with the effects of Hall current. The flow is also assumed to be in the x- direction which is taken along the plate in the upward direction and yaxis is normal to it. At time t > 0, the temperature and the species concentration at the plate are raised to T w and Cw, Tď&#x201A;Ľ and Cď&#x201A;Ľ being the temperature and species concentration of the uniform flow, and thereafter maintained constant. Following Ram 8, a strong magnetic field B is imposed in a direction that makes an angle ď Ą with the plane transverse to the plate

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170 (2010). which is assumed to be electrically non-conducting, such that B= (0,B0, (1-2)B0) where  = cos . Thus if =1 the imposed magnetic field is parallel to the y-axis and if =o the magnetic field is parallel to the plate. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is negligible compared to the applied magnetic field and the magnetic lines of force are fixed relative to the fluid, Shercliff10. The plate is assumed to be non-conducting hence Jy= 0 at the plate and hence zero everywhere. We have from Ohm’s law neglecting electron pressure and ion slip :

Jx 

pe B0  ( m u  w ), 1  m 2 2

 0 y

165

(2) 2

w w  w w  v   t y y 2  y y 2

p eB 0  ( m   w )  (1  m 2  2 )

T t

v

T y





 Cp

 2T

 C p y



(3)

 T 

 C p y y

   u  2  w  2            y    y  

C C  2C 1  C  v   2 t y Sc  y Sc  y  y

(4)

(5)

With the boundary conditions T q    , C  c  C w , at y  0  y k    u  0, w  0, T  T  , C  C  , as y     u  0, w  0,

pe B0  Jz  ( u  m ) 1  m 2 2 where, m=ee is the Hall parameter.

A simlarity parameter is introduced

It is assumed that the plate is infinite in extent and hence all physical quantities depend on y and t. Thus in accordance with the above assumptions and Boussinesq's approximation, the governing equations of the problem are :

in order to (6) make

A simlarity parameter is introduced in order to make the equations (1) to (5) similar as follows  =  (t) (7) Where,  is in fact a time dependent length

u u u   u v    2  g 0 (T  T  ) t y y y y  u p 2 e B 2 0   g 0  * (C  C  )   2 2 (u  m ) k  (1  m  ) (1) --

scale so that the governing equations could be transformed in to a similar form in time. Using this length scale the solution of Equation (2) is considered to be

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

166

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170 (2010).

 (8)  Where v0 is the suction parameter v  -v 0

Where

Gr 

g0  q 3 , U 0v  k

Gc 

g 0 * q 2 (C w  C ) , U 0 v

M

pe B0  2 , v 

Now, we introduce the following nondimensiona 1 quantities 

y u w , f( )  , g( )   U0 U0

 ( ) 

k (T  T  ) , q

 ( ) 

C -C w C w  C

(9)

(10)

Where U0 is a constant velocity. Viscosity and thermal conductivity of fluid are inverse linear functions of emperature14, so 1 1  [1   (T  T )],  

    c ,  c

1 k [1   1 (T  T ) ] k

(11)

k  k-  r  r

U Ec  0 , v

Sc 

Pr 



2 , k

 C p v k

 D

The corresponding boundary conditions are

f  0, g  0,   1,    -1 at   0

(15)

f  0, g  0,   0,   0 as    (16)

Introducing equations (7) to (10) in The similarity condition require that equations (1), (3), (4) and (5), we have  d the following non-dimensional equa-that 2 (17)   dt tions following the works of Sattar and     c d    c  f  v0 f   f  Hussain.   dt  



   c c f   Gr   Gc    c c   c  f  M

 g

 ( f  mg  ) (12) 1  m22

    c d    c   v0 g   g     c dt c   c

f   f  M Sc  

 ( f  mg  ) (13) 1  m 2 2

 d    c      v0   Sc       dt c   c 

(14)

RESULTS AND DISCUSSION The velocity profiles for x and z components of velocity commonly known as primary and the secondary are shown for different values of viscosity parameter, thermal conductivity parameter and the mass transfer parameter. In fig. 1 the primary velocity is presented for the viscosity parameter c=-1,-3, 9 and -20. The value of the

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170 (2010). Grashof number Gr=0.1, modified Grashof number Gc=0.1, =0.1 and magnetic Reynolds number M=0.3 has been taken. On substitution of these various values of the parameter it is observed that the fluid velocity increases with the increase of viscosity parameter c. Infig. 2 the secondary velocity profile is presented for viscosity parameter c=-0.2,-0.5,-1,-2.6. The values of =0.1 Hall parameter m=0.1 has been taken. Here it is also observed that the secondary velocity profile of the fluid increases with the decrease of the viscosity parameter c. In fig. 3 the temperature profile of the fluid is presented for the thermal conductivity parameter r =-1.1,-2.2,-4.4,-10, Pr=0.7Ec=0.1 and c=-10.The observations under boundary conditions show that he fluid temperature decreases with the increase of the thermal conductivity parameter r. In the forth figure it is observed the effect of Prandtl number Pr on the temperature profile. Substituting values for Pr=3.8, 4.9, 6.6, 9.9 and m=0.6, M=2, c =-10 we observe that the temperature profile asymptotically approaches the X-axis and the profile increases while the Prandtl number decreases. In fig. 5 the fluid concentration is presented for viscosity parameter c=-1,-2,-3,-4 and -10, Sc=1and =0.1. On substitutions of various values of the parameters it is observed that the concentration profile of the fluid decreases as the mass transfer parameter increase. In the fig. 6 the concentration profile has been observed for the changing values of the variable viscosity parameter

167

c. This has shown that the concentration profile decreases for increasing values of the viscosity parameter c when we introduce various values of the parameters like Pr=.73, r =-10, M=2, m=.1, E=5,  =0.5. In the 7th fig. the concentration profile for various values of the mass transfer parameter Sc has been observed. We introduce different values of the parameters like r =-10, c=-10,m=.2, M=3, E=1, Pr=.73. The observations under boundary conditions show that the concentration profile decreases with the increase of the mass transfer parameter Sc=1, 2, 4, 10 and asymptotically approaches to the X-axis. In the fig. 8 the observations has been made for the primary velocity profile with the variations of the thermal conductivity parameter r. And it is observed that for the values of c=10,m=.1,M=3,Pr=.73,  =0.1 and r =-1,-3,-20 the velocity profile decreases for the increasing value of the thermal conductivity parameter r. In the fig. nine we observe the effects of the thermal conductivity parameter r on the velocity profile. Substituting the values of r=-1,-3,6,-20; c=-10, m=.1, M=.1, E=1, Pr=.73 it was found that the velocity profile decreases with the increase of the thermal conductivity parameter r. From the above analysis we may conclude that for accurate results on Heat and mass transfer problem of MHD free convective flow through a porous medium along a porous medium along

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

168

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170

a porous vertical plate with Hall current and constant heat flux the effects of variable viscosity and thermal conductivity must be taken in to account.

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

(2010).

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170 (2010).

169

R E FE R E N C ES 1. Soundalgekar, V.M., Free Convection Effects on the Stokes Problem for an Infinite Vertical Plate, ASME J. Heat Transfer Vol. 99, pp.499-501, (1911). 2. Raptis, A. and Kafoussias, N. G., Magnetohydrodynamic Free Convection Flow and Mass Transfer Through Porous Medium Bounded by an Infinite Vertical Porous Plate with Constant Heat Flux, Can. J. Phys., Vol. 60, pp. 1725 - 1729, (1982). 3. Rahman, M. M. and Sattar, M. A., MHD Free Convection and Mass Transfer Flow with Oscillatory Plate Velocity and Constant Heat Source in a Rotating Frame of Reference, Dhaka Univ. J. Sci., Vol.a9(1), pp. 63-73 (1999). 4. Yih, K. A., Free Convection Effect on MHD Coupled Heat and Mass Transfer of a moving Permeable Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

170

Utpal Sarma et al., J.Comp.&Math.Sci. Vol.1(2), 163-170 (2010). Vertical Surface, Int. Comm. Heat Mass Transfer, Vol. 26, pp.95-104. (1999). 17] Sattar M.A. and Hossain M.M. Unsteady Hydromagnetic Free Convection Flow with Hall Current and Mass Transfer Along an Accelerated Porous Plate with Time Dependent Temperatrure and ConcentrationC, an. J. Phys., Vol. 70, pp. 369-374 (1992). Sattar, M. A., Rahman M. M. and Alam, M.M., Free Convection Flow and Heat Transfer Through a Porous Vertical Flat Plate Immersed n a Porous Medium with Variable Suction, J. Energy Heat and Mass Transfer, Yol. 22, pp.17-21 (2000). Kafoussias, N. G. and Willaiams, E.W., Thermal-diffusion and Diffusion-thermo Effects on Mixed Free Forced Convective and Mass Transfer Boundary Layer Flow with Temperature Dependent viscosity, Int. J. Engng, Sci., Vol. 33, pp. 13691384 (1995). Anghel, M., Takhar, H. S. and Pop, I., Dufour and Soret Effects on Free convection Boundary Layer Over a Vertical Surface Embedded in a Porous Medium, Studia Universitatis Babes-Bolyai, Mathematica Vol. XLV, pp.ll-21 (2000). Postelnicu, A., Influence of a Magnetic Field on Heat and Mass

10.

11.

12.

13.

14.

15.

Transfer by Natural Convection from Vertical Surfaces in porous Media Considering Soret and Dufour Effects, Int. J. Heat Mass Transfer, Yol.4â&#x20AC;&#x2122;7, pp. 1467-1472, (2004). Shercliff, J.A., A textbook of magneto hydrodynamics, pergamon press Inc.,New York (1965). Schlichting, H., Boundary layer theory, Mc. Graw Hill Book Co. Inc. New York. (1968). Aboeldahab, E. M. and Elbarbary, E. M. E., Hall Current Effect on Magnetohydrodynamic Free Convection Flow Past Asemi-infinite Vertical Plate with Mass Transfer, Int. J. Engng. Sci., Vol. 39, pp. 1641-1652 (2001). Megahead, A. A., Komy, S. R. and Afify, A. A., Similarity Analysis in Magnetohydrodynamics Hall Effects on Free Convection Flow and Mass Transfer Past a Semi-infinite Vertical Flat Plate, Iner. Jour. Non-linear Mecha., Vol. 38, pp. 513-520 (2003). Lai, F.C., Kulacki F.A., Int. J. Heat and Mass Transfer, Vol. 33, pp 10281032 (1990). Stokes, G. G., On the Effects of the Internal Friction of Fluids on the Motion of Pendulum. Trans. Combr. Phil. Soc., Vol. 9, pp.8-106, 1 856.

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)