Effects of variable viscosity and thermal conductivity on free convective heat and mass transfer flow with constant heat flux through a porous medium

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  with the plane transverse to the plate. A similarity parameter  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.

Soundalgekar

¹

studied free convection

effects on the stokes problem for an

infinite vertical plate, when it is cooled

or heated by the free convec tion

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

,

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

Rahman and Sattar

³

, Yih

⁴

, In the above stated papers, the diffusion-thermo term and thermal-diffusion term were ne glec t e d f rom t he e ne rgy a nd concentration equations respectively.

Kafoussias and Williams

⁷

studied thermal- diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with t em pe ra ture de pe ndent vis cosity.

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 convec- tion 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, aero- nautics and engineering many researchers studied the effects of magnetohy- drodynamics (MHD) free convection flow : some of them are Aboeldahab and Elbarbary

¹²

, Megahead et al.

¹³

. Sattar and Hussain

⁵

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 Alam

⁶

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 con- centration 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 the rma l conductivity on free convective heat and mass transfer flow a nd La i a nd Kula c k i

¹⁴

proba bly presented the expression for these two terms.

Mathematical Analysis

We consider an ele ctrically 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 y- axis is normal to it. At time t > 0, the temperature and the species concen- tration at the plate are raised to T

w

and C

w

, T



and C



being the temperature

and spe cies conc entration of the

uniform flow, and thereafter maintained

constant. Following Ram

⁸

, a strong

ma gnetic f ie ld B is im pos ed in a

direction that makes an angle  with

the plane transvers e to t he plate

which is assumed to be electrically non-conducting, such that B= (0,B

0

,

(1-

²

)B

0

) 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, Shercliff

¹⁰

. The plate is assumed to be non-con- ducting hence J

y

= 0 at the plate and hence zero everywhere. We have from Ohm’s law neglecting electron pressure and ion slip :

), 1

^{2 2}

(

0



  m u w

m B

J

_x

p

^e









) 1 (

J

_{z 2}⁰ ₂



  u m

m B p

_e

 





where, m=

e



e

is the Hall parameter.

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 assump- tions and Boussinesq's approximation, t he gove rning equa t ions of t he problem are :

) t (

u

2 0 2



 

 







 



 





T



T y g

u y

y u y

v u   

- -

) ) (

1 ) (

(

*

₂⁰₂

2 2

0



 







_

u m

m B p k C u C

g

^e











 

(1)

0

 

 y

 ₍₂₎

t w

2 2







 



 



 





y y w y

w y

v w 



) ) (

1

(

^{2 2}

0 2 2

 

 m w

m B p

e

 







(3 ) 1

t T

2 2







 



 



 





y y T C y

T C k y v T

p p







2 2

















 



 





 

 



 





 

y w y

u C

_p

 (4 )

1

t C

2 2

y y C Sc y

C Sc y v C







 



 



 



  

(5 )

With the boundary conditions

make order to in introduced is

parameter simlarity

A

y

as , C C , T T 0, w 0, u

0 y a t , C c C ,

0, w 0,

u

_w





 





























 

 







k q y T

(6) 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

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

-v

v 

₀





(8) Where v

₀

is the suction parameter Now, we introduce the following non- dimensiona 1 quantities

U ) w g(

U , ) u f(

y ,

0 0





  

 

(9) C

C - ) C ( ) , ) (

(

w w

C q

T T k





 

   

 



(10) Where U

₀

is a constant velocity.

Viscosity and thermal conductivity of fluid are inverse linear functions of emperature

¹⁴

, so

) ] (

1 k [ k 1 )], (

1 1 [ 1

1

T T

T

T 

 

 

_

 

^



_

 

^

- k ,

r r c

c

k













 

^{ }

 

 



(11)

Introducing equations (7) to (10) in equations (1), (3), (4) and (5), we have the following non-dimensional equa- tions

0





















f f dt v

f d

c c c

c

    

 

 

 



 











 f Gr Gc

c c c

c

c

  

 



 

)

1

^{2 2}

( 



  f mg

M m

f 

 

 (12)

0

 























g g dt v

g d

c c

c c

c



 





 

 





)

1

^{2 2}

( 



  f mg

M m f

f 

 



 (1 3 )

0





 



 





 









 





 





 

 c c

c

Sc dt v

Sc d

(14)

Where

g ,

Gr

0 3



0

 





k v U

q

,

) , (

Gc g

2

0 2

*

0

 



^



 





k

v U

C C

q

_w

, Pr

p ,

M

2 2 0 2

e



 













k

v C v

B

p

Sc v ,

Ec U

2

0



 

_



D

The corresponding boundary conditions are

0 at -1 1, 0, g ,

0     

   

f (15)

as 0 0, 0, g 0,

f          ⁽¹⁶⁾

The similarity condition require that

2 that

dt d  







(17) following the works of Sattar and Hussain.

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 pa ram et er and the mas s t ra nsf er parameter.

In fig. 1 the primary velocity is presented for the viscosity parameter

c=-1,-3, 9 and -20. The value of the

166 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 incre- ases 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 obser- vations under boundary conditions show that he fluid temperature decreases with the inc rea se of t he the rm al 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 obse rve that the t emperature 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 sub- stitutions of various values of the para- meters it is observed that the concen- tration 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

c. This has shown that the concen- tration 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 7

^th

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 para- meter Sc=1, 2, 4, 10 and asymp- totically 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

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

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

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