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Effects of variable viscosity and thermal conductivityon free convective heat and mass transfer

flow with constant heat flux througha porous medium

UTPAL SARMA andDr. 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 anglewith the plane transverseto the plate. A similarity parameter has been introduced andthe 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

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

INTRODUCTION

The hydrodynamic flow of a

viscous incompressible fluid past an

impulsively started infinite horizontal

plate was studied by Stokes15, and

because of its practical importance this

problem was extended to bodies of

different shapes by various authors.

Soundalgekar1studied 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 Kafoussias2,

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

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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 Williams7studied thermal-

diffusion and diffusion-thermo effectson mixed free-forced convective and

mass transfer boundary layer flow with

temperature dependent viscosity.

Recently, Takhar et al.8studied 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,

Postelnicu9 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 Elbarbary12, Megahead et al.13.

Sattar and Hussain5

studied the effectsof 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 Alam6have 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 Alam6 our aim is to study the

effects of variable viscosity and thermal

conductivity on various parameters likevelocity, 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 Kulacki14 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 y-axis is normal to it. At time t > 0, the

temperature and the species concen-

tration at the plate are raised to Twand Cw, Tand Cbeing the temperature

and species concentration of the

uniform flow, and thereafter maintained

constant. Following Ram8, a strong

magnetic field B is imposed in a

direction that makes an angle withthe plane transverse to the plate


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Utpal Sarma et al., J .Comp.&Math.Sci. Vol.1(2), 163-170 (2010). 165

which is assumed to be electrically

non-conducting, such that B= (0,B0,(1-2)B0) where = cos . Thus if= 1 the imposed magnetic f ield isparallel to the y-axis and if = o themagnetic field is parallel to the plate.

The magnetic Reynolds number of theflow 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-con-

ducting hence J y= 0 at the plate and

hence zero everywhere. We have from

Ohms law neglecting electron pressure

and ion slip :

),(1 22

0

wumm

BpJ ex

)(1

J22

0z mu

m

Bpe

w here, m=ee is the Hall parameter.

It is assumed that the plate isinfinite in extent and hence all physicalquantities depend on y and t. Thus in

accordance with the above assump-tions and Boussinesq's approximation,the governing equations of theproblem are :

)(t

u02

2

TTgy

u

yy

u

y

uv

)()1(

)(*22

022

0

mum

Bp

k

uCCg

e

(1)

0y

(2)

t

w2

2

yy

w

y

w

y

wv

)()1( 22

0

22

wm

mBp e

(3)

1

t

T2

2

yy

T

Cy

T

C

k

y

Tv

pp

22

y

w

y

u

Cp

(4)

1

t

C2

2

yy

C

Scy

C

Scy

Cv

(5)

With the boundary conditions

yas,CC,TT0, w0,u

0ya t,CcC,0, w0,uw

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


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-vv 0

(8)

Where v0is the suction parameter

Now, we introduce the following non-

dimensiona 1 quantities

U

w

)g(,U

u

)f(,

y

00 (9)

C

C-C)(,

)()(

w

w

Cq

TT

(10)

WhereU0 is a constant velocity.

Viscosity and thermal conductivity of

fluid are inverse linear functions of

emperature14, so

) ](1[k

1k)],(1[

111 TTTT

-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

ffvdt

df

c

c

c

c

GcGrf

c

c

c

c

c

)(1 22

mgf

mMf

(12)

0

ggvdt

dg

cc

c

c

c

)(

122

mgf

mMff

(13)

0

cc

c Scvdt

dSc

(14)

Where

,g

Gr0

30

kvU

q

,,)(g

Gc2

0

2*0

kvU

CCq w

,Pr,p

M22

0

2

e

k

vC

v

B p

Sc,v

UEc

2

0

D

The corresponding boundary conditions

are

0at-11,0,g,0 f (15)

as00,0,g0,f (16)

The similarity condition require that

2dt

d

(17)

f o l l ow ing the wo rks o f Sa t ta r and

Hussain.

RESULTS AND DISCUSSION

The velocity profiles for x andz components of velocity commonlyknown as primary and the secondaryare shown for different values ofviscosity parameter, thermal conductivityparameter and the mass transferparameter.

In fig. 1 the primary velocity ispresented for the viscosity parameterc= -1,-3, 9 and -20. The value of the



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Grashof number Gr=0.1, modifiedGrashof number Gc=0.1, = 0.1 andmagnetic Reynolds number M= 0.3 hasbeen taken. On substitution of thesevarious values of the parameter it isobserved that the fluid velocity incre-

ases with the increase of viscosityparameter c. Infig. 2 the secondaryvelocity profile is presented for viscosityparameter c=-0.2,-0.5,-1,-2.6. Thevalues of =0.1 Hall parameter m=0.1has been taken. Here it is also observedthat the secondary velocity profile ofthe fluid increases with the decreaseof the viscosity parameter c. In fig. 3the temperature profile of the fluid ispresented for the thermal conductivityparameter r = -1.1,-2.2,-4.4,-10,

Pr=0.7Ec=0.1 andc= -10.The obser-vations under boundary conditionsshow that he fluid temperature decreaseswith the increase of the thermalconductivity parameter r. In the forthfigure it is observed the effect of Prandtlnumber Pr on the temperature profile.Substituting values for Pr=3.8, 4.9,6.6, 9.9 and m= 0.6, M= 2, c = -10we observe that the temperatureprofile asymptotically approaches theX-axis and the profile increases whilethe Prandtl number decreases. In fig. 5the fluid concentration is presented forviscosity parameter c= -1,-2,-3,-4and -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 decreasesas the mass transfer parameter increase.In the fig. 6 the concentration profilehas been observed for the changingvalues of the variable viscosity parameter

c. This has shown that the concen-tration profile decreases for increasingvalues of the viscosity parameter cwhen we introduce various values ofthe parameters like Pr=.73, r = -10,M= 2, m=.1, E= 5, = 0.5. In the 7th

fig. the concentration profile for variousvalues of the mass transfer parameterSc has been observed. We introducedifferent values of the parameters liker =-10, c=-10,m=.2, M=3, E=1,Pr= .73. The observations underboundary conditions show that theconcentration profile decreases withthe increase of the mass transfer para-meter Sc= 1, 2, 4, 10 and asymp-totically approaches to the X-axis. Inthe fig. 8 the observations has been

made for the primary velocity profilewith the variations of the thermalconductivity parameter r. And it isobserved that for the values of c=-10,m=.1,M= 3,Pr= .73, = 0.1 andr = -1,-3,-20 the velocity profiledecreases for the increasing value ofthe thermal conductivity parameter r.

In the fig. nine we observe theeffects of the thermal conductivityparameter 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 velocityprofile decreases with the increase ofthe thermal conductivity parameter r.

From the above analysis we mayconclude that for accurate results on Heatand mass transfer problem of MHDfree convective flow through a porousmedium along a porous medium along

Utpal Sarma et al., J .Comp.&Math.Sci. Vol.1(2), 163-170 (2010). 167


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



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REFERENCES

1. Soundalgekar, V.M., Free Convection

Effects on the Stokes Problem foran Infinite Vertical Plate, ASME J .

Heat Transfer Vol. 99,pp.499-501,

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Magnetohydrodynamic Free Con-

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Through Porous Medium Bounded

by an Infinite Vertical Porous Plate

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(1982).3. Rahman, M. M. and Sattar, M. A.,

MHD Free Convection and Mass

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Velocity and Constant Heat Source

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Transfer of a moving Permeable

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13. Megahead, A. A., Komy, S. R. and

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