COMBINED HEAT AND MASS TRANSFER EFFECTSON MHD FREE CONVECTION FLOW PAST

AN OSCILLATING PLATE EMBEDDED IN POROUS MEDIUM

R. C. CHAUDHARY , ARPITA JAIN

Department of Mathematics, University of RajasthanJaipur-302004 (INDIA)

rcchaudhary@rediffmail.com

Received November 1, 2006

The objective of this paper is to study the MHD flow past an infinite verticaloscillating plate through porous medium, taking account of the presence of freeconvection and mass transfer. The governing equations are solved in closed form byLaplace-transform technique. The results are obtained for velocity, temperature,concentration, Nusselt number and skin-friction. The effects of various materialparameters are discussed on flow variables and presented by graphs.

Key words: Free convection, magnetohydrodynamic flows, porous medium,mass transfer, oscillating plate.

INTRODUCTION

Free convection flows are of great interest in a number of industrialapplications such as fiber and granular insulation, geothermal systems etc.Buoyancy is also of importance in an environment where differences betweenland and air temperatures can give rise to complicated flow patterns.Magnetohydrodynamic has attracted the attention of a large number of scholarsdue to its diverse applications. In astrophysics and geophysics, it is applied tostudy the stellar and solar structures, interstellar matter, radio propagationthrough the ionosphere etc. In engineering it finds its application in MHDpumps, MHD bearings etc. Convection in porous media has applications ingeothermal energy recovery, oil extraction, thermal energy storage and flowthrough filtering devices. The phenomena of mass transfer is also very commonin theory of stellar structure and observable effects are detectable, at least on thesolar surface. The study of effects of magnetic field on free convection flow isimportant in liquid-metals, electrolytes and ionized gases. The thermal physics ofhydromagnetic problems with mass transfer is of interest in power engineeringand metallurgy.

Free convection effects on flow past a vertical surface studied byVedhanayagam et al. [1], Martynenko et al. [2], Kolar et al. [3], Ramanaiah et al.

Rom. Journ. Phys., Vol. 52, Nos. 5–7 , P. 505–524, Bucharest, 2007

506 R. C. Chaudhary, Arpita Jain 2

[4] and Camargo et al. [5] with different boundary conditions. Many workerslike Revankar [6], Anwar [7] and Sahoo et al. [8] worked on hydromagneticnatural convection flow past a vertical surface.

Convective heat transfer through porous media has been a subject of greatinterest for the last three decades. Kim et al. [9] and Harris et al. [10] solved theproblem of natural convection flow through porous medium past a plate. Recently,Magyari et al. [11] have discussed analytical solutions for unsteady free convectionin porous media. The magnetic current in porous media considered by Raptiset al. [12] and Geindreau et al. [13].

Muthukumaraswamy et al. [14, 15] investigated mass diffusion effects onflow past a vertical surface. Mass diffusion and natural convection flow past aflat plate studied by researchers like Chandrasekhara et al. [16] and Panda et al.[17]. Magnetic effects on such a flow is investigated by Hossain et al. [18] andIsrael et al. [19]. Sahoo et al. [20] and Chamkha et al. [21] discussed MHD freeconvection flow past a vertical plate through porous medium in the presence offoreign mass.

Flows past a vertical plate oscillating in its own plane has many industrialapplications. The first exact solution of Navier-Stokes equation was given byStokes [22] which is concerned with flow of viscous incompressible fluid past anhorizontal plate oscillating in its own plane. Natural convection effects on Stokesproblem was first studied by Soundalgekar [23]. The same problem was consideredby Revankar [24] for an impulsively started or oscillating plate. Turbatu et al.[25] investigated the flow of an incompressible viscous fluid past an infiniteplate oscillating with increasing or decreasing velocity amplitude of oscillation.Recently, Gupta et al. [26] have analyzed flow in the Ekman layer on an oscillatingplate. Soundalgekar et al. [27] gave an exact solution for magnetic free convectionflow past an oscillating plate. Mass transfer effects on flow past a oscillatingplate considered by Lahurikar et al. [28].

This paper deals with the study of magnetic and mass diffusion effects onthe free convection flow, when the plate is made to oscillate with a specifiedvelocity. It is also assumed that the plate is embedded in porous medium.

FORMULATION OF THE PROBLEM

We consider unsteady, free convection two-dimensional flow of anincompressible and electrically conducting viscous fluid along an infinite non-conducting vertical flat plate through a porous medium. The x′ axis is takenalong the plate in the vertically upward direction and y′ axis is taken normal tothe plate. A magnetic field of uniform strength B0 is applied in the direction offlow and the induced magnetic field is neglected. Initially, the plate and the fluidare at same temperature T∞′ in a stationary condition with concentration level

3 Heat and mass transfer effects on MHD free convection 507

C∞′ at all points. At time t′ > 0 the plate starts oscillating in its own plane with avelocity U0cosω′t′. Its temperature is raised to wT ′ and the concentration level atthe plate is raised to .wC′ Using the Boussinesq approximation, the governingequations for the flow are given by

22 02 ( ) ( )C

B uu u ug T T g C Ct Ky ∞ ∞

′σ′ ′ ′∂ ∂ ν′ ′ ′ ′= ν + β − + β − − −′ ′∂ ρ′∂ (1)

2

2p

T Tt C y′ ′∂ κ ∂=′∂ ρ ′∂

(2)

2

2C CDt y′ ′∂ ∂=′∂ ′∂

(3)

The boundary conditions are given by

0, , for all , 0u T T C C y t∞ ∞′ ′ ′ ′ ′ ′ ′= = = ≤ (4)

0 cos , , at 0, 0

0, , as , 0w wu U t T T C C y t

u T T C C y t∞ ∞

′ ′ ′ ′ ′ ′ ′ ′ ′ ⎫= ω = = = > ⎪⎬

′ ′ ′ ′ ′ ′ ′= = = →∞ > ⎪⎭ (5)

Let us introduce the non-dimensional variables

2200 0

20

202 20 0

3 30 0

, , ,

, Pr ,

( ) ( )Gr , Gm

Sc , ,

p

T w C w

w w

U Kt U y Uuu t y KU

CBM

U U

g T T g C CU U

T T C CD T T C C

∞ ∞

∞ ∞

∞ ∞

′′ ′′= = = =ν ν ν

μσ ν ′ω ν= = ω =κρ

′ ′ ′ ′ν β − ν β −= =

′ ′ ′ ′− −ν= θ = φ =′ ′ ′ ′− −

(6)

where D is mass diffusivity, Gr is Grashof number, Gm is modified Grashofnumber, K is permeability parameter, M is magnetic parameter, Pr is Prandtlnumber, Sc is Schmidt number, βT is thermal expansion coefficient, βC isconcentration expansion coefficient and ω is frequency of oscillation. Otherphysical variables have their usual meanings.

With the help of (6), the governing equations with the boundary conditionsreduce to

( )2

21Gr Gmu u M u

t Ky∂ ∂= + θ + φ − +∂ ∂

(7)

508 R. C. Chaudhary, Arpita Jain 4

2

2Prt y

∂θ ∂ θ=∂ ∂

(8)

2

2Sct y

∂φ ∂ φ=∂ ∂

(9)

0, 0, 0 for all , 0u y t= θ = φ = ≤ (10)

cos , 1, 1 at 0, 0

0, 0, 0 as , 0

u t y t

u y t

= ω θ = φ = = > ⎫⎬= θ = φ = →∞ > ⎭

(11)

METHOD OF SOLUTION

We solve the governing equations in an exact form by using Laplace-transforms.

The Laplace transform of the equations (7), (8), (9) and the boundaryconditions (11) are given by

2

2d ( ) Gr Gmd

u p M uy

′− + = − θ − φ (12)

2

2d Pr 0d

pyθ − θ = (13)

2

2d

Sc 0d

pyφ − φ = (14)

where 1 ,M MK

′ = + p is the Laplace transformation parameter.

2 21, at 0, 0

0, 0, 0 as , 0

pu y t

pp

u y t

⎫= θ = φ = = > ⎪+ ω ⎬⎪= θ = φ = →∞ > ⎭

(15)

Solving equations (12), (13), (14) with the help of equation (15), we get

( )( )

( ) ( ){ }

( ) ( ) ( ){ }

2 2

exp ( ) Gr( , )(Pr 1)

Pr 1

exp ( ) exp Pr

Gm exp ( ) exp Sc(Sc 1)

Sc 1

p y p Mu y p

Mp p p

y p M y p

y p M y PMp p

′− += + ×

′+ ω − − −

′× − + − − +

′+ − + − −′− − −

(16)

5 Heat and mass transfer effects on MHD free convection 509

( )exp Pr( , )

y py p

p

−θ = (17)

( )exp Sc( , )

y py p

p

−φ = (18)

Inverting equations (16), (17) and (18), we getFor Pr = Sc ≠ 1

( ) ( ){( ) ( )}

( ) ( ){( ) ( )}

( ) ( ) ( ){( ) ( )}

1( , ) exp( ) exp 2 ( ) ( )4

exp 2 ( ) ( )

1 exp( ) exp 2 ( ) ( )4

exp 2 ( ) ( )

1 Gr Gm exp 22

exp 2

Gr e2

u y t i t M i t erfc M i t

M i t erfc M i t

i t M i t erfc M i t

M i t erfc M i t

M t erfc M tM

M t erfc M t

M

′ ′= ω − η + ω η− + ω +

′ ′+ η + ω η+ + ω +

′ ′+ − ω − η − ω η− − ω +

′ ′+ η − ω η+ − ω −

+ ′ ′− − η η− +′

′ ′+ η η+ +

+ ′ ( ) ( ) ( )( ) ( ) ( )

( ) ( )

Pr Prxp exp 2Pr 1 Pr 1 Pr 1

Pr PrPr exp 2Pr 1 Pr 1 Pr 1

ScGmPr exp exp 2Pr 1 2 Sc 1 Sc 1

M t M t M terfc

M t M t M terfc erfc

M tM t M terfcM

⎡ ⎧⎛ ⎞ ⎛ ⎞′ ′ ′− η η− −⎨⎢ ⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠⎩⎣⎫ ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′ ′− η − + η η+ −⎬ ⎨⎜ ⎟ ⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎭ ⎩

⎛ ⎞⎤ ′⎫⎛ ⎞ ⎛ ⎞′ ′− η + + − η⎜⎬⎥ ⎜ ⎟⎜ ⎟ ⎜′− − −⎝ ⎠⎝ ⎠⎭⎦ ⎝ ⎠

( )

( )

Sc Sc

Sc 1 Sc 1

Sc Sc exp 2 Sc

Sc 1 Sc 1 Sc 1

Gr Gm( Pr ) ( Sc)

M t M terfc erfc

M t M t M terfc erfc

erfc erfcM M

⎡×⎢ ⎟⎟⎢⎣

⎧ ⎫⎛ ⎞′ ⎛ ⎞⎪ ⎪⎛ ⎞ ′× η− − η − +⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭⎤⎧ ⎫⎛ ⎞ ⎛ ⎞′ ′ ⎛ ⎞⎪ ⎪⎛ ⎞ ⎛ ⎞ ′ ⎥+ η η+ − η + +⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎥⎝ ⎠⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭⎦

+ η + η′ ′

(19)

( )( , ) Pry t erfcθ = η (20)

( )( , )y t erfc Scφ = η (21)

510 R. C. Chaudhary, Arpita Jain 6

where2

yt

η =

And for Pr = Sc = 1

( ) ( ){( ) ( )}

( ) ( ){( ) ( )}

( ) ( ) ( ){( ) ( )}

1( , ) exp( ) exp 2 ( ) ( )4

exp 2 ( ) ( )

1 exp( ) exp 2 ( ) ( )4

exp 2 ( ) ( )

Gr Gm 2 ( ) exp 22

exp 2

u y t i t M i t erfc M i t

M i t erfc M i t

i t M i t erfc M i t

M i t erfc M i t

erfc M t erfc M tM

M t erfc M t

′ ′= ω − η + ω η− + ω +

′ ′+ η + ω η+ + ω +

′ ′+ − ω − η − ω η− − ω +

′ ′+ η − ω η+ − ω +

+ ′ ′+ η − − η η− −′

′ ′− η η+

(22)

In expression, erfc(x1 + iy1) is complementary error function of complexargument which can be calculated in terms of tabulated functions [29]. Thetables given in [29] do not give erfc(x1 + iy1) directly but an auxiliary functionW1(x1 + iy1) which is defined as

{ }21 1 1 1 1 1 1( i ) ( i )exp ( i )erfc x y W y x x y+ = − + − +

Some properties of W1(x1 + iy1) are

1 1 1 2 1 1( i ) ( i )W x y W x y− + = +

{ }21 1 1 1 1 2 1 1( i ) 2exp ( ) ( )W x y x iy W x iy− = − − − +

where W2(x1 + iy1) is complex conjugate of W1(x1 + iy1).SKIN-FRICTION: We now study skin-friction from velocity field. It is

given by

0y

uy ′=

′∂τ = −μ ′∂

which in virtue of (6) reduces to

0y

uy =

⎛ ⎞∂τ = −⎜ ⎟∂⎝ ⎠

For Pr = Sc ≠ 1

7 Heat and mass transfer effects on MHD free convection 511

( ){( )}

( )( ) ( ) ( )

( )

1 exp( ) ( ) ( )2

exp( ) ( ) ( )

1 Gr Gmexp( )

PrGr Pr expPr 1 Pr 1 Pr 1 Pr 1

Sc ScGm expSc 1 Sc 1 Sc 1

i t M i t erf M i tt

i t M i t erf M i t

M t M erf M tMt

M tM M t M terf erfM

M M tM t erfM

′ ′τ = ω + ω + ω +

′ ′+ − ω − ω − ω +

+′ ′ ′+ − − +′π⎧ ⎫′⎪ ⎪⎛ ⎞′ ′ ′+ − +⎨ ⎬⎜ ⎟′ − − − −⎝ ⎠⎪ ⎪⎩ ⎭

′ ′⎛ ⎞ ⎛′+ ⎜ ⎟ ⎜′ − − −⎝ ⎠ ⎝ ( )Sc 1M terf

⎧ ⎫⎪ ⎪⎞ ′−⎨ ⎬⎟ −⎠⎪ ⎪⎩ ⎭

(23)

And for Pr = Sc = 1

( ){( )}

( )

1 exp( ) ( ) ( )2

exp( ) ( ) ( )

1 Gr Gmexp( )

i t M i t erf M i tt

i t M i t erf M i t

M t M erf M tMt

′ ′τ = ω + ω + ω +

′ ′+ − ω − ω − ω +

+′ ′ ′+ − − ′π

(24)

NUSSELT NUMBER: In non-dimensional form, the rate of heat transfer isgiven by

0Nu

PrNu

yy

t

=

⎛ ⎞∂θ= −⎜ ⎟∂⎝ ⎠

=π

(25)

DISCUSSION

In order to point out the effects of various parameters on flowcharacteristic, the following discussion is set out. The values of the Prandtlnumber are chosen Pr = 7 (water) and Pr = 0.71 (air). The values of the Schmidtnumber are chosen to represent the presence of species by hydrogen (0.22),water vapour (0.60), ammonia (0.78) and carbondioxide (0.96). Fig. 1 representsthe velocity profiles due to the variations in ωt and Pr. It is evident from figurethat the velocity near the plate exceeds at the plate i.e. the velocity overshootoccurs. Furthermore, the magnitude of the velocity decreases with increasingphase angle (ωt) for both air and water. The velocity for Pr = 0.71 is greater thanPr = 7. Physically, it is possible because fluids with high Prandtl number have high

512 R. C. Chaudhary, Arpita Jain 8

Fig. 1 – Velocity profiles when Gr = 5, Gm = 10, Sc = 0.22, t = 0.2, M = 5, K = 1.

viscosity and hence move slowly. Figs. 2 and 3 reveal the velocity variationswith Gr, Gm and Pr in cases of cooling and heating of the surface respectively. It is

9 Heat and mass transfer effects on MHD free convection 513

Fig. 2 – Velocity profiles when Sc = 0.22, t = 0.2, ωt = π/2, M = 5, K = 1.

observed that greater cooling of surface (an increase in Gr) and increase in Gmresults in an increase in the velocity for both air and water. It is due to the factincrease in the values of Grashof number and modified Grashof number has thetendency to increase the thermal and mass buoyancy effect. This gives rise to anincrease in the induced flow. The reverse effect is observed in case of heating ofthe plate (Gr < 0). Figs. 4 and 5 illustrate the influences of M, K, Pr in cases of

514 R. C. Chaudhary, Arpita Jain 10

Fig. 3 – Velocity profiles when Sc = 0.22, t = 0.2, ωt = π/2, M = 5, K = 1.

cooling and heating of the plate respectively. In case of cooling of the plate, thevelocity near the plate is greater than at the plate. The maximum velocity attainsnear the plate and is in the neighbourhood of point η = 0.4. After η > 0.4, thevelocity decreases and tends to zero as η → ∞. Again it is found that the velocitydecreases with increasing magnetic parameter for both Pr = 7 and 0.71. It isbecause that the application of transverse magnetic field will result a resistitive

11 Heat and mass transfer effects on MHD free convection 515

Fig. 4 – Velocity profiles when Gr = 15, Gm = 10, ωt = π/2, Sc = 0.22, t = 0.2.

516 R. C. Chaudhary, Arpita Jain 12

Fig. 5 – Velocity profiles when Gr = –15, ωt = π/2, Gm = –10, Sc = 0.22, t = 0.2.

type force (Lorentz force) similar to drag force which tends to resist the fluidflow and thus reducing its velocity. The presence of a porous medium increasesthe resistance to flow resulting in decrease in the flow velocity. This behaviour isdepicted by the decrease in the velocity as K decreases and when K = ∞ (i.e. theporous medium effect is vanished) the velocity is greater in the flow field. InFig. 5, the opposite phenomenon is observed for heating of the plate. Figs. 6 and 7display the effects of Sc (Schmidt number), Pr (Prandtl number) and t (time) onthe velocity field for the cases Gr > 0, Gm > 0 and Gr < 0, Gm < 0 respectively.

13 Heat and mass transfer effects on MHD free convection 517

Fig. 6 – Velocity profiles when Gr = 5, Gm = 10, ωt = π/2, M = 5, K = 1.

In case of cooling of the plate, the velocity near the plate increases owing to thepresence of foreign gases (such as hydrogen, water vapour and ammonia) in theflow field. We again noticed that although there is a rise in the velocity due topresence of water vapour and ammonia, but it is not so high as in the case ofhydrogen. The magnitude of the velocity for ammonia increases with timefor both air and water. The reverse effect is observed in case of heating of the plate.

518 R. C. Chaudhary, Arpita Jain 14

Fig. 7 – Velocity profiles when Gr = –5, Gm = –10, ωt = π/2, M = 5, K = 1.

Fig. 8 depicts the temperature profiles against η (distance from plate). Themagnitude of temperature is maximum at the plate and then decays to zeroasymptotically. The temperature for air is greater than that of water, this is due tothe fact that thermal conductivity of fluid decreases with increasing Pr, resultinga decrease in thermal boundary layer thickness. Fig. 9 concerns with the effect of

15 Heat and mass transfer effects on MHD free convection 519

Fig. 8 – Temperature profiles.

Sc on the concentration. Like temperature, the concentration is maximum at thesurface and falls exponentially. The Concentration decreases with an increase inSc. Further, it is noted that concentration falls slowly and steadily for hydrogenin comparison to other gases. Fig. 10 depicts skin-fiction against time t for different

520 R. C. Chaudhary, Arpita Jain 16

Fig. 9 – Concentration profiles.

values of parameters. The Skin-fiction increases with an increase in Sc. Further,the skin-friction increases with M due to enhanced Lorentz force which importsadditional momentum in the boundary layer. On the other hand, the skin-fictiondecreases with increasing K, Gm and Gr. The magnitude of skin-friction for

17 Heat and mass transfer effects on MHD free convection 521

Pr = 0.71 is less than that of Pr = 7. Nusselt number is presented in Fig. 11against time. It decreases with time. Nusselt number for Pr = 7 is higher than thatof Pr = 0.71. The reason is that smaller values of Pr are equivalent to increasingthe thermal conductivities and therefore heat is able to diffuse away from theplate more rapidly than higher values of Pr, hence the rate of heat transfer isreduced.

Fig. 10 – Skin-friction when ωt = π/4.

522 R. C. Chaudhary, Arpita Jain 18

Fig. 11 – Nusselt number.

19 Heat and mass transfer effects on MHD free convection 523

CONCLUSIONS

In this paper, combined heat and mass transfer effects on MHD freeconvection flow past an oscillating plate embedded in porous medium ispresented. Results are presented graphically to illustrate the variation of velocity,temperature, concentration, skin-friction and Nusselt number with variousparameters. In this study, the following conclusions are set out:

(1) In case of cooling of the plate (Gr > 0), the velocity decreases with anincrease in phase angle, magnetic parameter, Schmidt number andPrandtl number. On the other hand, it increases with an increase in thevalue of Grashof number and modified Grashof number, permeabilityparameter and time.

(2) In case of heating of the plate (Gr < 0), the velocity increases with anincrease in magnetic parameter, Schmidt number and Prandtl number.On the other hand, it decreases with an increase in the value of Grashofnumber and modified Grashof number, permeability parameter and time.

(3) The concentration decreases with an increase in Schmidt number.(4) The skin-friction increases with an increase in Schmidt number, Prandtl

number, magnetic parameter, while it decreases with an increase in thevalue of Grashof number, modified Grashof number, permeabilityparameter and time.

(5) Nusselt number increases with an increase in Prandtl number whiletemperature decreases with an increase in the value Prandtl number.

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