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GENERAL PHYSICS – MAGNETOHYDRODYNAMICS

HALL EFFECT ON MHD MIXED CONVECTIVE FLOWOF A VISCOUS INCOMPRESSIBLE FLUID PAST

A VERTICAL POROUS PLATE IMMERSED IN POROUS MEDIUMWITH HEAT SOURCE/SINK

BHUPENDRA KUMAR SHARMA, ABHAY KUMAR JHA, R. C. CHAUDHARY

Department of Mathematics, University of Rajasthan, Jaipur-302004 , Indiae-mail: [email protected]

Received June 26, 2006

An analysis is presented with unsteady MHD mixed convective flow of aviscous incompressible fluid past an infinite vertical porous flat plate in the presenceof a heat source/sink with Hall effect. The solution is found to be dependent on thegoverning parameters including the Hartman number, the Prandtl number, theGrashof number, the Schmidt number, the Hall parameter, rotating parameter and theheat source/sink parameter. Representative results are presented and discussed forvelocity, temperature, concentration species, surface heat flux, mass flux and skinfriction coefficient with the help of graphs and tables.

Key words: MHD, mass transfer, heat source/sink.

1. INTRODUCTION

The hydromagnetic convection with heat and mass transfer in porousmedium has been studied due to its importance in the design of MHD generatorsand accelerators in geophysics, in design of under ground water energy storagesystem, soil-sciences, astrophysics, nuclear power reactors and so on.Magnetohydrodynamics is currently undergoing a period of great enlargementand differentiation of subject matter. The interest in these new problemsgenerates from their importance in liquid metals, electrolytes and ionized gases.On account of their varied importance, these flows have been studied by severalauthors – notable amongst them are Shercliff [1], Ferraro and Plumpton [2] andCrammer and Pai [3]. Elbashbeshy [4] studied heat and mass transfer along avertical plate in the presence of a magnetic field. Hossain and Rees [5] examinedthe effects of combined buoyancy forces from thermal and mass diffusion bynatural convection flow from a vertical wavy surface. Numerical results wereobtained to show the evolution of the surface shear stress, rate of heat transferand surface concentration gradient. Recently, combined heat and mass transfer inMHD free convection from a vertical surface has been studied by Chen [6].

Rom. Journ. Phys., Vol. 52, Nos. 5–7 , P. 487–503, Bucharest, 2007

488 B. K. Sharma, A. K. Jha, R. C. Chaudhary 2

Further, the effect of Hall current on the fluid flow with variable concentrationhas many applications in MHD power generation, in several astrophysical andmeteorological studies as well as in plasma flow through MHD powergenerators. From the point of application, model studies on the Hall effect onfree and forced convection flows have been made by several investigators.Aboeldahab [7], Datta et al. [8], Acharya et al. [9] and Biswal et al. [10] havestudied the Hall effect on the MHD free and forced convection heat and masstransfer over a vertical surface.

In the above mentioned studies the heat source/sink effect is ignored. Dueto its great applicability to ceramic tiles production problems, the study of heattransfer in the presence of a heat source/sink has acquired newer dimensions. Anumber of analytical studies have been carried made of for various forms of heatgeneration (Ostrach [11–13], Raptis [14]). Singh [15] analysed MHD freeconvection and mass transfer flow with heat source and thermal diffusion.

The problem investigated here is the study of the Hall Effects on thecombined heat and mass transfer unsteady flow which occur due to buoyancyforces caused by thermal diffusion (temperature differences) and mass diffusion(concentration differences) of comparable magnitude past a vertical porous platewhich is immersed in porous medium with a constant magnetic field and heatsource/sink applied perpendicular to the plate.

2. FORMULATION

Consider the unsteady flow of an electrically conducting fluid past aninfinite vertical porous flat plate coinciding with the plane y = 0 such that thex-axis is along the plate and y-axis is normal to it. A uniform magnetic field B0 isapplied in the direction y-axis and the plate is taken as electrically non-conducting.Taking z-axis normal to xy-plane and assuming that the velocity V and themagnetic field H have components (u, v, w) and (Hx, Hy, Hz) respectively, the

equation of continuity 0V∇ ⋅ = and solenoidal relation 0H =∇ ⋅ give

0 0v constant, 0.V V= − > (1)

From Maxwell’s electromagnetic field equations,

0yHy

∂=

∂. (2)

If the magnetic Reynold number is small, induced magnetic field isnegligible in comparison with the applied magnetic field, so that [16]

00 and (constant).x z yH H H B= = =

3 Hall effect on MHD mixed convective flow 489

If (Jx, Jy, Jz) are the components of electric current density .J The equation

of conservation of electric charge 0J∇ ⋅ = gives,

Jy = constant. (3)

Since the plate is non-conducting, Jy = 0 at the plate and hence zeroeverywhere in the flow.

Neglecting polarization effect, we get

0.E = (4)Hence

0 0( , 0, ), (0, , 0), ( , , ).x zJ J J H B V u V w= = = (5)

The generalized Ohm’s law, taking Hall effect into account, is given by

( )0

1 ,e ee

eJ J H E V B p

B eω τ ⎛ ⎞+ × = σ + × + ∇⎜ ⎟η⎝ ⎠

(6)

where V is the velocity vector, ωe is the electron frequency, τe is the electroncollision time, e is the electron charge, ηe is the number density of electron, pe is

the electron pressure, σ is the electric conductivity and E is the electric field. Inwriting equation (6) ion slip, thermoelectric effects and polarization effects areneglected. Further, it is also assumed that ~ 0e eω τ and 1,i iω τ ≤ where ωe, ωi

are cyclone frequencies of electrons and ions, and τe, τi are collision times ofelectrons and ions.

Equations (5) and (6) yield,

02

02

( ),1

( ),1

x

z

BJ mu w

mB

J u mwm

σ ⎫= − ⎪⎪+⎬σ ⎪= +⎪+ ⎭

(7)

where u and w are the x-component and z- component of V and e em = ω τ is theHall parameter.

The equations of motion, energy and concentration governing the flowunder the usual Boussinesq approximation are

22 0 *2 2

( )v ( ) ( ) ,

(1 )B u mwu u u ug T T g C C

t y ky m ∞ ∞σ +∂ ∂ ∂ ν+ = ν − + β − + β − −

∂ ∂ ∂ ρ +(8)

22 02 2

( )v ,

(1 )B mu ww w w w

t y ky mσ −∂ ∂ ∂ ν+ = ν + −

∂ ∂ ∂ ρ +(9)


20

2) ) )

v ( ),p

KT T T T T TS T T

t y c y∞ ∞ ∞

∞∂( − ∂( − ∂ ( −

+ = + −∂ ∂ ρ ∂

(10)

2

2( ) ( ) ( )

v .C C C C C C

Dt y y

∞ ∞ ∞∂ − ∂ − ∂ −+ =

∂ ∂ ∂(11)

where g is the acceleration due to gravity, T is the temperature of the fluid withinthe boundary layer, C is the specis concentration, ρ is the fluid density ofboundary layer, ν is the kinematic viscosity, K0 is the thermal conductivity, cp isthe specific heat at constant pressure, D is the chemical and molecular diffusivity, kis the permeability of porous medium and S is the source sink parameter.

In equation (10) the viscous dissipation and ohmic dissipation are neglectedand in equation (11), the term due to chemical reaction is assumed to be absent.Now using

0v , ( , ) ( , )V T y t T y t∞= − − = θ and *( , ) ( , )C y t C C y t∞− =

Subjecting to the initial and boundary conditions

*( , ) ( , ) 0, 0, 0 for all t u y t w y t C y≤ = = θ = =

*

*

(0, ) (0, ) 0, (0, ) , (0, ) at 0

( , ) ( , ) 0, ( , ) 0, ( , ) 0 at

( 0).

i t i tu t w t t ae C t be y

u t w t t C t y

t

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

(12)

Introducing the following non-dimensional quantities

2 *0 0

0 0

2*0

3 3 300 0 0

20

2 20

, , , , , ,4

44 4, , , Pr ,

Sc , , .4

V y V T u w Ct u w CV V a b

cBg a g b pG G Mc KV V V

V k Sk SD V

θ′ ′ ′ ′ ′η = = = = θ = =ν ν

νρσνβν β ν= = = =ρ

ν ν′ ′= = =ν

(13)

Equations (8) to (11) are transformed to their corresponding non-dimensional forms (dropping the dashes) as

2

2 24 4 ( ) ,1

u u u M umw u G G Cct km∂ ∂ ∂− = − + + θ + −∂ ∂η ∂η +

(14)

2

2 24 4 ( ) ,1

w w w M wmu wt km

∂ ∂ ∂− = + − −∂ ∂η ∂η +

(15)


2

244 ,Pr

St

∂θ ∂θ ∂ θ− = + θ∂ ∂η ∂η

(16)

2

244 ,C C C

t Sc∂ ∂ ∂− =∂ ∂η ∂η

(17)

where V0 is some reference velocity, G is the Grashof number for heat transfer,Gc is the Grashof number for mass transfer, Pr is the Prandtl number, Sc is theSchmidt number, M is the magnetic field parameter and S is the source/sinkparameter.

Dropping dashes, the modified boundary conditions become,

0 : ( , ) ( , ) 0, 0, 0 for all t u t w t C≤ η = η = θ = = η

(0, ) (0, ) 0, (0, ) , (0, ) at 00

( , ) ( , ) 0, ( , ) 0, ( , ) 0 at .

i t i tu t w t t e C t et

u t w t t C t

ω ω⎧ = = θ = = η =⎪> ⎨∞ = ∞ = θ ∞ = ∞ = η → ∞⎪⎩

(18)

3. SOLUTION

The equations (14) and (15) can be combined using the complex variable

,u iwψ = + (19)giving

2

2 21 1 1 1 1(1 ) .4 4 4 41

M im G G Cct km∂ ψ ∂ψ ∂ψ ⎡ ⎤+ − − − + ψ = − θ −⎢ ⎥∂η ∂∂η +⎣ ⎦

(20)

Introducing the non-dimensional parameter 20

4VνωΩ = and using equation

(19), the boundary conditions in (18) are transferred to

(0, ) ( , ) 0, (0, ) ,

(0, ) , ( , ) 0, ( , ) 0.

i t

i t

t t C t e

t e t C t

Ω

Ω

⎫ψ = ψ ∞ = = ⎪⎬

θ = θ ∞ = ∞ = ⎪⎭(21)

Putting ( , ) ( )i tt e fΩθ η = η in equation (16), we get

Pr Pr( ) Pr ( ) 0,4 4

i Sf f Ω′′ ′η + η − + = (22)

which has to be solved under the boundary conditions,

(0) 1, ( ) 0.f f= ∞ = (23)

Hence,


2

2

1 Pr Pr Pr Pr2

Pr Pr Pr Pr2

( ) ,

( , ) .

i S

i t i S

f e

t e

⎡ ⎤− − + Ω − η⎣ ⎦

η⎡ ⎤Ω − + + Ω −⎣ ⎦

η =

⇒ θ η =(24)

Separating real and imaginary parts, the real part is given by,

{ } ( )1Pr cos2 2

1( , ) cos sin 2 2

Rr t t R e

η α− +η αθ η = Ω − (25)

where

( )1/ 2 2 2 1/ 4 11 Pr [(Pr ) ] , tan

PrR S

S− Ω= − + Ω α =

−. (26)

Putting ( , ) ( )i tC t e gΩη = η in equation (17), we get

( ) ( ) ( ) 0,4

i Scg Sg gΩ′′ ′η + η − η = (27)

which has to be solved under the boundary conditions,

(0) 1, g( ) 0.g = ∞ = (28)Hence

2

2

12 ,

2

( )

( , ) .

Sc Sc i Sc

i t Sc Sc i Sc

g e

C t e

⎡ ⎤− − + Ω η⎣ ⎦

η⎡ ⎤Ω − + + Ω⎣ ⎦

η =

⇒ η =(29)

Separating real and imaginary parts, the real part is given by

{ } 2 cos2 2

2( , ) cos sin ,2 2

Sc R

rC t t R eη β⎛ ⎞− +⎜ ⎟

⎝ ⎠η βη = Ω − (30)

( )1/ 2 2 2 1/ 4 12 Sc (Sc ) , tan .

ScR − Ω= + Ω β = (31)

In order to solve equation (20), substituting ( )i te fΩψ = η and usingboundary conditions

(0) 0, ( ) 0F F= ∞ = , (32)

and then separating real and imaginary parts, we have

19

1 2

8 9

19

/ 217 20 18 20

/ 2 / 221 2 22 2 25

/ 2 / 223 9 24 9 26

/ 218 20 17 20

[{ cos( / 2) sin( / 2)}

{ cos( / 2) sin( / 2) }

{ cos( / 2) sin( / 2) } ]cos

[{ cos( / 2) sin( / 2)}

A

A A

A A

A

u A A A A e

A t A e A t A e A

A t A e A t A e A t

A A A A e

− η

− η − η

− η − η

− η

= η + η −

− Ω − η + Ω − η −

− Ω − η + Ω − η Ω −

− η − η −

(33)


2 1

9 8

/ 2 / 221 2 22 2 25

/ 2 / 223 9 24 9 26

{ sin( / 2) cos( / 2) }

{ sin( / 2) cos( / 2) } ]sin .

A A

A A

A t A e A t A e A

A t A e A t A e A t

− η − η

− η − η

− Ω − η − Ω − η −

− Ω − η − Ω − η Ω

19

1 2

8 9

19

/ 217 20 18 20

/ 2 / 221 2 22 2 25

/ 2 / 223 9 24 9 26

/ 218 20 17 20

[{ cos( / 2) sin( / 2)}

{ cos( / 2) sin( / 2) }

{ cos( / 2) sin( / 2) } ]sin

[{ cos( / 2) sin( / 2)}

A

A A

A A

A

w A A A A e

A t A e A t A e A

A t A e A t A e A t

A A A A e

− η

− η − η

− η − η

− η

= η + η −

− Ω − η + Ω − η −

− Ω − η + Ω − η Ω +

+ η − η −

− 2 1

9 8

/ 2 / 221 2 22 2 25

/ 2 / 223 9 24 9 26

{ sin( / 2) cos( / 2) }

{ sin( / 2) cos( / 2) } ]cos .

A A

A A

A t A e A t A e A

A t A e A t A e A t

− η − η

− η − η

Ω − η − Ω − η −

− Ω − η − Ω − η Ω

(34)

The shearing stress at the wall along x axis is given by

10

,uη=

⎛ ⎞∂τ = ⎜ ⎟∂η⎝ ⎠(35)

and the shearing stress at the wall along z axis is given by

20

,wη=

⎛ ⎞∂τ = ⎜ ⎟∂η⎝ ⎠(36)

where,

1/ 2 2 2 1/ 41 Pr [(Pr ) ] ,R S= − + Ω 1/ 2 2 2 1/ 4

2 Sc (Sc ) ,R = + Ω

1/ 42 2

3 2 21 1 ,

1 1M MmR

k m m

⎡ ⎤⎛ ⎞ ⎛ ⎞= + + + Ω −⎜ ⎟ ⎜ ⎟⎢ ⎥+ +⎝ ⎠ ⎝ ⎠⎣ ⎦

( )1tan ,Pr S

− Ωα =−

( )1 tan ,Sc

− Ωβ = ( ) ( )12 2

1tan 11 1

Mm Mkm m

− ⎛ ⎞γ = Ω − + +⎜ ⎟+ +⎝ ⎠

1 1Pr cos ,2

A R α= + 2 1 sin ,2

A R α= 3 21 ,

1MA

k m= +

+ 4 2 ,

1MmA

m= Ω −

+

2 25 1 2 1 3( 2 ),A A A A A= − − − 6 1 2 2 4(2 2 ),A A A A A= − − 2 2

7 5 6 ,A A A= +

8 2 cos / 2,A Sc R= + β 9 2 sin / 2,A R= β 2 210 8 9 8 3( 2 ),A A A A A= − − −

11 8 9 9 4(2 2 ),A A A A A= − − 12 5 6 25( cos sin ) ,A GA t GA t A= Ω + Ω

13 5 6 25( sin cos ) ,A GA t GA t A= Ω − Ω 14 10 11 26( cos sin ) ,A G A t G A t Ac c= Ω + Ω

15 10 11 26( sin cos ) ,A G A t G A t Ac c= Ω − Ω 2 216 10 11 ,A A A= + 17 12 14 ,A A A= +


18 13 15,A A A= + 19 31 cos ,2

A Rγ= + 20 3 sin ,

2A R

γ= 21 5 ,A GA=

22 5 23 10 24 11 25 267 16

1 1, , , , .A GA A G A A G A A Ac c A A= = = = =

4. DISCUSSION

A study of the velocity field, variations of temperature and concentration,shearing stresses, surface heat flux and mass flux in oscillatory hydromagneticmixed convective flow past an infinite vertical porous plate through porousmedium with Hall effect and heat source sink has been carried out in thepreceding sections. Approximate solutions are obtained for various flow variables.In order to get physical insight into the problem, the velocity, temperature,concentration fields, shear stress, rate of heat transfer and rate of mass transferhave been discussed by assigning numerical values of M (Magnetic parameter),G (Grashof number for heat transfer), Sc (Schmidt number), Ω (Frequencyparameter). The values of Pr (Prandtl number) are taken as 0.71 and 7.0 thatrepresent air and water at 20°C temperature and one atmosphere pressurerespectively. The Grashof number for heat transfer is chosen to be G = 2.0 andmodified Grashof number Gc = 5.0 corresponding to a cooled plate (Gc > 0)

Fig. 1 – Variation of velocity component for G = 2.0, Gc = 5.0 and Ωt = π/2.


while Gc = –5.0 corresponding to a heated plate (Gc < 0). The values of Hallparameter m = 0.5, 1.0, permeability k = 1.0, magnetic parameter M = 5, 10, heatsource S = 1 (S > 0) and heat sink S = –1 (S < 0) are chosen arbitrarily. Thenumerical results obtained are illustrated in Figs. 1 to 10 and Tables 1 and 2.

Figs. 1 and 2 depict the variation of the velocity component for Gc = 5.0(cooled Newtonian fluid Gc > 0) and Gc = –5.0 (heated Newtonian fluid Gc < 0)respectively. These velocity profiles are shown for Pr = 0.71 (air) and Pr = 7.0(water), taking the different values of m, M, Sc and S.

Fig. 2 – Variation of velocity component for G = 2.0, Gc = –5.0 and Ωt = π/2.

Case I: When Gc > 0 (i.e. flow on a cooled plate)

It is observed that an increase in Hall parameter (m) leads to an decrease inthe velocity for both air and water, while a reverse effect is observed for theapplied magnetic field (M). The velocity is higher for Ammonia (Sc = 0.78, at25°C temperature and one atmosphere pressure) than Helium (Sc = 0.30, at 25°Ctemperature and one atmosphere pressure) for both air and water. We notice thatthe velocity is higher for water than that for air for different parameters. It is alsonoticed that the velocity is higher for a heat sink (S < 0) than that a heat sourcefor air, while the reverse effect is observed for water.

Case II: When Gc < 0 (i.e. flow on a heated plate)

It is noticed that an increase in Hall parameter (m) leads to an increase inthe velocity for air, while the reverse phenomenon is observed for water. It is


found that the values of velocity decrease as the values of M increase for air,while, there is a rise in the values of velocity in water. The velocity is less forHelium than for Ammonia in air, but this behavior is reversed in water. Thevalues of velocity are positive for air while negative for water.

A comparative studies of the curves reveal that the values of velocity aremore for heat source than heat sink for both air and water.

Figs. 3 and 4 represent the variation of the velocity component w forGc = 5.0 and Gc = –5.0, respectively.

Fig. 3 – Variation of velocity component w for G = 2.0, Gc = 5.0 and Ωt = π/2.


An increase in m and M leads to a rise in the velocity for both air andwater. The values of velocity are greater for air than that of water. The velocityis greater for Helium than Ammonia for air, but the reverse effect is observed forwater. It is also noticed that velocity is higher for a sink than that of a source forboth air and water. Further, it is observed that the velocity increases graduallynear the plate and then decreases slowly far away from the plate.


Fig. 4 – Variation of velocity component w for G = 2.0, Gc = –5.0 and Ωt = π/2.


In the case of a heated plate, a comparison of velocity profiles shows thatvelocity increases near the plate and thereafter decreases for air. The velocitydecreases due to an increase in m and M for air, while, the reverse effect isobserved for water. For both the cases, air and water, it is noticed that thevelocity is greater for heavier (Ammonia) particles and heat a sink than forlighter particles (Helium) and heat a source.

The temperature profiles are shown in Fig. 5. The effects of Pr, Ω and S onthe temperature profiles are drawn from this figure. An increase in Ω (frequencyparameter) leads to a fall in temperature for both air and water. The temperatureprofiles remain positive for a heat sink and negative for a heat source in bothsituations (air and water) near the plate and vanish far away from the plate.Fig. 6 gives the concentration for Helium (Sc = 0.30) and Ammonia (Sc = 0.78)and Ω = 1 and 3. We notice that the effect of increasing values of Sc is to decrease


the concentration profile in the flow field. Physically, the increase of Sc meansdecrease of molecular diffusivity D. This results in a decrease of concentrationboundary layer. Hence, the concentration of the species is higher for small valuesof Sc and lower for larger values of Sc. The values of the concentration increasesgradually near the plate and then decrease slowly for away from the plate.Further it is also noticed that the values of the concentration increases at eachpoint with variation of the frequency parameter Ω.

Figs. 7 and 8 present the shearing stress τ1 for Gc = 5.0 and Gc = –5.0,respectively.

Fig. 7 – Variation of shearing stress τ1 for G = 2.0, Gc = 5.0 and Ωt = π/2.


Comparison of shearing stress shows that τ1 increase near the plate andthereafter remains constant far away from the plate. It is observed that anincrease in M increases τ1 for both air and water, whereas, the values of τ1 are

higher in Helium than Ammonia for both air and water. We also noticed that τ1

is higher for a heat sink for air, while the reverse phenomenon occurs in the caseof water.


Fig. 8 – Variation of shearing stress τ1 for G = 2.0, Gc = –5.0 and Ωt = π/2.

Case II : When Gc < 0 (i.e. flow on a heated plate)

It is seen that τ1 decrease near the plate and thereafter remains constant faraway from the plate for air, while in the case of water, the reverse effect isobserved. We observed that an increase in M decreases τ1 for air but in the caseof water reverse phenomenon is noticed. The values of τ1 are higher for Heliumfor both air and water. Further, it is noted that τ1 is greater for a heat source thana sink in air, while the reverse effect is observed.

The shearing stress τ2 for Gc = 5.0 and Gc = –5.0 has been shown in Figs. 9and 10, respectively.


It is observed that an increase in M and Sc decreases τ2 for both air andwater. We also noticed the values of τ2 are more for a sink than for a source forboth cases.


Comparison of shearing stress shown that τ2 decreases with increasing Mand Sc for both cases (air and water). The values of τ2 are more for a source than


sink for both air and water. Further, in the vicinity of the plate, the values of τ2

are higher in water than in air and thereafter the reverse effect is observed faraway from the plate.

Table 1 gives the mass transfer [C(t)] for different cases such as Hydrogen(Sc = 0.22), Helium (Sc = 0.30), Water vapour (Sc = 0.60) and Ammonia(Sc = 0.78) at 25°C temperature and one atmosphere pressure. It is observed thatC(t) decreases with increasing Ω. The value of C(t) increases with increasing Sc.

The data in Table 2 represents heat transfer for Pr, S and Ω. It can be seenthat with an increase in Ω the values of Q(t) increase in air for a heat sourcewhile the reverse effect is observed for heat sink. Further the values of Q(t)decreases with increasing Ω for both heat source and sink in the case of water. Itis interesting to note that the values of Q(t) are higher for a heat sink than for aheat source for both air and water.

Table 1

Variation of C(t)

C(t)

Ω Sc 0.22 0.30 0.60 0.78

0.2 0.223 0.301 0.590 0.774

0.4 0.221 0.299 0.580 0.756

0.6 0.218 0.290 0.560 0.720

Table 2

Variation of Q(t)

Ω Q(t)

Pr 0.71 7.0 0.71 7.0

S 1.0 1.0 –1.0 –1.0

0.2 0.59 6.7 0.89 7.2

0.4 0.63 6.58 0.87 7.07

0.6 0.67 6.39 0.84 6.87

Acknowledgement. One of the authors Abhay Kumar Jha thank to the UGC, New Delhi forawarding a Senior Research Fellowship.

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15. Atul Kumar Singh, MHD free convection and mass transfer flow with heat source and thermaldiffusion, J. of Energy, Heat and Mass Transfer, 23, 227–249, 2001.

16. T. G. Cowling, Magneto-Hydrodynamics, Wiley Interscience, New York, 1957.

Fig. 5 – Variation of temperature θr for ωt = π/2.

Fig. 6 – Variation of concentration Cr for Ωt = π/2.

Fig. 9 – Variation of shearing stress τ2 for G = 2.0, Gc = 5.0 and Ωt = π/2.

Fig. 10 – Variation of shearing stress τ2 for G = 2.0, Gc = –5.0 and Ωt = π/2.

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