Soret and Dufour effects on Unsteady MHD flow Past an Oscillating Vertical Plate

IJIRST –International Journal for Innovative Research in Science & Technology| Volume 2 | Issue 08 | January 2016 ISSN (online): 2349-6010

Soret And Dufour Effects on Unsteady MHD Flow Past an Oscillating Vertical Plate B R Sharma Professor Department of Mathematics Dibrugarh University, Dibrugarh-786004, Assam

Bismeeta Buragohain Research Scholar Department of Mathematics Dibrugarh University, Dibrugarh-786004, Assam

Abstract In this paper, the unsteady magneto hydrodynamic flow past an oscillating vertical plate under the influence of Soret and Dufour effects has been investigated. The nonlinear partial differential equations governing the flow are solved by an iterative technique based on finite difference scheme. The flow phenomenon is characterized with the help of velocity, temperature and concentration profiles for different parameters.

Keywords: Soret effect, Dufour effect, MHD, oscillating vertical plate, finite difference method _______________________________________________________________________________________________________

I. INTRODUCTION The science of magnetohydrodynamics (MHD) concerns with geophysical, astrophysical and engineering problems for a number of years. The MHD flow of Newtonian fluids on plate, cone, and disks has been one of the subjects which has attracted attention of many workers. Atul [1] investigated the effects of mass transfer on free convection in MHD flow of a viscous fluid. ELKabeir and Abdou [2] studied the effects of chemical reaction and heat and mass transfer on MHD flow over a vertical isothermal cone surface in micropolar fluid1s with heat generation/absorption effects. Palani and Srikanth [3] studied the MHD flow of an electrically conducting fluid over a semi-infinite vertical plate under the influence of the transversely magnetic field.

Ellahi and Riaz [4] investigated analytically MHD flow in a third grade fluid with variable viscosity. Makinde [5] investigated the MHD boundary layer flow with the heat and mass transfer over a moving vertical plate in presence of magnetic field and convective heat exchange at the surface. Kishore et. al [6] studied the effects of thermal radiation and viscous dissipation on MHD heat and mass diffusion flow past an oscillating vertical plate embedded in a porous medium with variable surface conditions.

The Soret and Dufour effects are very important when the temperature and concentration gradients are high. The flux of mass caused due to temperature gradient is known as the Soret effect or the thermal-diffusion effect. The energy flux caused by a concentration gradient is referred to as the Dufour effect or diffusion-thermo effect. Postelnicu [7] studied simultaneous heat and mass transfer by natural convection from a vertical plate embedded in an electrically conducting fluid saturated porous medium in presence of Soret and Dufour effects using Darcy-Boussinesq model. Srihari et al. [8] discussed Soret effect on unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with oscillatory suction velocity and heat sink.

Gaikwad et al. [9] investigated the onset of double diffusive convection in two component couple of stress fluid layer with Soret and Dufour effects using both linear and nonlinear stability analysis.

II. FORMULATION OF THE PROBLEM

Fig. 1: physical configuration and coordinate system

An unsteady two-dimensional magneto hydrodynamic flow of a viscous incompressible fluid past an oscillating vertical plate under the influence of Soret and Dufour effects is investigated numerically. The plate is taken along � ′ axis in vertically upward direction and � ′ axis is taken normal to the plate.

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101

Soret And Dufour Effects on Unsteady MHD Flow Past an Oscillating Vertical Plate (IJIRST/ Volume 2 / Issue 08/ 017)

Let, đ?‘˘â€˛ be the velocity component of the fluid generated along đ?‘Ľ ′ axis due to the oscillation of the plate. It is assumed that initially the plate and the fluid are at the same temperature đ?‘‡ ′ ∞ and concentration level đ??ś ′ ∞ . At time t ′ > 0, the plate is oscillated with a velocity đ?‘˘â€˛ = đ?‘ˆđ?‘… đ?‘?đ?‘œđ?‘ ω′ t ′ in its own plane and the temperature of the plate is raised to đ?‘‡ ′ đ?‘¤ and concentration level is also raised to đ??ś ′ đ?‘¤ . Where đ?‘ˆđ?‘… is reference velocity, đ?œ”′ đ?‘Ą ′ is the phase angle.

Then by Boussinesqs’ approximation, the unsteady flow is governed by the following equations: đ?œ•đ?‘˘â€˛ đ?œ•đ?‘Ą ′ đ?œ•đ?‘‡ ′ đ?œ•đ?‘Ą ′ đ?œ•đ??ś ′ đ?œ•đ?‘Ą ′

=đ?œˆ =( =

đ?œ•2 đ?‘˘â€˛ 2 đ?œ•đ?‘Ś ′

k

)

+ gđ?›˝đ?‘‡ (đ?‘‡ ′ − đ?‘‡ ′ ∞ ) + gđ?›˝đ??ś (đ??ś ′ − đ??ś ′ ∞) –

đ?œ•2 đ?‘‡ ′

1

-

(

đ?œ•đ?‘žđ?‘&#x;

)+

đ??ˇđ?‘šđ?‘˜đ?‘Ą

đ?œŒđ??śđ?‘? đ?œ•đ?‘Ś ′ 2 đ?œŒđ??śđ?‘? đ?œ•đ?‘Ś ′ đ?œ•2 đ??ś ′ đ??ˇ đ?‘˜ đ?œ•2 đ?‘‡ ′ đ??ˇđ?‘š ′ 2 + đ?‘š đ?‘Ą ′ 2 đ?‘‡đ?‘š đ?œ•đ?‘Ś đ?œ•đ?‘Ś

đ?œŽđ??ľ0 2 đ?‘˘â€˛

(1)

đ?œŒ

đ?œ•2 đ??ś ′

(2)

đ??śđ?‘ đ??śđ?‘? đ?œ•đ?‘Ś ′ 2

(3)

where the appropriate initial and boundary conditions for the velocity, temperature and concentration fields are

In the above equations all the physical variables are functions of y′ and t′ alone as the plate is infinite. where, đ??ľ0 is the magnetic field component along y′ axis, đ??ˇđ?‘š is mass diffusivity, đ??śđ?‘? is the specific heat at constant pressure, đ??śđ?‘ is concentration susceptibility, đ?‘‡đ?‘š is the mean fluid temperature, đ?‘žđ?‘&#x; is the radiation heat flux, đ?‘˜đ?‘Ą is the thermal diffusivity ratio, đ?œŽ is the electrical conductivity, g is the acceleration due to gravity, đ?›˝đ?‘‡ , đ?›˝đ??ś are coefficients of thermal expansion and concentration expansion, k is the thermal conductivity, đ?œŒ is the density.

Invoking Rosseland approximation for radiative heat flux we get đ?‘žđ?‘&#x; =

−4đ?œŽđ?‘ đ?œ•đ?‘‡ ′

′4

4

(5)

đ?œ•đ?‘Ś ′

3đ?‘˜đ?‘

and expanding đ?‘‡ by Taylor’s expansion neglecting higher order terms we get 4 3 4 đ?‘‡ ′ = 4đ?‘‡ ′ ∞ đ?‘‡ ′ -3đ?‘‡ ′ ∞ Using equation (5) and (6), equation (2) reduces to đ?œ•đ?‘‡ ′ đ?œ•đ?‘Ą ′

=(

k đ?œŒđ??śđ?‘?

)

đ?œ•2 đ?‘‡ ′ 2 đ?œ•đ?‘Ś ′

+

16đ?œŽđ?‘ 3đ?œŒđ??śđ?‘? đ?‘˜đ?‘

đ?‘‡ â€˛âˆž

3 đ?œ•2 đ?‘‡ ′ 2 đ?œ•đ?‘Ś ′

+

(6)

đ??ˇđ?‘š đ?‘˜đ?‘Ą đ?œ•2 đ??ś ′

(7)

đ??śđ?‘ đ??śđ?‘? đ?œ•đ?‘Ś ′ 2

On introducing the following non dimensional quantities, t=

�′ ��

Gm =

,y=

�′

�′

,u=

đ??żđ?‘… đ?‘ˆđ?‘… đ?œˆđ?‘”đ?›˝đ??ś (đ??ś ′đ?‘¤ −đ??ś â€˛âˆž )

, đ?œ” = đ?œ”′ đ?‘Ąđ?‘… , Pr = −2â „ 3,

N=

đ??ˇđ?‘š đ?‘˜đ?‘Ą (đ??ś ′ đ?‘¤ −đ??ś ′ ∞ ) đ??śđ?‘ đ??śđ?‘? đ?œˆ(đ?‘‡ ′đ?‘¤ −đ?‘‡ â€˛âˆž )

đ?‘˜đ?‘ đ?‘˜ 4đ?œŽđ?‘ đ?‘‡ â€˛âˆž

3

,M=

đ?œŽđ??ľ0 2 đ?œˆ đ?œŒđ?‘ˆđ?‘… 2

, Sc=

đ?œˆ

,đ?œƒ=

đ?‘‡ â€˛âˆ’đ?‘‡ â€˛âˆž

, Gr =

đ?‘ˆđ?‘… 3

đ??ˇđ?‘š

, Sr =

đ?‘‡ ′đ?‘¤ −đ?‘‡ â€˛âˆž

,C=

đ??ś ′ −đ??ś ′ ∞

đ??ś ′ đ?‘¤ −đ??ś â€˛âˆž 1 −1 1 đ?‘‡ ′ ∞ )} â „3 , đ??żđ?‘… = {đ?œˆ −2 đ?‘”đ?›˝đ?‘‡ (đ?‘‡ ′ đ?‘¤ − đ?‘‡ ′ ∞ )} â „3 , A= , đ?‘Ąđ?‘… đ?œˆđ?‘”đ?›˝đ?‘‡ (đ?‘‡ ′đ?‘¤ −đ?‘‡ ′ ∞ ) đ??ˇđ?‘š đ?‘˜đ?‘Ą (đ?‘‡ ′ đ?‘¤ −đ?‘‡ â€˛âˆž )

đ?‘˜

, đ?‘ˆđ?‘… ={đ?œˆđ?‘”đ?›˝đ?‘‡ (đ?‘‡ ′ đ?‘¤ −

đ?‘ˆđ?‘… 3

đ?‘Ąđ?‘… ={đ?‘”đ?›˝đ?‘‡ (đ?‘‡ ′ đ?‘¤ − đ?‘‡ ′ ∞ )} Du =

đ?œ‡đ??śđ?‘?

đ?‘‡đ?‘šđ?œˆ(đ??ś ′đ?‘¤ −đ??ś â€˛âˆž )

,

,

Where Pr is prandtl number, Gr is Grashof number, Gm is mass Grashof number, M is magnetic parameter, Sc is Schmidt number, t is time in dimensional coordinate, N is radiation parameter, đ??żđ?‘… is reference length, đ?‘Ąđ?‘… is reference time, Sr is Soret number, Du is Dufour number, đ?œ” is frequency of oscillation, Equations (1), (7), (3) in the non-dimensional form can be written as đ?œ•đ?‘˘ đ?œ•đ?‘Ą đ?œ•đ?œƒ đ?œ•đ?‘Ą đ?œ•đ??ś đ?œ•đ?‘Ą

= = =

đ?œ•2 đ?‘˘ đ?œ•đ?‘Ś 2 1

+ Grđ?œƒ + GmC – Mu 4

(1+ )

đ?œ•2 đ?œƒ

+ Du

Pr 3N đ?œ•đ?‘Ś 2 1 đ?œ•2 đ??ś đ?œ•2 đ?œƒ Sc đ?œ•đ?‘Ś 2

+ Sr

(8)

đ?œ•2 đ??ś

(9)

đ?œ•đ?‘Ś 2

(10)

đ?œ•đ?‘Ś 2

The appropriate initial and boundary conditions equation (4) for the velocity, temperature and concentration fields in nondimensional form are u = 0, đ?œƒ = 0, C = 0 at y, t ≤ 0 (11) u = cos đ?œ”đ?‘Ą, đ?œƒ = đ?‘Ą, C = t at y = 0, t > 0 (12) u → 0, đ?œƒ → 0, C → 0 as y → ∞, t > 0 (13)

III. METHOD OF SOLUTION The dimensionless governing differential equations (8)-(10) subject to the initial and boundary conditions (11)-(13) are reduced to a system of difference equations using the following finite difference scheme, and then the system of difference equations is solved numerically by an iterative method. The scheme for a variable v is given by,

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102

Soret And Dufour Effects on Unsteady MHD Flow Past an Oscillating Vertical Plate (IJIRST/ Volume 2 / Issue 08/ 017) ∂v ∂y

=

vi+1 j −vi j ∆y

∂2 v

,

∂y2

=

vi+1 j + vi−1 j − 2vi j (∆y)2

IV. RESULT AND DISCUSSION The Soret and Dufour effects on the velocity, temperature and concentration profiles are studied then by supplying various value of the parameters. Numerical calculations have been carried out for different values of time t, Sr, Du and for fixed values of Pr, đ?œ‹ Sc, Gr, Gm, M, N, đ?œ”đ?‘Ą. The value of Pr, Sc, Gr, Gm, M, N, đ?œ”đ?‘Ą is taken to be 0.71, 0.6, 0.5, 0.5, 1, 1, respectively.

6

t=0.1 t=0.2 t=0.3 t=0.4

0.8

f'

0.6 0.4 0.2 0

0

1

2

3

4

y

Fig. 1: velocity profile for different values of t 0.4 t=0.1 t=0.2 t=0.3 t=0.4

ď ą

0.3

0.2

0.1

0

0

0.5

1

1.5 y

2

2.5

3

Fig. 2: Temperature profile for different values of t 0.5 t=0.1 t=0.2 t=0.3 t=0.4

0.4

ď Ś

0.3

0.2

0.1

0

0

0.5

1

1.5 y

2

2.5

3

Fig. 3: Concentration profile for different values of t Sr=0.6 Sr=1.2 Sr=2

0.8

f'

0.6 0.4 0.2 0

0

1

2

3

4

y

Fig. 4: velocity profile for different values of Sr

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103

Soret And Dufour Effects on Unsteady MHD Flow Past an Oscillating Vertical Plate (IJIRST/ Volume 2 / Issue 08/ 017)

1 Sr=0.6 Sr=1.2 Sr=2

0.8



0.6 0.4 0.2 0

0

1

2 y

3

4

Fig. 5: Temperature profile for different values of Sr

1.5 Sr=0.6 Sr=1.2 Sr=2



1 0.5 0

0

1

2

3

4

5

y Fig. 6: Concentration profile for different values of Sr Du=0.6 Du=1.2 Du=2

0.7 0.6 0.5

f'

0.4 0.3 0.2 0.1 0

0

1

2

3

4

y

Fig. 7: velocity profile for different values of Du 1 Du=0.6 Du=1.2 Du=2

0.8



0.6 0.4 0.2 0

0

1

2

3

4

5

y

Fig. 8: Temperature profile for different values of Du 1 Du=0.6 Du=1.2 Du=2

0.8



0.6

0.4

0.2

0

0

1

2

3

4

5

y

Fig. 9: Concentration profile for different values of Du

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104

Soret And Dufour Effects on Unsteady MHD Flow Past an Oscillating Vertical Plate (IJIRST/ Volume 2 / Issue 08/ 017)

Effect of Time (t): The effect of t on velocity, temperature and concentration profile is depicted in Figs. 1 to 3. From these figure it is observed that velocity, temperature and concentration increase for increasing values of t.

Effect of Soret Number (Sr): The effect of Sr on velocity, temperature and concentration profile is depicted in Figs. 4 to 6. It can be seen that an increase in Sr both velocity and concentration increases but noticeable decreasing effect on the temperature profile.

Effect of Dufour Number (Du): The effect of Du on velocity, temperature and concentration profile is depicted in Figs. 7 to 9. It can be seen that an increase in Du both velocity and temperature increases but noticeable decreasing effect on the concentration profile.

V. CONCLUSION In this study, effects of Soret, Dufour and time are examined on magneto hydrodynamic flow of a viscous incompressible fluid past an oscillating vertical plate. The leading governing equations are solved by an iterative technique based on finite difference scheme. The effect of Soret, Dufour and time has been shown graphically. It has been found that with increasing Soret number there is an increase in the flow and the concentration but decrease in the temperature values. Increasing Dufour number serves to accelerate the flow and the temperature but reduce concentration values. In general, with increasing time the flow, the temperature and the concentration values increases.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

Atul Kumar Singh, “Effects of mass transfer on free convection in MHD flow of a viscous fluid,” Indian Journal of Pure and Applied Physics, Vol. 41, 2003, pp. 262 – 274.

S. M. M. El-Kabeir and M. M. M. Abdou, “Chemical reaction, heat and mass transfer on MHD flow over a vertical isothermal cone surface in micropolar fluids with heat generation/absorption,” Applied Mathematical Sciences: Journal for Theory and Applications, vol. 1, 2007, pp. 1663–1674.

G.Palani and U. Srikanth, “MHD flow past a semi-infinite vertical plate with mass transfer,” Nonlinear Analysis, Modelling and control, Vol. 14, 2009, pp.

345-356.

R. Ellahi, and A. Riaz, “Analytical solution for MHD flow in a third grade fluid with variable viscosity,” Mathematical and Computer Modelling, Vol. 52, 2010, pp.1783-1793.

O. D. Makinde, “On MHD heat and mass transfer over a moving vertical plate with a convective surface boundary condition,” The Canadian Journal of Chemical Engineering, Vol. 88, 2010, pp. 983-990.

P.M.Kishore,V.Rajesh and S.Vijayakumar Verma, “The effects of thermal radiation and viscous dissipation on MHD heat and mass diffusion flow past an oscillating vertical plate embedded in a porous medium with variable surface conditions,” Theoret. Appl. Mech., Vol.39, 2012, pp. 99–125.

A. Postelnicu, “Influence of a magnetic field on heat and mass transfer by natural convection for vertical surfaces in porous media considering Soret and Dufour effects,” International Journal of Heat and Mass Transfer, Vol.47, 2004, pp. 1467-1472.

K. Srihari, S. R. Reddy and J. A. Rao, “Soret effect on unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with oscillatory suction velocity and heat sink,” International Journal of Applied Mathematical Analysis and Applications, Vol. 1, 2006, pp. 239 – 259.

S. N. Gaikwad, M. S. Malashetty and K. R. Prasad, “An analytical study of linear and nonlinear double diffusive convection with Soret and Dufour effects in couple stress fluid,” International Journal of Nonlinear Mechanics, Vol. 42, 2007, pp. 903-913

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