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Finite difference solution of unsteady MHD free convective mass transfer flow past an infinite, vertical porous plate with variable suction and Soret effect.

Introduction

Convective heat transfer in a porous media is a topic of rapidly growing interest due to its application to geophysics, geothermal reservoirs, thermal insulation engineering, exploration of petroleum and gas fields, water movements in geothermal reservoirs, etc. A comprehensive review of the study of convective heat transfer mechanisms through porous media in relation to the applications to the above areas has been made by Nield and Bejan [7].

Several researchers are attracted to the unsteady free convective flow past an infinite or semi-vertical plates due to its important technological applications. As presence of suction being more important and appropriate from the technological point of view, Nanda and Sarma [6], Schetz and Eichhorn [9], Soundalgekar [10], [11] and Kafousias [5] have studied unsteady free convective flow past vertical plates with suction.

Recently, the study of free convective mass transfer flow has become the object of extensive research as the effects of heat transfer along with mass transfer effects are dominant features in many engineering applications such as rocket nozzles, cooling of nuclear reactors, high sinks in turbine blades, high speed aircrafts and their atmospheric re-entry, chemical devices and process equipments. Soundalgekar [12], [13], Hossain and Begum [3], Subhashini et al [15] have discussed unsteady free convective mass transfer flow past vertical porous plates. But in all these papers thermal diffusion effects have been neglected, whereas in a convective fluid when the flow of mass is caused by a temperature difference, thermal diffusion effects cannot be neglected. In view of the importance of this diffusion-thermo effect, Jha and Singh [4] presented an analytical study for free convection and mass transfer flow past an infinite vertical plate moving impulsively in its own plane taking Soret effects into account. In all the above studies, the effect of the viscous dissipative heat was ignored in free-convection flow. However, Gebhart [1], Gebhart and Mollendorf [2] have shown that when the temperature difference is small or in high Prandtl number fluids or when the gravitational field is of high intensity, viscous dissipative heat should be taken into account in free convection flow past a semi-infinite vertical plate.

The unsteady free convection flow of a viscous incompressible fluid past an infinite vertical plate with constant heat flux is considered on taking into account viscous dissipative heat, under the influence of a transverse magnetic field studied by Srihari. K et al [14]. Ramana Kumari.,C.V., and Bhaskara Reddy.N.,[8] have studied a two-dimensional unsteady MHD free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate with variable suction.

The object of the present paper is to study the MHD effects as well as Soret effects on the unsteady free convective mass transfer flow past an infinite vertical plate with variable suction, where the plate temperature oscillates with the same frequency as that of variable suction velocity. The governing equations are solved by using finite difference method.

Mathematical Analysis

Unsteady flow of an incompressible, electrically conducting viscous fluid past an infinite vertical porous plate under the influence of a uniform transverse magnetic field is considered. Here the origin of the co-ordinate system is taken to be at any point of the plate. Let the components of velocity along x' and y' axes be u' and v' which are chosen in the upward direction along the plate and normal to the plate respectively. The polarization effects are assumed to be negligible and hence the electric field is also negligible. Hence the governing equations of the problem are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Here, the status of an equation of state is that of equation p' = constant.

This means that the density variations produced by the pressure, temperature and concentration variations are sufficiently small to be unimportant. Variations of all fluid properties other than the variations of density except in so far as they give rise to a body force, are ignored completely (Boussinesq approximation). All the physical variables are functions of y' and t' only as the plate is infinite. It is also assumed that the variation of expansion coefficient is negligibly small and the pressure and influence of the pressure on the density are negligible.

In a convective fluid the flow of mass is caused by a temperature difference, the thermal diffusion (Soret effect) cannot be neglected. With in the framework of above assumptions the governing equations reduce to

[[partial derivative]v'/[partial derivative]t'] = 0 (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and the corresponding boundary conditions are

u' = 0, T' = [T'.sub.w] = 1 + [epsilon] [e.sup.i,[omega]'t'], C' = [C'.sub.w] at y' = 0 u' [right arrow] 0, T' [right arrow] [T'.sub.[infinity]], C' [right arrow] [C'.sub.[infinity]] as y' [right arrow] [infinity] (9)

From the continuity equation, it can be seen that v' is either a constant or a function of time. So, assuming suction velocity to be oscillatory about a non-zero constant mean, one can write

v' = -[v.sub.0] (1 + [epsilon] A [e.sup.i,[omega]'t'])

where [v.sub.0] is the mean suction velocity and [epsilon], A are small such that [epsilon] A << 1. The negative sign indicates that the suction velocity is directed towards the plate.

To write non-dimensional form of above equations we introduce the following non-dimensional quantities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, using the above non-dimensional quantities, the equations (6)-(9) in the nondimensional form can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

and

u = 0, T = [T.sub.w](t) = 1 + [epsilon] [e.sup.i[omega]t], C = 1 at y = 0 u [right arrow] 0, T [right arrow] o, C [right arrow] 0 as y [right arrow] [infinity] (13)

Here, t' is the time, g the acceleration due to gravity, [beta] the coefficient of volume expansion, [[beta].sup.*] the coefficient of thermal expansion with concentration, T' the temperature of the fluid, [T'.sub.[infinity]] the temperature of the fluid far away from the plate, C' the species concentration, [C'.sub.[infinity]] the species concentration of the fluid far away from the plate, [T'.sub.w] the plate temperature, [C'.sub.w] the species concentration near the plate, v the kinematic viscosity, [rho] the density, [c.sub.p] the specific heat at constant pressure, [K.sub.T] the thermal conductivity, D the chemical molecular diffusivity, [mu] the coefficient of viscosity, M the magnetic field parameter, So the Soret number, Sc the Schmidt number and E the Eckert number.

All the physical quantities have their usual meaning. The second term on the right hand side of equation (11) represents the Joule's dissipative heat.

Method of Solution

Applying the Crank-Nicolson finite difference formula for equations (10) to (12), the following system of equations are obtained

-2r [u.sup.j+1.sub.i-1] + (1 + 4r) [u.sup.j+1.sub.i] - 2r [u.sup.j+1.sub.i+1] = [D.sup.i.sub.3] (i) (14)

-2r [T.sup.j+1.sub.i-1] + (P + 4r) [T.sup.j+1.sub.i] - 2r [T.sup.j+1.sub.i+1] = [D.sup.j.sub.2] (i) (15)

-2r [C.sup.j+1.sub.i-1] + (Sc + 4r) [C.sup.j+1.sub.i] - 2r [C.sup.j+1.sub.i+1] = [D.sup.j.sub.3] (i) (16)

where

[D.sup.j.sub.1] (i) = 2r[u.sup.j.sub.i-1] + [A.sub.1][u.sup.j.sub.i] + [A.sub.2][u.sup.j.sub.i+1] + [A.sup.j.sub.3] (i)

[D.sup.j.sub.2] (i) = 2r[T.sup.j.sub.i-1] + [B.sub.1][T.sup.j.sub.i] + [B.sub.2][T.sup.j.sub.i+1] + [B.sup.j.sub.3] (i)

[D.sup.j.sub.3] (i) = 2r[C.sup.j.sub.i-1] + [C.sub.1][C.sup.j.sub.i] + [C.sub.2][C.sup.j.sub.i+1] + [C.sup.j.sub.3] (i)

[A.sub.1] (i) = 1 - 4rhF - 4r - 4r[h.sup.2] M;

[A.sub.3] = 2r + 4rh;

[A.sub.3.sub.j] (i) = 4r[h.sup.2]Gr[T.sup.j] (i) + 4r[h.sup.2]Gc[C.sup.j] (i)

F = 1 + [epsilon] A Cos ([omega]kj);

[B.sub.1] = P - 4r - 4rhPF;

[B.sub.2] = 2r + 4rhPF;

[B.sup.j.sub.3] (i) = 4r[h.sup.2]PEM[u.sup.j.sub.i] * [u.sup.j.sub.i]

[C.sub.1] = Sc - 4r - 4rhScF;

[C.sub.2] = 2r + 4rhScF;

[C.sup.j.sub.3] (i) = 4ScSo[[[T.sup.j.sub.i-1] - 2[T.sup.j.sub.i] + [T.sup.j.sub.i+1]/[h.sup.2]]] * [h.sup.2] * r

the boundary conditions (13) becomes

[U.sup.j.sub.0] = 0; [T.sup.j.sub.0] = 1 + [epsilon] Cos[omega]kj; [C.sup.j.sub.0] = 1

[U.sup.j.sub.[infinity]] = 0; [T.sup.j.sub.[infinity]] = 0; [C.sup.j.sub.[infinity]] = 0

where h, k are mesh sizes along space direction and time direction respectively, r = [k/[h.sup.2]]. Index i refers to space and j refers to time. The mesh system which consists of h = 0.2 and k = 0.01, has been considered for computations. The solutions of the equations (14)-(16) have been solved by Gauss-Siedel method. In order to prove convergence of finite difference scheme, the computation is carried out for slightly changed values of h and k, by computing the same program. Negligible change is observed. Thus it is concluded that, the finite difference schme is convergent and stable.

Results and Discussion

In order to get a physical insight into the problem, numerical calculations are carried out for different values of [omega], Gr, Gc, Sc, P, So, M and A and are represented in figures and tables. The motion being free convective, E is very small and the value is taken as 0.001 to 0.01 as this will be more appropriate from practical point of view. The values of Gr and Gc, M, So, E, A and [omega] are arbitrarily chosen whereas Sc and P are chosen in such a way that they represent realistic case. Thus values of Sc are chosen such that they represent hydrogen (Sc = 0.24) and water (Sc = 0.6) and the values of P are chosen as 0.71 and 7 for air and water respectively.

It is observed from fig.(1) that as Eckert number(E) is increased, velocity is increasing. Velocity is also increasing as Gr and Gc, [omega], So are increasing, which is observed from figures(2), (5), (6) where as velocity is decreasing as Sc, P, M are increasing observed from figures (3), (4), (7). Further, it is noticed that from fig. (8), the velocity is higher in the case of constant suction (A=0) than in the case of variable suction (A = 0.2, A = 0.4).

From physical point of view, Gr > 0 and E > 0 corresponds to an externally cooling of the plate. It is observed that the velocity increases with the greater cooling of the plate. Physically this can be justified, because in the process of externally cooling of the plate, the free convection currents travel away from the plate. As the fluid is also moving in the upward direction, the free convection currents tend to help the velocity to increase.

From fig. (9) it is observed that temperature increases with the increase of E. Also, observed that from fig. (10) temperature increases as Gr and Gc increases whereas the temperature decreases with the increase of P which is observed from fig. (11). The temperature profiles, with the effect of Sc, So, M and A have not been shown as there is no significant effect on it.

Figures (12) to (17) represent the concentration profiles. As seen from fig. (12) that as E is increasing concentration is decreasing. From figures (15), (13), (16), (17) it is observed that concentration increases as P, Gr & Gc, [omega], So are increased, where as it decreases when Sc is increased as seen in fig. (14). There is no significant effect on the concentration with the increase of M and A and hence they have not been shown by graphs.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

References

[1] Gebhart, B., Effects of viscous dissipation in natural convection, J. Fluid Mech., 14, 225 (1962).

[2] Gebhart, B., and Mollendorf, J., Viscous dissipation in external natural convective flows, J. Fluid Mech., 38, 97 (1969).

[3] Hossain, M.A. and Begum, R.A., The effects of mass transfer on the unsteady free-convection flow past an accelerated vertical porous plate with variable suction, Astrophys. Space Sci., 145, 115 (1985).

[4] Jha, B.K., and Singh, A.K., Soret effects on free-convection and mass transfer flow in the stokes problem for an infinite vertical plate, Astrophys. Space Sci., 173, 251 (1990).

[5] Kafousias, N.G., Nanousis, N.D. and Georgantopoulos, G.A., The effects of free convective currents on the flow field of an incompressible viscous fliud past an impulsively started infinite vertical porous plate with constant suction, Astrops. Space Sci., 64, 391 (1979).

[6] Nanda, R.S. and Sharma, V.P., Possibility similarity solution of unsteady free convection flow past a vertical plate with suction, J. Phys. Soc. Japan, 17, 1651 (1962).

[7] Nield, D.A., and Bejan, A., (1998) Convection in porous media, 2nd edition, Springer-Verlag, Berlin.

[8] Ramana Kumari, C.V. and Bhaskara Reddy, N., Mass Tranfer effects on unsteady free convective flow past an infinite, vertical porous plate with variable suction, Journal of Energy, Heat and Mass Transfer, Vol. 16, pp. 279-287 (1994).

[9] Schetz, J.A. and Eichhorn, R., Unsteady natural convection in the vicinity of a doubly infinite vertical plate, J.Heat Transfer, Trans. ASME, 84c, pp. 334 (1962).

[10] Soundelgekar, V.M., Free convective effects on the oscillatory flow past an infinite, vertical, porous plate with constant suction I, Proc. R. Soc., A333, 25 (1973a).

[11] Soundelgekar, V.M., Free convective effects on the oscillatory flow past an infinite, vertical, porous plate with constant suction II, Proc. R. Soc., A333, 37 (1973b).

[12] Soundalgekar, V.M., Effects of mass transfer and free convection currents on the flow past an impulsively started vertical plate, J. Appl. Mech. Trans. ASME, 46, 757 (1979).

[13] Soundalgekar, V.M. and Ganesan, P., Transient free convective flow past a semi-infinite vertical plate with mass transfer, Reg. J. Energy Heat and Mass Transfer, 2, No.1, 83 (1980).

[14] Srihari, K., Anand Rao, J. and Kishan, N. MHD free convection flow of an incompressible viscous dissipative fluid in an infinite vertical oscillating plate with constant heat flux, Jl. Energy, Heat and Mass Transfer, vol 28, pp. 19-28, 2006.

[15] Subhashini, A., Bhaskara Reddy, N. and Ramana Kumari, C.V., Mass transfer effects on the flow past a vertical porous plate, J. Energy Heat and Mass Transfer, 15, 221 (1993).

S. Renuka, * N. Kishan and J. Anand Rao

Depertment of Mathematics, University College of Science, Osmania University, Hyderabad, A.P., India.

* Corresponding Author
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Author:Renuka, S.; Kishan, N.; Rao, J. Anand
Publication:International Journal of Petroleum Science and Technology
Date:Jan 1, 2009
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