Effect of Uniform Horizontal Magnetic Field on Thermal Convection in a Rotating Fluid Saturating aPorous Medium

Rovin Kumar and Vijay Mehta

Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur – 342001 (Raj.) INDIA.
email: rovinpradhan86@gmail.com email:vijaymehtanrs@gmail.com

(Received on: November 1, Accepted: November 3, 2017)

ABSTRACT

In this paper we study thermal convection in a rotating fluid saturating a porous medium in uniform horizontal magnetic field and obtained a dispersion relation. Using normal mode analysis, from this dispersion relation we observed that the medium permeability k1 has a stabilizing effect, in the absence of rotation the medium permeability has destabilizing effect. In the absence of magnetic field, the medium permeability has stabilizing effect, the rotation has stabilizing effect, whatever the magnetic field is applied and magnetic field has a stabilizing effect and a necessary condition for oscillatory mode is obtained with the condition
1<pm< and Ω2o<EPr3/4π2k13

Keywords:Thermal convection, Porous medium, Magnetic field, Rotation.

INTRODUCTION

The problem of convection in a horizontal layer of fluid heated from below (referred to as thermal instability problem), under varying assumptions of hydrodynamics and hydromagnetics, has been discussed in detail by Chandrasekhar (1961). The effect of Hall currents on the thermal instability of a horizontal layer of conducting fluid has been studied by Gupta (1967). Lapwood (1948) has investigated the stability of convective flow in hydrodynamics in a porous medium using Rayleigh's procedure. Sastry and Rao (1983) noticed that when the fluid, is heated from below, the rotation of the system delays the onset of stability. Recently, Sharma et al. (2006) have analyzed the effect of magnetic field and rotation on the stability of stratified elastico-viscous fluid in porous medium. The effect of Hall currents on the thermal instability of electrically conducting fluid in the presence of a uniform vertical magnetic field has been studied by Gupta (1967). Sharma and Kumar (1997) have studied thermosolutal convection in Rivlin–Erickson fluid in hydromagnetics saturating a porous medium. Bhatia and Steiner (1972) have studied the problem of thermal instability of a Maxwellian visco-elastic fluid in the presence of rotation and found that rotation has a destabilizing influence in contrast to the stabilizing effect on a viscous Newtonian fluid. Just as in hydrodynamics, when a conducting fluid permeates a porous material in the presence of a magnetic field the actual path of an individual particle of fluid can not be followed analytically. The gross effect, as the fluid slowly percolates through the pores of the rock, must be represented by a macroscopic law applying to masses of fluid which is the usual Darcy's law. The usual viscous term in the equations of fluid motion will be replaced by the resistance term (μ/k1)q , where μ, is the viscosity of the fluid, k1 the permeability of the medium (which has the dimension of length squared), and q -> the velocity of the fluid, calculated from Darcy's law. In all the above studies, the Boussinesq approximation has been used which means that density variations are disregarded in all the terms in the equations of motion except the one in the external force. The equations governing the system become quite complicated when the fluids are compressible. To simplify them, Boussinesq tried to justify the approximation for compressible fluids when the density variations arise principally from thermal effects. For stationary convection the medium permeability has stabilizing effect under the condition((Pmk2x0/2πΩoPr)<√π2+a2<2πΩok1/∈)and the magnetic field has a stabilizing effect under the condition b∈2/k12>4π2Ωo2
In the present paper, I studied the effect of uniform horizontal magnetic field on thermal convection in a rotating fluid saturating a porous medium. To the best of my knowledge if it uninvestigated so far.

MATHEMATICAL FORMULATION

In this problem, we consider an infinite, horizontal, electrically non-conducting incompressible fluid layer of thickness d. This layer is heated from below such that the lower boundary is held at constant temperature T = To and the upper boundary is held at fixed temperature T = T1 so that T0 > T1, therefore a uniform temperature gradient β|dT/dz| is maintained. The physical structure of the problem is one of infinite extent in x and y directions bounded by the planes z = 0 and z = d . The fluid layer is assumed to be flowing through an isotropic and homogenous porous medium of poricity ∈ and the medium permeability k,
which is acted upon a uniform rotation Ω(0, 0, Ωo) and gravity field g = (0, 0,-g). A uniform magnetic field H (Ho, 0, 0) = is applied along x-axis. The magnetic Raynold number is assumed to very small so that the induced magnetic field can be neglected in comparison to the applied field. We also assumed that both the boundaries are free and no external couples and the heat sources are present.

Fig. 1 Geometry of the Problem

The equation governing the motion of rotating fluids saturating a porous medium. Following Boussinesq approximation are as follow:
The equation of continuity for a incompressible fluid is

The equation of momentum, following Darcy law is given by

Where q , p ,ρ, ρo, s, μ, μe, k, T, t, T , To, Cv, Cs and e^z denote respectively filter velocity, pressure, fluid density, reference density, density of solid matrix, fluid viscosity, magnetic permeability, medium permeability, temperature, time, thermal conductivity, reference temperature, specific heat at constant volume, specific heat of solid matrix and unit vector along z-direction.
The Maxwell's equation become

where γ m is the magnetic viscosity.

BASIC STATE OF THE PROBLEM

The basic state of the problem is taken as

PERTURBATION EQUATIONS

Making the equation (10)-(13) into non-dimensional linearlized form by using the following non-dimensional variable and dropping the stars. We have

BOUNDARY CONDITIONS

Here, we consider the case when both boundaries are free as well as being perfect conductor of heat, while the adjoining medium is perfectly conducting, then we have

DISPERSION RELATIONS

Applying curl twice, to equation (17) and taking z-component, we have

Apply curl once to equation (17) and taking z-component, we have,

Applying curl once to equation (19) and taking z-component, we have

Taking z-component of the equation (19), we get

NORMAL MODE ANALYSIS

The normal mode analysis can be defined as follows:


From the equation (31) we observe that
D(2n)W = 0, n is a positive integer at z = 0, 1
Therefore the proper solution W characterizing the lowest mode is
W = Wo sinπ z
Where Wo is a constant
Substituting for W in equation (34) we have

STATIONARY CONVECTION

For stationary marginal state we put σ = 0 in (35) we get

When Q = 0 (in the absence of magnetic field) we have

When Ωo = 0 (In the absence of rotation we have)

From (36) we get

From (39), we obtain

In the absence of rotation (Ωo = 0) , equation reduces to

Which is always negative, thus in this case the medium permeability has a destabilizing effect in the absence of rotation.
In the absence of magnetic field Q = 0 , equation (40) reduce to

When
Thus, in the absence of magnetic field the medium permeability has stabilizing effect When
From (39), we have
Which is always positive, thus the rotation has stabilizing effect
From (39) we have

Thus, the magnetic field has a stabilizing effect when

OSCILLATORY CONVECTION

Equation (35) can be rewritten


CONCLUSIONS

For Stationary Convection

  1. The critical Rayleigh number increases as the medium permeability increases for stationary convection under the condition (52). Thus the medium permeability has a stabilizing effect when.

    In the absence of rotation, the critical Rayleigh number decreases as the medium permeability increases. In the absence of rotation the medium permeability has destabilizing effect.
    In the absence of magnetic field, the critical Rayleigh number increases the medium permeability increases under the conditionthe medium permeability stabilizing effects when
  2. The critical Rayleigh number increases with the increase of rotation, thus the rotation has stabilizing effect, whatever the magnetic field is applied.
  3. The critical Rayleigh number increases with the increase of Chandrashekhar number, thus the magnetic field has a stabilizing effect under the condition

For oscillatory Convection:The necessary condition for the existence of oscillatory mode are given by condition

REFERENCES

  1. Bhatia, P.K., Steiner, J.M., Convective instability in a rotating viscoelastic fluid layer, Z.Angew. Math. Mech. 52 321–327 (1972).
  2. Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Oxford University Press,Chaps. 2-5 (1961).
  3. Gupta, A.S. Rev. Roumaine Math. Pures Appl. 12, 665 (1967).
  4. Gupta, A.S., Hall effects on thermal instability, Rev. Roum. Math. Pures Appl. 12 665-677 (1967).
  5. Lapwood, E.R. Proc. Camb. Phil. Soc. 44, 508 (1948).
  6. Sharma, R.C., Kumar, P., On the micropolar fluid heated from below in hydromagnetics in porous medium, Czechoslovak J. Phys. 47 637 (1997).
  7. Sharma, V., Sunil, Gupta, U., Stability of stratified elastico-viscous walters (Model B')fluid in the presence of horizontal magnetic field and rotation in a porous medium, Arch. Mech. 58 (2) 187–197 (2006).
  8. Shastry, V.U.K., and Rao, V.R., Int. J. Eng. Sci. 5, 449 (1983).
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