INSTABILITY THROUGH POROUS MEDIUM OF TWO VISCOUS SUPERPOSED CONDUCTING FLUIDS

The stability of the plane interface separating two viscous superposed conducting fluids through porous medium is studied when the whole system is immersed in a uniform horizontal magnetic field. The stability analysis is carried out for two highly viscous fluids of equal kinematic viscosities, for mathematical simplicity. It is found that the stability criterion is independent of the effects of viscosity and porosity of the medium and is dependent on the orientation and magnitude of the magnetic field. The magnetic field is found to stabilize a certain wave number range of the unstable configuration. The behaviour of growth rates with respect to viscosity, porosity and medium permeability are examined analytically.


INTRODUCTION.
The instability of the plane interface between two fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been discussed by Chandrasekhar   [i].Bhatia [2] has studied the influence of viscosity on the stability of the plane interface separating two incompressible superposed conducting fluids of uniform densities, when the whole system is immersed in a uniform horizontal magnetic field.He has carried out the stability analysis for two highly viscous fluids of equal kinematic viscosities and different uniform densities.When the fluid slowly percolates through the pores of the rock, the gross effect is represented by Darcy's law which states that the usual viscous term in the equations of fluid motion is replaced by the resistance term (/k 1) q, where is the viscosity of the fluid, k I the permeability of the medium and q the velocity of the fluid.Wooding [3] has experimentally observed, in the absence of viscous dissipation and consider- ing only Darcy resistance, that convection sets on as a fairly regular cellular pattern in the horizontal.This problem in the case of a conducting fluid consider- ing both Darcy and viscous resistances has been investigated by Prabhamani and   Rudraiah [4].Saville  [5] has studied the stability of motions involving fluid interfaces in porous media.Various problems of fluid flows through porous medium have been treated by Saffman and Taylor [6], Chouke et al [7], Scheidegger [8], Yih [9], Nayfeh [i0] and Rudraiah and Prabhamani [ii].
The instability of two viscous superposed conducting fluids through porous medium may find applications in geophysics.It is therefore the motivation of this study to examine the effects of viscosity and medium permeability on "the stability of the plane interface separating two incompressible superposed conducting fluids of uniform densities, when the whole system is immersed in a uniform horizontal magnetic field.We examine the roles of viscosity, medium permeability and magnetic field on the instability problem.This aspect forms the subject matter of the present study wherein we have carried out the stability analysis for two highly viscous fluids of equal kinematic viscosities and different uniform densities.
Consider the motion of an incompressible, infinitely conducting viscous fluid (of variable viscosity o(Z)) in the presence of a uniform magnetic field H(Hx,Hy,0).Let q(u,v,w), g0, gP and h(hx,hy,hz) denote the perturbations in velocity, density pressure p and magnetic field H respectively.Then the linearized perturbation equations of a fluid flowing throuaporous medium when both Darcy as well as viscous resistances are present are: 0 q Vp + g60 + (Vh) 09 1 e t q + (Vq).V+(V-V)q], (2.1) e p + (q.V)p 0. (2.4) Equation ( 4) ensures that the density of every particle remains unchanged as we follow it with its motion, v(=/O) denotes the kinematic viscosity of the fluid, e is porosity (0<e<l) and g (0,0,-g) is the acceleration due to gravity, e i and k I correspond to nonporous medium.Analyzing the disturbances into normal modes, we assume that the perturbed quantities have the space (x,y,z) and time (t) where k k are horizontal wave numbers (k , n is the growth rate of x' y x y the harmonic disturbance and f(z) is some function of z.
TWO SUPERPOSED VISCOUS FLUIDS OF UNIFORM DENSITIES.
Here we consider the case when two superposed fluids of uniform densities O 1 and 02 and uniform viscosities I and 2 are separated by a horizontal boundary at z 0. The subscripts I and 2 distinguish the lower and upper fluids respectively.
(i) Stable case For the potentially stable arrangement i>2 we find by applying Hurwitz' criterion to (3.20), that (as all the coefficients in (3.20) are then positive) all the roots of n are either real and negative or there are complex roots with negative real parts.The system is therefore stable in each case.The potentially stable configuration, therefore, remains stable whether the effects of viscosity and medium porosity are included or not.
(ii) Unstable case For the potentially unstable arrangement 2>i the system is unstable in the hydrodynamic case for all wave numbers k in the presence of viscosity effects and in the absence of porosity effects (Chandrasekhar [I]).Also the system, in the present case, is unstable if 2 gk 2(k'VA) < e (a2 i)" In the present hydromagnetic case we find, by applying Hurwitz' criterion to (3.20) when 2>i that the system is stable for all wave numbers which satisfy the inequality 2(.V+A)2 > gk (2 i ), (3.21) i.e. 2k(VlCOS@ + V2sin@)2> (2 el )' (3.22)where V I and V 2 are the Alfvn velocities in the x and y directions and G is the angle between k and H x The stability criterion (3.22) is independent of the effects of viscosity and medium porosity.The magnetic field stabilizes a certain wave number range k > k* where k* g(2-l)/2g(VlCOS@+V2sin@)2, (3.23) of the unstable configuration even in the presence of the effects of viscosity and medium porosity.The critical wave number k*, above which the system is stabilized, is dependent on the magnitudes V I and V 2 of the magnetic field as well as the orientation of the magnetic field 8.
vanish both when z (in the lower fluid) and z + (in the upper fluid), the solutions appropriate to the two regions can be written as +k H )2}. K2 /k2 + q 4n202 x x y y (3.5) (3.6)In writing the solutions (3.3) and (3.4) it is assumed that K 1 and K 2 are so defined that their real parts are positive.The solutions (3.3) and (3.4) must satisfy certain boundary conditions.The boundary conditions to be satisfied at the interface z 0 are (Chandrasekhar [i], p.432) .)Dw 2 -)Dw 2 + (k H +k H Dw 2} Dw I --(D2-k2)DWl + -(k H +k H Dw I} (Dw) are the unique values of these quantities at z 0. O O Applying the conditions (3.7)-(3.10) to the solutions (3.3) and (3.4), we obtain A I + B 1 A 2 + B 2 kA I + KIBI -kA 2 K2B2, HI {2k2Al + (KI2+k2)BI 2 {2k2A2 + (K22+k2) B2}' AI,A2,BI,B 2 from (3.11)-(3.14),we obtain roots, the dispersion relation (3.16) is quite complex.We therefore carry out the stability analysis for highly viscous fluids.Under this assumption of highly viscous fluids, 18), (3.19) in (3.16) and putting (the case of equal kinematic viscosities for mathematical simplicity as inChandrasekhar [l]),we obtain the following dispersiom relation