Hydromagnetic parametric resonance instability of two superposed conducting fluids in porous medium

Abstract

Rayleigh–Taylor instability of a heavy fluid supported by a lighter one through porous medium, in the presence of a uniform, horizontal and oscillating magnetic field is studied. The fluids are taken as viscous (obeying Darcy's law), uniform, incompressible, and infinitely conducting. The amplitude of the oscillating part of the field is taken to be small compared with its steady part. The dispersion relation is obtained in the form of a third-order differential equation, with time as the independent variable and with periodic coefficients, for the vertical displacement of the surface of separation of the two fluids from its equilibrium position. The oscillatory magnetic field of frequency ω and steady part H 0 has a stabilizing influence on a mode of disturbance which is unstable in a steady magnetic field of strength H 0 . It is found that the oscillatory magnetic field and porosity of the porous medium have stabilizing effects, while the medium permeability has a destabilizing influence on the considered system. For a constant value of any of these physical parameters, the system has been found to be unstable (for small wavenumbers) as well as stable afterwards after a definite wavenumber value. The marginal stability case of parametric resonance holds when M 1 = M 2 = 0 (and hence m = 0 ), in which the characteristic exponents, and the corresponding solutions for u break down, is also investigated in detail. It is found, to order ϵ , that the effect of an oscillating magnetic field has no stabilizing influence on a disturbance which is marginally stable in the steady magnetic field; while to order ϵ 2 , and when the magnetic field oscillates, a resonance between this mode of disturbances and the oscillating field leads to instability when ρ 2 > ρ 1 . It is found also, in this resonant case, that all the constant or varied physical parameters, mentioned above, have destabilizing influences on the considered system. Finally, the other two resonance points appear in non-porous media (i.e., when m = ± i ω and m = ± 2 i ω ), are disappeared here due to the presence of the porous medium.