Abstract
The current study concerns, the effect of a horizontal magnetic field on the stability of three horizontal finite layers of immiscible fluids in porous media. The problem examines few representatives of porous media, in which the porous media are assumed to be uniform, homogeneous and isotropic. The dispersion relations are derived using suitable boundary and surface conditions in the form of two simultaneous Mathieu equations of damping terms having complex coefficients. The stability conditions of the perturbed system of linear evolution equations are investigated both analytically and numerically and stability diagrams are obtained. The stability diagrams are discussed in detail in terms of various parameters governing the flow on the stability behavior of the system such as the streaming velocity, permeability of the porous medium and the magnetic properties. In the special case of uniform velocity, the fluid motion has been displayed in terms of streamlines concept, in which the streamlines contours are plotted. In the uniform velocity motion, a fourth order polynomial equation with complex coefficients is obtained. According to the complexity of the mathematical treatments, when the periodicity of the velocity is taken into account, the method of multiple scales is applied to obtain stability solution for the considered system.<br />It is found that a stability effect is found for increasing, the magnetic permeability ratio, the magnetic field, and the permeability parameter while the opposite influence is observed for increasing the upper layer velocity.
Highlights
The flow instability of a plane interface between two superposed fluids of different densities through porous media is of considerable interest for petroleum engineers and in geophysical fluid dynamicists
It is found that a stability effect is found for increasing, the magnetic permeability ratio, the magnetic field, and the permeability parameter while the opposite influence is observed for increasing the upper layer velocity
Theoretical and numerical analysis of linear stability of a fluid sheet of finite thickness embedded between two bounded layers of fluids through porous media are carried out
Summary
The flow instability of a plane interface between two superposed fluids of different densities through porous media is of considerable interest for petroleum engineers and in geophysical fluid dynamicists. In paper (Zakaria et al, 2009), the instability properties of streaming superposed conducting fluids through porous media under the influence of uniform magnetic field have been investigated, where the system is composed of a middle fluid sheet of finite thickness embedded between two semi-infinite fluids. Bhatia (1974) has studied the influence of viscosity on the stability of the plane interface separating two incompressible superposed fluids of uniform densities, when the whole system is immersed in a uniform horizontal magnetic field. He has developed the stability analysis for two fluids of equal kinematic viscosities and different uniform densities. The results and some important conclusions are outlined in last section of this work
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