Stable Configuration of Double Horizontal Interfaces via the He-Multiple Scales Method

Abstract

The present work deals with the temporal instability of three horizontal superposed conducting incompressible fluids. The system is stressed upon by uniform tangential magnetic fields. These fields admit a presence of free-surface currents. In accordance with the importance of the porous media in many applications, the study is carried throughout porous media. To avoid the mathematical manipulation, the viscous potential theory is utilized. Therefore, the viscosity contributions could be demonstrated only on the boundary conditions. The linear stability approach together with the normal modes analysis reveal two coupled differential equations, with complex coefficients, of the Ince’s type. Away from the symmetric and anti-symmetric modes of perturbations, the present study presents a general case of the amplitudes of the interface surface waves. To relax the calculations, the matrix approach is used. The stability criteria of the resonance as well as the non-resonance modes are, theoretically, discussed. The analytical perturbed solutions of the interfaces are derived. A set of graphs is depicted to identify the influences of the various parameters on the stability picture. A non-dimension analysis is adopted before the numerical calculations. It observed that the tangential magnetic fields and the porosity have stabilizing effect. In contrast, the streaming has a destabilizing influence.