Nonlinear Rayleigh-Taylor instability for hydromagnetic Darcian flow: effect of free surface currents

Capillary-gravity waves of permanent form at the interface between two unbounded magnetic fluids in porous media are investigated. The system is influenced by the horizontal direction of the magnetic field to the separation face of two semi-infinite homogeneous and incompressible fluids, so that the fields allow free-surface currents at the interface. The solutions of the linearized equations of motion under nonlinear boundary conditions lead to derivation of a nonlinear equation governing the interfacial displacement. This equation is accomplished by using the cubic nonlinearity. Taylor theory is used to expand the governing nonlinear equation in the light of the multiple scales in both space and time. The perturbation analysis leads to imposition of two levels of solvability conditions, which are used to construct the well-known nonlinear Ginzburg-Landau equation. The stability criteria are discussed theoretically and numerically and stability diagrams are obtained. Regions of stability and instability are identified for the surface current density. It is found that the stabilizing role for the magnetic field is retarded when the flow is in porous media. Moreover, the increase in the values of resistance parameters plays a dual role, in stability behavior and in the increase in surface current density.