Nonlinear hydromagnetic Rayleigh-Taylor instability for strong viscous fluids in porous media

In the present work a weakly nonlinear stability for magnetic fluid is discussed. The research of an interface between two strong viscous homogeneous incompressible fluids through porous medium is investigated theoretically and graphically. The effect of the vertical magnetic field has been demonstrated in this study. The linear form of equation of motion is solved in the light of the nonlinear boundary conditions. The boundary value problem leads to construct nonlinear characteristic equation having complex coefficients in elevation function. The nonlinearity is kept to third-order expansion. The nonlinear characteristic equation leads to derive the well-known nonlinear Schroedinger equation. This equation having complex coefficients of the disturbance amplitude varies in both space and time. Stability criteria have been performed for nonlinear Chanderasekhar dispersion relation including the porous effects. Stability conditions are discussed through the assumption of equal kinematic viscosity. The calculation shows that the stratification of the vertical magnetic fields H(1) and H(2) plays a destabilizing role. On the other side, the kinematic viscosities play a stabilizing role when the fluid flows through a porous media, while a destabilizing influence is recorded when the fluid flows through non-porous media. The investigation has shown that the porous permeability plays a dual role in the stability behaviour