Nonlinear gravitational stability of streaming in an electrified viscous flow through porous media

Weakly nonlinear Kelvin–Helmholtz instability for two viscous fluids streaming through porous media is investigated. The electro-gravitational stability of the horizontal plane interface is examined. A vertical or a horizontal electric field stresses the system. The linear form of equation of motion is solved in the light of the nonlinear boundary conditions. The present boundary value problem leads to construct nonlinear characteristic equation. This nonlinear characteristic equation has complex coefficients for the elevation function. The nonlinearity is kept to the third order. The method of multiple scales, in both space and time, is used. The use of the Taylor expansion through the multiple scale scheme leads to the derivation of the well-known nonlinear Schrödinger equation with complex coefficients from the nonlinear characteristic equation. This equation describes the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrödinger equation that occurs in the non-resonance case. The relation between the stratified kinematic viscosity and the porous permeability is performed in order to control the marginal state representation. This marginality is utilised in order to relax the complexity of the linear dispersion relation. Stability conditions are discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. It is found that the porosity of the media increases the destabilizing influence for the fluid density. In nonlinear scope, a destabilizing influence for the upper porous permeability is recorded, while a stabilizing influence is found for the lower porous permeability. Both the horizontal and vertical electric fields are still playing the same roles in linear and nonlinear examinations as in the non-porous media. Two opposite roles are presented for the variation of the stratified fluid velocity V. A stabilizing influence for V 1 and a destabilizing effect for V>1 are illustrated in this examination. A dual role in the nonlinear examination is recorded for values of the wave-frequency.