Nonlinear waves on the surface of a magnetic fluid jet in porous media

A weakly nonlinear theory of capillary waves on the surface of a magnetic fluid jet is investigated. The magnetic fluids are uniformly streaming through porous media. The linear forms of the equations of motion are solved in the light of the nonlinear boundary conditions. The boundary-value problem leads to a nonlinear characteristic equation governing the interfacial displacement and having complex coefficients, where the nonlinearity is kept up to the third order. Taylor theory is adopted to expand the governing nonlinear equation through the multiple scales, in both space and time, to yield the well-known Schrödinger equation with complex coefficients. This equation describes the evolution of the wave train, it may be regarded as the counter parts of the single nonlinear Schrödinger equation that occurs in the non-resonance case. The stability criteria are discussed theoretically and illustrated graphically in which stability diagrams are obtained. Regions of stability and instability are identified for the stratified magnetic fields versus the wave number for the wave train of the disturbance. It is found that the porous media have a destabilizing influence. This influenced is enhanced when the Darcy's coefficients are different. New instability regions in the parameter space, which appear due to the nonlinear effects, are shown.