Nonlinear Kelvin–Helmholtz instability of Oldroydian viscoelastic fluid in porous media

The Kelvin–Helmholtz instability of two semi-infinite Oldroydian fluids in a porous medium has been considered. The system is influenced by a vertical electric field. A stability analysis has been carried out. The solutions of the linearized equations of motion under nonlinear boundary conditions lead to a nonlinear complex equation which govern the interfacial displacement. Taylor theory is adopted to expand the governing nonlinear equation in the light of the multiple time scales. This scheme leads to imposing of two levels of the solvability conditions, which are used to construct the well-known nonlinear Schrödinger equation with complex coefficients. The nonlinear Schrödinger equation generally describes the competition between nonlinearity and dispersion. The stability criteria are theoretically discussed. Stability diagrams are obtained for different sets of physical parameters. New instability regions in the parameter space, which appear due to nonlinear effects, are shown.