New analytical approximation forms for non-linear instability of electric porous media

We study the Kelvin–Helmholtz instability for parallel flow between two dielectric fluids in porous media. A normal electric field stresses the system. A non-linear perturbation technique is introduced. This technique is based on the Fourier transform and the multiple scales method. We discussed the stability conditions for non-Darcian and Darcian flows. A linear analytical dispersion relation and several coupled non-linear equations contain non-linear correction terms in uniform analytical descriptions derived. For non-Darcian, a complex non-linear Ginzburg–Landau equation is used to analyse the stability of the problem obtained. We introduce a simple analytical technique for calculating the non-linear cutoff wavenumber, and a new analytical solution form is derived. The newly derived analytical solution is compared with several previously obtained approximation solutions. The analytical solution is more realistic than the previously proposed descriptions, which were widely used in the past. For some special physical parameters, a non-linear modified diffusion equation for low Reynolds number is obtained. Also, for high Reynolds number we obtain a new analytical non-linear dispersion relation (complex non-linear modified Ginzburg–Landau equation) in terms of the linear analytical dispersion relation. Numerical results and stability diagrams support the analytical proofs. Regions of stability and instability are identified. The non-linear numerical results showed that the linear model is inadequate.