Structural stability of a porous channel of electrical flow affected by periodic velocities

This paper investigates the effect of periodic velocity on the stability of two interfacial waves propagating between three layers of immiscible incompressible fluids. The flow propagates saturated in porous media under the influence of an electric field. The viscous potential theory is used to simplify the mathematical procedure, by which viscosity is accumulated on the separating surface rather than in the bulk of fluids. The dispersion relation is shown to be the result of coupled simultaneous Mathieu equations with complex coefficients. The mathematical difficulty in delaying the two Mathieu equations was overcome using the method of multiple time scales. The stability was discussed analytically and numerically by drawing some diagrams and obtaining the transition curves. In the presence and absence of stream periodicity, the linear stage of interface progress is visualized. Depending on the physical quantities used, it has been determined that the upper fluid speed dampens the wave more than the lower and middle velocities. It was discovered that the viscosity ratio of the upper interface, as opposed to the lower viscosity ratio, helps stabilize the liquid sheet. The upper layer porosity has a dual role depending on the sensitivity and significance of the sheet thickness.