New Methodology in Scrutinizing Nonlinear Interfacial Instability Between Two Walters’ B/Rivlin–Ericksen Fluids Exposed to Periodic Electric Fields

The paper proposes a new mathematical framework in explaining the effect of periodic electric fields on the nonlinear interfacial instability emerging between two Walters’ B/Rivlin–Ericksen of non-Newtonian fluids. The suggested approach is designed to increase the prediction and control of electrically induced instability phenomena observed in advanced Electrohydrodynamics. Accordingly, under the impact of periodic EFs, the instability properties between the two superposed, electrically conducting viscoelastic fluids passing through a porous medium are examined. Furthermore, the fluids differ in their densities, electrical conductivities, permeabilities, and viscoelastic characteristics, surface tension and are supposed to performance at the disturbed interface. To decrease the mathematical complexity, viscous potential theory is adopted. By combining the pertinent nonlinear boundary conditions with the governing linearized equations of motion, more simplifications are made. The methodology leads to a nonlinear Mathieu oscillator characterizing the interfacial displacement. Within the scope of the non-perturbative approach, the resulting nonlinear ordinary differential equation is converted into an equivalent linear representation. A non-dimensional analysis yields a set of typical dimensionless parameters, significantly reducing the number of governing variables and facilitating physical interpretation. The stability criteria are numerically studied under complex conditions, indicating that the fundamental stability mechanism stays unchanged for both real and imaginary coefficients of the nonlinear characteristic equation regulating the interfacial motion.