Temporal instability of a viscous liquid film on an oscillating plane in the presence of static electric field

This work examined the linear instability of a two-dimensional liquid film on an oscillating plane theoretically in the presence of a static electric field perpendicular to the plane. The liquid was considered to be viscous and perfectly conducting, while the gas was considered to be viscous and dielectric. The viscous potential theory and Floquet theory were adopted to obtain the dispersion equation. The influences of electric field intensity, forcing amplitude, forcing frequency, and viscosity were obtained. The results showed that with there is more than one instability region with a non-zero forcing amplitude. The first instability region is called inherent instability, which is due to the electric force overcoming the viscous dissipation, surface tension, and gravity, while the other regions are due to the parametric instability induced by the oscillating plane, forming famous Faraday waves. It was found that increasing the forcing amplitude stabilizes the inherent instability but has a destabilizing impact on Faraday waves. In addition, the increase in the electric field intensity and forcing frequency enhances the inherent instability but has a complex impact on the Faraday waves, depending on the intensity of the viscous dissipation, which is approximately proportional to the square of the wavenumber.