Nonlinear Dynamics of Electrostatic Faraday Instability in Thin Films

A nonlinear inertial lubrication model is developed for the evolution of an interface between a perfect conducting liquid and a perfect dielectric gas subject to periodic electrostatic forcing. Inertial thin films are shown to become unstable to finite-wavelength Faraday modes at the onset, prior to the long-wave pillaring instability reported in the full lubrication limit. The pillaring-mode instability is subcritical with the interface approaching either the top or bottom wall, depending on the liquid-gas holdup. On the other hand, the Faraday modes exhibit subharmonic or harmonic oscillations that saturate to standing waves at low-forcing amplitudes. Unlike the pillaring mode, wherein the interface approaches the wall, Faraday modes may exhibit saturated standing waves when the instability is subcritical. At higher forcing amplitudes the interface may approach either wall, again depending on the liquid-gas holdup. Also, a gravitationally unstable configuration of such thin films cannot be stabilized by periodic electrostatic forcing, unlike mechanical Faraday forcing. In this case, it is observed that the interface exhibits oscillatory sliding behavior, approaching the wall in an “earthworm-like" motion.