Electrified thin-film flow over inclined topography

We consider both a long-wave model and a first-order weighted-residual integral boundary layer (WIBL) model in the investigation of thin film flow down a topographical incline whilst under the effects of a normal electric field. The liquid is assumed to be a perfect dielectric, although is trivially extended to the case of a perfect conductor. The perfect dielectric case with no topography includes a simple modified electric Weber number which incorporates the relative electrical permittivity constant into itself. Linear stability analysis is carried out for both models, and critical Reynolds numbers which depend on the electric Weber number and the capillary number are produced. Regions of stability, convective instability and absolute instability are then determined for both models in terms of our electric Weber number and Reynolds number parameters in the case of no topography. Time-dependent simulations are produced to corroborate the aforementioned regions and investigate the effect of normal electric field strength in addition to sinusoidal and rectangular topographical amplitude on our system for various domain sizes. For the time-dependent simulations we find strong agreement with the linear stability analysis, and the results suggest that the inclusion of a normal electric field may have some stabilising properties in the long-wave model which are absent in the case of a flat wall, for which the electric field is always linearly destabilising. This stabilising effect is not observed for the same parameters in the WIBL model with a sinusoidal wall, although a similar effect is noticed in the WIBL model with a rectangular wall. We also investigate the simultaneous effect of domain size, wall amplitude and electric field strength on the critical Reynolds numbers for both models, and find that increasing the electric field strength can make large-amplitude sinusoidal topography stabilising rather than destabilising for the long-wave model. Continuation curves of steady solutions and bifurcation diagrams are also produced, and comparisons between the two models are made for various parameter values, which show excellent agreement with the literature. Subharmonic branches and time-periodic solutions are additionally included, similarly showing very good agreement with the literature.