Viscous electrified film flow over step topography

The steady, gravity-driven flow of a liquid film over a topographically structured substrate is investigated. The analysis is based on a model nonlinear equation for the film thickness derived on the basis of long-wave asymptotics. The free-surface shape is expanded in a regular asymptotic expansion in powers of the topography amplitude, and solutions are obtained up to second order. Solutions are constructed for downward steps, upward steps, and rectangular trenches, and the results are compared favorably with numerical solutions of the nonlinear model equation. The results indicate that all of the salient features previously found for film flows over steps and into trenches are captured by the small-step asymptotics, including the capillary ridge formed just above a downward step and oscillations upstream of an upward step. We derive analytical expressions for the period and amplitude of these oscillations. The effect of a normal electric on the film surface shape is also investigated on the assumption that both the film and the medium above the film behave as perfect dielectrics. Again, the small-amplitude asymptotics describe the essential characteristics of the free surface, including oscillations downstream of a downwards step with a quantifiable period and amplitude. It is established analytically and numerically that the amplitude of interfacial oscillations just upstream of a step decreases with an increase of the electric field strength in the case of perfect conductors, but increases to a limiting value for perfect dielectrics. It is found that nonlinear solutions are in excellent agreement with the small-amplitude theory even for relatively large topography amplitudes.