Electrified film flow over step topography at zero Reynolds number

The flow of a liquid film over step topography under the influence of an electric field is considered in the limit of zero Reynolds number. The particular topographies considered include a flat wall with a downward step or an upward step, or a flat wall which is indented with a rectangular trench. A uniform electric field is imposed at infinity in the direction normal to the flatwall. The air above the film is treated as a perfect dielectric. The liquid in the film is assumed to behave either as a perfect conductor or as a perfect dielectric whose dielectric constant in general differs from that in the air. Asymptotic results are derived on the assumption of small step height, and formulas are presented for the first-order correction to the free-surface deformation due to the topography. It is demonstrated that, in an appropriate long-wave limit, the solutions approach those obtained using the lubrication approximation. Finally, the small-step asymptotics are favourably compared with numerical solutions for Stokes flow over steps of arbitrary height computed using the boundary-element method. In summary, it is shown that asymptotic models based on small-amplitude step topography provide simple formulas which are effective in describing the flow even for moderate step amplitudes, making them an efficient analytical tool for solving practical film-flow problems.