Electrified falling film flow over topography in the presence of a finite electrode

The effect of an electric field on a liquid film flowing down an inclined wall is examined. The liquid film is treated as a perfect conductor and the air above the film is treated as a perfect dielectric. The electric field is created by a single or periodically repeated electrode of arbitrary shape charged to establish a prescribed potential difference between itself and the liquid-film surface. The steady deformation of the free surface in the presence of the electric field is computed first on the assumption of a thin film and next within the Stokes-flow regime. Calculations are performed for flow over a plane wall and over a step of asymptotically small height. In the latter case, the focused electric field obtained by positioning a circular electrode directly above the step is found to eliminate the capillary ridge identified by previous authors without significantly disrupting the flow away from the step. This result is confirmed numerically for Stokes flow over a step of arbitrary height using the boundary-element method