Electrostatic Effects on Linear and Nonlinear Waves in Hanging Film Flows

The dynamics of a liquid film wetting the underside of a semi-infinite horizontal substrate are investigated. Gravity causes the film to be unstable and the objective of this work is to control such instabilities by using horizontal electric fields. The film wets a semi-infinite dielectric solid block and is bounded below by air. The electric fields from these two regions are coupled to those in the liquid film and contribute to the hydrodynamics through the interfacial boundary conditions by introducing electric Maxwell stresses in the stress tensor. As a result novel nonlocal terms arise and their effect is studied both in the linear regime as well as nonlinearly through a derivation of a system of long-wave one-dimensional evolution equations describing the interfacial position and the leading order horizontal velocity in the film. When surface tension is present there exists a band of unstable wavenumbers rendering the film long-wave unstable. It is shown that the size of this band decreases asymptotically to zero as the electric field strength is increased (physically this means that any system of finite horizontal length can be fully stabilized by a sufficiently strong electric field). The derived nonlinear system supports nonlinear coherent structures in the form of traveling waves of permanent form, and such solutions are computed numerically for a range of parameters. The production of such traveling waves may be useful in enhancing the transport properties in hanging film systems.