Nonlinear Dynamics of Electrified Thin Liquid Films

We study a nonlinear nonlocal evolution equation describing the hydrodynamics of thin films in the presence of normal electric fields. The liquid film is assumed to be perfectly conducting and to completely wet the upper or lower surface of a horizontal flat plate. The flat plate is held at constant voltage, and a vertical electric field is generated by a second parallel electrode kept at a different constant voltage and placed at a large vertical distance from the bottom plate. The fluid is viscous, and gravity and surface tension act. The equation is derived using lubrication theory and contains an additional nonlinear nonlocal term representing the electric field. The electric field is linearly destabilizing and is particularly important in producing nontrivial dynamics in the case when the film rests on the upper side of the plate. We give rigorous results on the global boundedness of positive periodic smooth solutions, using an appropriate energy functional. We also implement a fully implicit numerical scheme and perform extensive numerical experiments. Through a combination of analysis and numerical experiments we present evidence for the global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena, which are also observed in hanging films when electric fields are absent.