Instabilities and Saturation of Electrified Thin Liquid Films
Demetrios T. Papageorgiou, Department of Mathematical Sciences, New Jersey Institute of Technology, 609 Cullimore Hall, 323 Martin Luther King Jr. Blvd, Newark, NJ 07102-1982 and Dmitri Tseluiko, Mathematical Sciences, New Jersey Institute of Technology, 609 Cullimore Hall, 323 Martin Luther King Jr. Blvd, Newark, NJ 07102-1982.
Under normal circumstances, a viscous liquid film that wets the upper side of a horizontal flat plate is stable. On the other hand, if the film wets the underside of the plate, gravity destabilizes the flow and different nonlinear dynamics can develop depending on the surface tension coefficient and the initial mass of the liquid layer. In the present study we consider the physical effect of stressing the liquid-air interface with an electric field applied normal to the plate. The liquid is modeled as a perfect conductor and the hydrodynamically passive region above it as a perfect dielectric. The plate is an electrode and the field is generated by placing a second parallel electrode at infinity so that there is a uniform vertical electric field at large lateral distances. We derive a long wave fully nonlinear model to describe the interfacial evolution. The equation has an additional nonlinear-nonlocal term due to the contribution of the electric Maxwell stresses to the stress tensor. This contribution is always destabilizing and can, in particular, destabilize the flow of a film wetting the upper side of the plate, and enhance the instability of the underlying film. The electric instability dominates at short waves but surface tension enters to keep the problem well-posed. We will present several theoretical findings including accurate numerical calculations that indicate that the solution (for both overlying and underlying films) remains positive and bounded for all time and touches down locally in infinite time. A theorem has been proven which states that positive smooth solutions satisfy global boundedness – the proof is based on estimating an appropriate energy functional.