Three-dimensional wave evolution on electrified falling films

We consider the full 3D dynamics of a thin falling liquid film on a flat plate inclined at some non-zero angle to the horizontal. In addition to gravitational effects, the flow is driven by an electric field, which is normal to the substrate far from the flow. We study both the cases of overlying and hanging films, where the liquid rests above and below the substrate respectively. Starting with the Navier-Stokes equations coupled with electrostatics, a fully nonlinear 2D Benney equation for the interfacial dynamics is derived valid for waves that are long compared to the film thickness. The weakly nonlinear evolution is governed by a Kuramoto-Sivashinsky equation with a non-local term due to the electric field effect. The electric field term is linearly destabilising and produces growth rates proportional to the cube of the size of the wavenumber vector of the perturbations. It is found that transverse gravitational instabilities are always present for hanging films and lead to unboundedness of solutions even in the absence of electric fields. For overlying films and a restriction on the strength of the electric field, the equation possesses bounded solutions. This 2D equation is studied numerically for the case of periodic boundary conditions in order to assess the effects of inertia, electric field strength the dimensions of the periodic domain. Rich dynamical behaviours are observed and classified in various parameter windows. For subcritical Reynolds number flows, a sufficiently strong electric field can promote non-trivial dynamics for some choices of domain dimensions, leading to fully 2D evolutions for the interface. These dynamics are also found to produce spatiotemporal chaos on sufficiently large domains. For supercritical flows, such 2D chaotic dynamics emerge in the absence of a field, and its presence enhances the amplitude of the fluctuations and broadens their spectrum.