Three-dimensional wave evolution on electrified falling films

Abstract

We consider the full three-dimensional 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. This extends the work of Tseluiko & Papageorgiou (J. Fluid Mech., vol. 556, 2006b, pp. 361–386) by including transverse dynamics. We study both the cases of overlying and hanging films, where the liquid lies above or below the substrate, respectively. Starting with the Navier–Stokes equations coupled with electrostatics, a fully nonlinear two-dimensional 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 $|\unicode[STIX]{x1D743}|^{3}$, where $\unicode[STIX]{x1D743}$ is the wavenumber vector of the perturbations. It is found that transverse gravitational instabilities are always present for hanging films, and this leads to unboundedness of nonlinear solutions even in the absence of electric fields – this is due to the anisotropy of the nonlinearity. For overlying films and a restriction on the strength of the electric field, the equation is well-posed in the sense that it possesses bounded solutions. This two-dimensional equation is studied numerically for the case of periodic boundary conditions in order to assess the effects of inertia, electric field strength and the size of the periodic domain. Rich dynamical behaviours are observed and reported. For subcritical Reynolds number flows, a sufficiently strong electric field can promote non-trivial dynamics for some choices of domain size, leading to fully two-dimensional evolutions of the interface. We also observe two-dimensional spatiotemporal chaos on sufficiently large domains. For supercritical flows, such two-dimensional chaotic dynamics emerges in the absence of a field, and its presence enhances the amplitude of the fluctuations and broadens their spectrum.