Derivation of an Evolution Equation for Two-Dimensional Waves on Thin Films

Abstract

We examine wave propagation on thin liquid films subjected to gravity, fluid friction, surface tension, and Marangoni effects. The physical configuration is a thin liquid layer on a planar incline. Following previous studies, the Marangoni effect is incorporated by a constant surface tension gradient and yields a non-convex flux function in our thin film equation. We extend previous studies by deriving the thin film equation governing two-dimensional waves on the liquid layer. We then derive a simplified evolution equation governing weakly nonlinear, quasi-planar, and weakly dissipative waves on the layer. When the undisturbed state is in the vicinity of an inflection point in the streamwise component of the flux function, the mixed nonlinearity, fourth order dissipation and the transverse modulations interact over time scales on the order of the scaled amplitude to the negative second power. The effect the transverse modulations is found to be intrinsically nonlinear.