Nonlinear dynamics of a thin liquid film deposited on a laterally oscillating corrugated surface in the high-frequency limit

Abstract

The nonlinear dynamics of thin liquid films deposited on various periodically corrugated substrates, both left-right symmetric and asymmetric, subjected to lateral vibration in the high-frequency limit is investigated. The method used to derive the governing evolution equation is based on the long-wave approximation, multiscale time expansion, and averaging over the fast time scale. The resulting evolution equation contains the effects of gravity, capillarity, vibration, and the substrate topology. The initial-boundary-value problem associated with this evolution equation is numerically solved and the system behavior is investigated for a variety of parameter sets. Typical patterns emerging as a result of the film evolution include hump formation within the troughs of the substrate and homogenized coatings whose configuration resembles that of the substrate, as well as the possibility of film rupture. We show that the choice of the vibration parameters and the topological features of the substrate may be used for controlling the shape of the film interface as well as its properties such as the amplitude, continuity, or rupture. Together with the film profiles stationary in terms of the averaged film interface with respect to the slow time scale, time evolution of the total (comprising of the averaged and pulsating components) interfacial profiles and streamline maps is presented to illustrate the film flow. We carry out Floquet stability analysis of periodically replicated steady states for the time-independent problem, linear stability analysis based on a reduced low-order projection approximation for the time-dependent problem, and stability analysis with respect to disturbances of a larger wavelength. We have found also that in the case of two-dimensional corrugated substrates, the vast majority of two-dimensional steady-state flows in terms of the averaged film interface exhibit stability in three dimensions with respect to small perturbations in the transverse spatial direction.