Nonlinear dynamics of temporally excited falling liquid films

The two-dimensional spatiotemporal dynamics of falling thin liquid films on a solid vertical wall periodically oscillating in its own plane is studied within the framework of long-wave theory. A pertinent nonlinear evolution equation referred to as the temporally modulated Benney equation (TMBE) is derived and its solutions are investigated numerically. The bifurcation diagram of the Benney equation (BE) describing the film dynamics in the unforced regime is computed depicting the domains of linearly stable, linearly unstable bounded, and unbounded behaviors. The solutions obtained for film dynamics via the BE are compared to those documented for direct numerical simulations of the Navier–Stokes equations (NSE). The comparison demonstrates that the BE constitutes an accurate asymptotic reduction of the NSE in the domain preceding the transition to the regime of its unbounded solutions. It is found that periodic planar boundary excitation does not alter the fundamental unforced bifurcation structure and the spatial topological structure of the interfacial waves. The film evolution as described by TMBE is found to be primarily of quasiperiodic tori complemented by several types of strange attractors. In the case of relatively thick films increase of either the amplitude or the frequency of wall oscillation results in significant decrease of the peak-to-peak size of interfacial waves.