Numerical analysis of a weighted-residual integral boundary-layer model for nonlinear dynamics of falling liquid films

The nonlinear dynamics of thin liquid films falling on a vertical plane is investigated numerically using the first-order time-dependent weighted-residual integral boundary layer (WRIBL) equations derived by Ruyer-Quil and Manneville (2000). We validate the WRIBL equations by comparison of its solutions with those of its second-order version, solutions obtained by both stationary and time-dependent direct numerical simulation and experiments. We find that sufficiently close to the stability threshold of the system with periodic boundary conditions, the emerging waves are of γ1-type. However, beyond a secondary bifurcation threshold, γ2-type waves emerge and can coexist with γ1-waves. The analysis of the WRIBL equations reveals, similar to the first-order Benney equation (BE), the existence of both periodic traveling wave (TW) and aperiodic non-stationary wave (NSW) flows. It is shown that although the WRIBL equations display bounded solutions for significantly larger values of the Reynolds number than the BE, they may exhibit negative flow rate which consists of reverse flow against gravity. The threshold for emergence of these solutions is a zero local flow rate that appears, for sufficiently large Kapitza numbers, to correspond to a specific value of the normalized wave height.