Weighted-residual integral boundary-layer model of temporally excited falling liquid films

The first-order time-modulated weighted-residual integral boundary layer (TMBL) equations are derived and analyzed. The non-linear dynamics of thin liquid films falling on a vertical oscillating plane is investigated numerically using these equations. The set of TMBL equations admits a solution corresponding to a flat film and a temporally periodic volumetric flow rate. Using Floquet theory we investigate stability of this solution. In the region of instability of time-periodic flow, forcing of the traveling wave and non-stationary wave regimes arising in the unmodulated system, results in the emergence of quasiperiodic and apparently chaotic regimes, respectively. The wave regimes of the γ1- and γ2-types survive the forcing imparted by plane oscillation. The possibility of the emergence of flow reversal is also addressed. The existence of regions of asymptotical stability of the flat film with a temporally periodic flow rate provides the window for reduction or even suppression of the waviness of the film interface. Numerical investigation of the non-linear dynamics of the modulated film confirms the analytical results arising from Floquet theory.