Dynamics of nonlinear air-blown waves on falling viscous liquid films

The dynamics of thin viscous liquid films flowing down an inclined wall under gravity in the presence of an upward flowing high-speed air stream is considered. The air stream induces nonlinear waves on the interface and asymptotic solutions are developed to derive a non-local evolution equation forced by the air pressure which is obtained analytically, and incorporating a constant tangential stress. Benney equations in the capillary (strong surface tension) and inertio-capillary regimes are derived and studied. The air stream produces Turing-type short wave instabilities in sub-critical Reynolds number regimes that would be stable in the absence of the outer flow. Extensive numerical experiments are carried out to elucidate the rich dynamics in the above-mentioned short-wave regime. The stability of different branches of solutions of non-uniform steady states is carried out, along with time-dependent nonlinear computations that are used to track the large-time behaviour of attractors. A fairly complete picture of different solution types are categorised in parameter space. The effect of the Reynolds number on the wave characteristics in the inertio-capillary regime is also investigated. It is observed that, for each value of the slenderness parameter $\delta$ , there exists a critical Reynolds number $R_c$ above which the solutions become unbounded by encountering finite-time singularities. Increasing the air speed significantly decreases $R_c$ , making the system more prone to large amplitude singular events even at low Reynolds numbers when the system would have been stable in the absence of the air stream.