Celerity-Amplitude Relations for Solitary Waves on Vertically Falling Films

Abstract

Experimental data relating the amplitude and celerity of solitary waves on vertically falling and naturally excited films are presented over a wide of range of fluid properties and flow rates (2 < Ka < 3883, 0.08 < We < 35). It is found that for a fixed set of fluid properties (Ka) and flow rate (1/We), the celerity-amplitude relationship for all finite amplitude waves that exist on the film is nearly linear with a slope that decreases with the flow rate (1/We). For viscous fluids (2 < Ka < 129), the observed normalized maximum wave amplitude increased monotonically with the flow rate (1/We) and saturated at values around 3 (hmax ≈ 3) and found to be nearly independent of the fluid viscosity (Ka). For less viscous fluids (206 < Ka < 3883) the waves were found to be continuously evolving with maximum normalized amplitudes of about 10. A recently developed two equation h–q model is used to analyze the spatio-temporal dynamics of waves on the film. The model predictions corroborate the hypothesis that the existence of very large amplitude waves for less viscous fluids is due to the phenomenon of flow reversal (or existence of up-flows) near the wall. The model predicts a critical Ka value below which no flow reversal occurs. It is also shown that once the wave amplitude exceeds a critical value, the amplitude-celerity relationship is nearly linear for all solitary waves that exist on the film and is independent of the inlet forcing frequency or amplitude or initial conditions. Local bifurcation analysis in the traveling wave coordinate, computational results on extended domains with periodic inlet forcing, and computational results on finite domains with periodic boundary conditions are used to explain the experimentally observed wave celerity-ampliude relations on naturally excited films.