Phase diagram for the onset of circulating waves and flow reversal in inclined falling films

The onset of circulating waves, i.e. waves with a circulating eddy in the main wave hump, and the onset of flow reversal, i.e. a vortex in the first capillary minimum, in inclined falling films is investigated as a function of the Reynolds number and inclination number using the weighted integral boundary layer (WIBL) model and direct numerical simulations (DNS). Analytical criteria for the onset of circulating waves and flow reversal based on the wave celerity, the average film thickness and the maximum and minimum film thickness have been approximated using self-similar parabolic velocity profiles. This approximation has been validated by second-order WIBL and DNS simulations. It is shown that the onset of circulating waves in the phase diagram for homoclinic solutions (waves of infinite wavelength) is strongly dependent on the inclination, but independent of the streamwise viscous dissipation effect. On the contrary, the onset of flow reversal shows a clear dependence on the viscous dissipation. Furthermore, simulation results for limit cycles (finite wavelength) reveal a strong increase of the corresponding critical Reynolds number with the excitation frequency. Additionally, a critical ratio between the maximum and substrate film thickness (value of approximately 2.5) was found for the onset of circulating waves, which is independent of wavelength, inclination, viscous dissipation and Reynolds number.