Long-wave instabilities of evaporating/condensing viscous film flowing down a wavy inclined wall: Interfacial phase change effect of uniform layers

Abstract

The interfacial phase change effect on a thin film flowing down an undulated wall has been investigated in the present study. The study is performed for a general periodic undulated bottom of moderate steepness that is long compared to the film thickness, followed by a case study over the sinusoidal bottom. The long-wave instabilities of the ununiform film are used by deriving a nonlinear evolution equation in the classical long-wave expansion method framework. The one-equation model can track the free surface evolution and involve the bottom undulation, viscosity, gravity, surface tension, and phase change (evaporation/condensation) effects. Linear stability analysis shows that the bottom steepness ζ has a dual role. In the downhill region, increasing ζ destabilizes, whereas increasing ζ stabilizes in the uphill region. Weakly nonlinear waves are studied using the method of multiple scales to obtain the complex Ginzburg–Landau equation. The results show that both supercritical and subcritical solutions are possible for evaporating and condensate film. Interestingly, while one subcritical region is visible for an evaporating film, two subcritical unstable regions are found for condensate film. The numerical solution of the free-surface equation demonstrates the finite-amplitude behavior that tends to dry out for an evaporating film. For condensate film, the thickness increases rapidly. The rupture dynamics highly depend on the initial perturbation, and the bottom steepness has a negligible effect on it. Kutateladze number has a significant impact on the stability characteristic of the film flow as it represents a sort of efficiency of phase change that occurs at the interface.