Linear stability analysis of evaporating falling liquid films

Abstract

We consider the linear stability of evaporating thin films falling down an inclined plate. The one sided-model presented first by “Burelbach, J.P., Bankoff, S.G., Davis, S.H., 1988, Nonlinear stability of evaporating/condensing liquid films, Journal of Fluid Mechanics 195, 463–494. ” was implemented to decouple the dynamics of the liquid than those of the vapor at the interface, at which the evaporation is modeled based on a thermal equilibrium approach. We consider the base state solution derived by “Joo, S., Davis, S.H., Bankoff, S., 1991, Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers, Journal of Fluid Mechanics 230, 117–146. ” which is based on the slow evaporation assumption. In previous works, only low dimensional models. i.e. the long wave theory, have been analysed for evaporating liquid films. Conversely in this paper, we extend the Orr-Sommerfeld eigenvalue problem for a film falling down a heated wall to include evaporation effects namely, vapor recoil and mass loss. As expected, we observe that the long wave theory fails in predicting the correct behavior when the inertia is strong or the wavenumber k is large. We confirm that the instability induced by vapor recoil (E-mode) behaves in a similar fashion to the instability due to the thermocapillary effect (S-mode). Both the S-mode and the E-mode can enhance each other, specially, at low Reynolds numbers Re. Moreover, we examine the perturbation energy budget to have an insight into the instabilities mechanism. We show that the presence of evaporation adds a new term corresponding to the work done by vapor recoil at the interface (VRE). We also find that the main contributor to the perturbation kinetic energy in the unstable E-mode is the work done by shear stress while VRE is negligible, unless Re << 1. Simpler analytical expressions of the energy balance terms near the instability threshold are obtained by using a long wave approximation.