Viscous flow of a volatile liquid on an inclined heated surface

We investigate the effects of evaporation on a gravity-driven flow of a viscous liquid on a heated solid surface. Vapor molecules are adsorbed on the dry areas of the solid and form a microscopic adsorbed film. The thickness of this film is calculated from the formulas for disjoining pressure and the principles of equilibrium thermodynamics. A lubrication-type approach is used to derive an evolution equation capable of describing both the macroscopic shape of the vapor-liquid interface and the adsorbed film on the vapor-solid interface. Under the conditions of negligible evaporation, the numerical solution of the evolution equation predicts translational motion and formation of capillary ridge, in agreement with previous investigations. Moderate evaporation is shown to slow down the flow and decrease the height of the capillary ridge, which implies a stabilizing effect of evaporation on the well-known instability observed in gravity-driven thin film flows. We also study the combined effects of evaporation and thermocapillary stresses and show that the latter act to reduce the velocity of the downward motion, but increase the height of the capillary ridge. Apparent contact angles are found from the solution and shown to increase with evaporation and contact line speed. For strong evaporation, steady state solutions are found such that evaporation balances the downward motion of the interface under the action of gravity.