The dynamics of liquid films, as described by the diffuse-interface model

The dynamics of a thin layer of liquid between a flat solid substrate and an infinitely thick layer of saturated vapor is examined. The liquid and vapor are two phases of the same fluid governed by the diffuse-interface model. The substrate is maintained at a fixed temperature, but in the bulk of the fluid, the temperature is allowed to vary. The slope epsilon of the liquid/vapor interface is assumed to be small, as is the ratio of its thickness to that of the film. Three asymptotic regimes are identified, depending on the vapor-to-liquid density ratio rho v/rho (l). If rho v/rho (l) similar to 1 (which implies that the temperature is comparable, but not necessarily close, to the critical value), the evolution of the interface is driven by the vertical flow due to liquid/vapor phase transition, with the horizontal flow being negligible. In the limit rho v/rho (l) -> 0, it is the other way around, and there exists an intermediate regime, rho v/rho (l) similar to epsilon (4/3), where the two effects are of the same order. Only the rho v/rho (l) -> 0 limit is mathematically similar to the case of incompressible (Navier-Stokes) liquids, whereas the asymptotic equations governing the other two regimes are of different types.