Abstract
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 ε 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 ρv/ρl. If ρv/ρl ∼ 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 ρv/ρl → 0, it is the other way around, and there exists an intermediate regime, ρv/ρl ∼ ε4/3, where the two effects are of the same order. Only the ρv/ρl → 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.
Highlights
The diffuse-interface model (DIM) originates from the idea of van der Waals1 and Korteweg2 that intermolecular attraction in fluids can be modeled by relating it to macroscopic variations of the fluid density
This approach was incorporated into hydrodynamics: more comprehensive models have been developed for multi-component fluids with variable temperature3,4 and simpler ones for single-component isothermal fluids5 or singlecomponent isothermal and incompressible fluids6–9 [in the last case, the van der Waals force does not depend on the density, but on a certain “order parameter” satisfying the Cahn–Hilliard equation]
Various versions of the DIM have been used in applications, such as nucleation, growth, and collapse of vapor bubbles,10–13 drops impacting on a solid wall,14 and contact lines in fluids
Summary
The diffuse-interface model (DIM) originates from the idea of van der Waals and Korteweg that intermolecular attraction in fluids can be modeled by relating it to macroscopic variations of the fluid density. A boundary condition describing the interaction of the fluid and substrate is needed Two versions of such a condition have been suggested: one involving the nearsubstrate density and its normal derivative, and another prescribing just the density.. Assuming that the flow is isothermal and the saturated-vapor density ρv is much smaller than the liquid density ρl, Pismen and Pomeau derived an asymptotic version of the DIM similar to the thin-film approximation of the Navier–Stokes equations for incompressible fluids. It has been argued, that in some, if not most, common fluids, including water, liquid/vapor interfaces are not isothermal.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
