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

TL;DR

This paper analyzes the behavior of thin liquid films using the diffuse-interface model, identifying three regimes based on vapor-to-liquid density ratios and deriving corresponding asymptotic equations.

Contribution

It introduces a detailed asymptotic analysis of liquid-vapor interface dynamics across different density ratio regimes within the diffuse-interface framework.

Findings

Three asymptotic regimes identified based on vapor-to-liquid density ratio.

Different mathematical models derived for each regime.

In the limit of low vapor density, the model aligns with incompressible Navier-Stokes equations.

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 $\varepsilon$ 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 $\rho_{v}/\rho_{l}\sim1$ (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 $\rho_{v}/\rho_{l}\rightarrow0$, it is the other way around, and there exists an intermediate regime, $\rho_{v}/\rho_{l}\sim\varepsilon^{4/3}$, where the two effects are of the same order. Only the $\rho_{v}/\rho_{l}\rightarrow0$ 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.