Vapor-Liquid Steady Meniscus at a Superheated Wall: Asymptotics in an Intermediate Zone Near the Contact Line

Abstract

The study concerns steady configurations of a perfectly wetting liquid in contact with its pure vapor and a superheated substrate/wall maintained at a constant temperature. Despite the perfect wetting, the system is characterized by a finite apparent contact angle formed at a microscale, within a steady microstructure of the contact line, the finiteness owing itself to an actually dynamic situation caused by the evaporation process. The angle is assumed to be small here, which is the case for sufficiently small superheats. When macroscopically treating a steady meniscus, one typically implies that the wall is met at the contact angle given by the microstructure. This remains somewhat an intuitive, heuristic approach unless a more rigorous asymptotic matching is carried out between the meniscus and the microstructure, which is accomplished in the present paper by studying an intermediate zone connecting these two regions. The analysis, based upon a standard one-sided planar model of an evaporating liquid layer in the lubrication approximation, confirms the validity of the mentioned approach. A possible uncertainty in the definition of the contact angle is shown to be small given that the macroscopic curvature (i.e. that of the meniscus and of the wall) is small on the scale of the contact line microstructure.