Can hydrodynamic contact line paradox be solved by evaporation--condensation?

TL;DR

This paper explores whether evaporation and condensation can resolve the hydrodynamic contact line singularity, demonstrating mathematically that phase exchange can regularize the problem, but the relevant length scales are extremely small in practice.

Contribution

The study introduces a multi-scale analysis showing that evaporation-condensation can mathematically regularize the contact line singularity in partial wetting.

Findings

Phase exchange relieves the contact line singularity mathematically.

Mass conservation is maintained through balanced evaporation and condensation.

Characteristic length scales are extremely small for common fluids.

Abstract

We investigate a possibility to regularize the hydrodynamic contact line singularity in the configuration of partial wetting (liquid wedge on a solid substrate) via evaporation-condensation, when an inert gas is present in the atmosphere above the liquid. The no-slip condition is imposed at the solid-liquid interface and the system is assumed to be isothermal. The mass exchange dynamics is controlled by vapor diffusion in the inert gas and interfacial kinetic resistance. The coupling between the liquid meniscus curvature and mass exchange is provided by the Kelvin effect. The atmosphere is saturated and the substrate moves at a steady velocity with respect to the liquid wedge. A multi-scale analysis is performed. The liquid dynamics description in the phase-change-controlled microregion and visco-capillary intermediate region is based on the lubrication equations. The vapor diffusion is considered in the gas phase. It is shown that from the mathematical point of view, the phase exchange relieves the contact line singularity. The liquid mass is conserved: evaporation existing on a part of the meniscus and condensation occurring over another part compensate exactly each other. However, numerical estimations carried out for three common fluids (ethanol, water and glycerol) at the ambient conditions show that the characteristic length scales are tiny.