Boundary conditions for dynamic wetting -- A mathematical analysis

TL;DR

This paper uses compatibility analysis to examine boundary conditions in dynamic wetting models, revealing conditions under which regular solutions are possible and identifying potential singularities at the contact line.

Contribution

It applies compatibility analysis to continuum models of dynamic wetting, providing explicit expressions for pressure and curvature at the contact line, and assesses the regularity of solutions.

Findings

Explicit pressure and curvature expressions at the contact line.

Regular solutions may still be singular in some models.

Compatibility conditions are crucial for regularity in dynamic wetting models.

Abstract

The moving contact line paradox discussed in the famous paper by Huh and Scriven has lead to an extensive scientific discussion about singularities in continuum mechanical models of dynamic wetting in the framework of the two-phase Navier-Stokes equations. Since the no-slip condition introduces a non-integrable and therefore unphysical singularity into the model, various models to relax the singularity have been proposed. Many of the relaxation mechanisms still retain a weak (integrable) singularity, while other approaches look for completely regular solutions with finite curvature and pressure at the moving contact line. In particular, the model introduced recently in (Lukyanov, Pryer, Langmuir 2017) aims for regular solutions through modified boundary conditions. The present work applies the mathematical tool of compatibility analysis to continuum models of dynamic wetting. The basic idea is that the boundary conditions have to be compatible at the contact line in order to allow for regular solutions. Remarkably, the method allows to compute explicit expressions for the pressure and the curvature locally at the moving contact line for regular solutions to the model of Lukyanov and Pryer. It is found that solutions may still be singular for the latter model.