Moving contact line with balanced stress singularities

TL;DR

This paper reexamines the moving contact-line problem using a finite-width interface model, showing how stress singularities can be balanced and deriving the dynamic contact angle based on physical parameters.

Contribution

It introduces a smoothed interface model that balances stress singularities and derives the dynamic contact angle considering finite interface width and slip conditions.

Findings

Stress singularity can be balanced by unbalanced surface stress.

Dynamic contact angle depends on surface tension, viscosity, and a non-dimensional parameter.

Navier boundary condition and Cox's hypothesis are derived from the same framework.

Abstract

A difficulty in the classical hydrodynamic analysis of moving contact-line problems, associated with the no-slip wall boundary condition resulting in an unbalanced divergence of the viscous stresses, is reexamined with a smoothed, finite-width interface model. The analysis in the sharp-interface limit shows that the singularity of the viscous stress can be balanced by another singularity of the unbalanced surface stress. The dynamic contact angle is determined by surface tension, viscosity, contact-line velocity and a single non-dimensional parameter reflecting the length-scale ratio between interface width and the thickness of the first molecule layer at the wall surface. The widely used Navier boundary condition and Cox's hypothesis are also derived following the same procedure by permitting finite-wall slip.