Viscous flows in corner regions: Singularities and hidden eigensolutions

TL;DR

This paper investigates numerical challenges in simulating viscous flows in corner regions, identifying singularities and eigensolutions that affect solution stability, and proposes a method to incorporate eigensolutions for mesh-independent results.

Contribution

It introduces a novel approach to include eigensolutions in finite element methods to address corner flow singularities and mesh dependence issues.

Findings

Eigensolutions dominate for supercritical corner angles.

Incorporating eigensolutions stabilizes numerical results.

Standard methods become mesh-dependent above a critical angle.

Abstract

Numerical issues arising in computations of viscous flows in corners formed by a liquid-fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the `moving contact-line problem' and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite-element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh-dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh-independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite element method in the context of the present problem are discussed.