Viscous flows in corner regions: Singularities and hidden eigensolutions

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.

Submitted to the International Journal for Numerical Methods in Fluids

References in corpus (2)

• Viscous Flow Over a Chemically Patterned Surface
• A continuum model for the flow of thin liquid films over intermittently chemically patterned surfaces

Cited by in corpus (4)

• Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process
• A Kinematic Evolution Equation for the Dynamic Contact Angle and some Consequences
• Stability and Bifurcation of Dynamic Contact Lines in Two Dimensions
• Putting the micro into the macro: A molecularly-augmented hydrodynamic model of dynamic wetting applied to flow instabilities during forced dewetting