Numerical estimate of the viscous damping of capillary-gravity waves: A macroscopic depth-dependent slip-length model

TL;DR

This paper introduces a numerical method using a depth-dependent slip-length model to regularize contact line singularities in viscous capillary-gravity wave computations, providing physically meaningful damping estimates.

Contribution

It presents a novel numerical regularization scheme with a macroscopic depth-dependent slip condition to accurately estimate viscous damping in capillary-gravity waves.

Findings

The method yields consistent results with existing asymptotic approaches.

The approach effectively regularizes the contact line singularity.

It provides physically meaningful estimates of viscous damping.

Abstract

We propose a numerical approach to regularize the contact line singularity appearing in the computation of viscous capillary-gravity waves with moving contact line in cylindrical containers. The linearized Navier-Stokes equations are complemented by a macroscopic Navier-like slip condition on the container side wall, with a depth-varying slip-length bridging a free-slip condition at the contact line to a no-slip condition further away from the meniscus. In accordance with suggestions from the literature, this characteristic penetration depth is chosen as the typical Stokes layer thickness pertaining to the eigenfrequency. Since the latter is unknown, the resulting nonlinear eigenvalue problem is first simplified using the inviscid eigenfrequency to estimate the Stokes layer thickness. The solution is then shown to provide consistent results when compared to the asymptotic approaches of the literature using the inviscid eigenmodes to estimate the different sources of dissipation with the exception of the neglected corner regions. This demonstrates that such numerical regularization scheme is suitable to retrieve physically meaningful results.