Rate independent capillary motion on a narrow Wilhelmy plate

TL;DR

This paper analyzes a rate-independent model of capillary motion in a Wilhelmy plate setup, demonstrating stability and asymptotic behavior as the container width becomes very large, leading to a simplified volume-unconstrained evolution.

Contribution

It establishes the stability of energy solutions in the large container limit and derives asymptotics for the energy independent of the container-to-plate ratio.

Findings

Stability of energy solutions as container width increases

Disappearance of volume constraint in the limit

Development of asymptotic energy estimates

Abstract

We study a rate independent energetic model of the Wilhelmy plate experiment in capillarity. The evolution is driven by vertical motions of the plate. We show stability of energy solutions to the evolution, in the sense used in the rate-independent systems literature, as the ratio between container width and plate width goes to infinity. In particular, we show that the volume-constraint for the finite-ratio problem disappears in the limit. This leads to a volume-unconstrained Dirichlet-forced evolution, a setting where monotonicity, uniqueness, and contact line regularity properties have been established in previous literature. Our result is based on using comparison principle techniques for the prescribed mean curvature equation with capillary contact angle condition that characterizes the liquid surface at equilibrium. Through barrier arguments, we are able to develop asymptotics for the energy which give us control independent of the container-to-plate ratio.