An obstacle approach to rate independent droplet evolution

TL;DR

This paper introduces a new obstacle problem-based solution concept for rate independent droplet motion with contact angle hysteresis, establishing uniqueness, regularity, and convergence results in a star-shaped setting.

Contribution

It develops a novel obstacle problem framework for droplet evolution, proving comparison principles, regularity, and convergence of schemes, which were not previously available.

Findings

Obstacle solutions are uniquely characterized by stability and slope conditions.

Contact line regularity is nearly optimal at $C^{1,1/2-}$.

Minimizing movements schemes converge to obstacle solutions.

Abstract

We consider a toy model of rate independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We introduce a notion of solutions based on an obstacle problem. These solutions jump ``as late and as little as possible", a physically natural property that energy solutions do not satisfy. When the initial data is star-shaped, we show that obstacle solutions are uniquely characterized by satisfying the local stability and dynamic slope conditions. This is proved via a novel comparison principle, which is one of the main new technical results of the paper. In this setting we can also show the (almost) optimal $C^{1,1/2-}$-spatial regularity of the contact line. This regularity result explains the asymptotic profile of the contact line as it de-pins via tangential motion similar to de-lamination. Finally we apply our comparison principle to show the convergence of minimizing movements schemes to the same obstacle solution, again in the star-shaped setting.