On the geometry of rate independent droplet evolution

TL;DR

This paper introduces a simplified model for droplet motion on surfaces with contact angle hysteresis, establishing existence of solutions and deriving PDE conditions that describe the dynamic contact angle behavior over time.

Contribution

It presents a novel toy model based on the Bernoulli free boundary problem, demonstrating existence of energy solutions and characterizing their PDE properties.

Findings

Energy solutions satisfy the dynamic contact angle condition almost everywhere.

Existence of solutions is proven via a minimizing movement scheme.

The model captures key features of droplet evolution with hysteresis.

Abstract

We introduce a toy model for rate-independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We consider a notion of energy solutions and show existence by a minimizing movement scheme. The main result of the paper is on the PDE conditions satisfied by general energy solutions: we show that the solutions satisfy the dynamic contact angle condition $\mathcal{H}^{d-1}$-a.e. along the contact line at every time.