Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface

TL;DR

This paper develops and analyzes unconditionally stable first and second order numerical schemes for simulating droplet motion driven by mean curvature and contact line dynamics on rough surfaces, using a geometric gradient flow approach.

Contribution

It introduces novel numerical schemes based on explicit moving boundaries, semi-Lagrangian methods, and predictor-corrector techniques for geometric droplet motion.

Findings

Schemes are proven stable and convergent for quasi-static dynamics.

Numerical examples validate accuracy and long-term behavior.

Methods effectively handle complex surface inhomogeneities.

Abstract

We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a Hilbert manifold. We propose unconditionally stable first/second order numerical schemes to simulate this geometric motion of the droplet, which is described using motion by mean curvature coupled with moving contact lines. The schemes are based on (i) explicit moving boundaries, which decouple the dynamic updates of the contact lines and the capillary surface, (ii) a semi-Lagrangian method on moving grids and (iii) a predictor-corrector method with a nonlinear elliptic solver upto second order accuracy. For the case of quasi-static dynamics with continuous spatial variable in the numerical schemes, we prove the stability and convergence of the first/second order numerical schemes. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples - including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets - are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs).