Global dynamic stability of contact lines in fluids: 2-D droplet problem

TL;DR

This paper studies the stability of fluid contact lines in a 2D droplet model, proving that solutions near equilibrium exist globally and converge exponentially, considering gravity, surface tension, and dynamic contact angles.

Contribution

It develops a framework of a priori estimates and demonstrates exponential convergence of solutions to equilibrium in a complex contact line fluid model.

Findings

Global solutions exist for initial data close to equilibrium.

Solutions converge exponentially fast to equilibrium.

The model accounts for dynamic contact angles and full contact point motion.

Abstract

In this paper, we investigate the dynamics of an incompressible viscous Navier-Stokes fluid evolving above a one-dimensional flat surface. The fluid is subject to a uniform gravitational field and capillary forces acting along the free boundary. The interface between the fluid and the surrounding air is a free surface whose motion is driven by gravity, surface tension, and the fluid velocity field. The triple-phase intersections where the fluid, the air above the vessel, and the solid vessel wall meet are referred to as contact points, and the angles formed there are called contact angles. The model under consideration incorporates boundary conditions that allow for full motion of the contact points and dynamic contact angles. Under these conditions, \cite{Yang} established the existence of equilibrium configurations for the model. These equilibria consist of a quiescent fluid occupying a domain whose upper boundary can be represented as the graph of a function in polar coordinates, minimizing a gravity-capillary energy functional subject to a fixed mass constraint. The equilibrium contact angles may take any value in $(0,\pi)$ depending on the choice of capillary parameters. In the present work, we develop a framework of a priori estimates for this model. We prove that, for initial data sufficiently close to equilibrium, the system admits global solutions that converge exponentially fast to a (horizontally) shifted equilibrium state.