Stability of contact lines in fluids: 2D Stokes Flow

TL;DR

This paper analyzes the stability of contact lines in a 2D viscous fluid flow, demonstrating that solutions near equilibrium decay exponentially, advancing understanding of free boundary problems with dynamic contact points.

Contribution

It introduces a model with fully dynamic contact points and angles, providing a priori estimates and proving exponential decay of solutions near equilibrium.

Findings

Global solutions exist for initial data close to equilibrium.

Solutions decay exponentially fast to equilibrium.

The model accommodates fully dynamic contact points and angles.

Abstract

In an effort to study the stability of contact lines in fluids, we consider the dynamics of an incompressible viscous Stokes fluid evolving in a two-dimensional open-top vessel under the influence of gravity. This is a free boundary problem: the interface between the fluid in the vessel and the air above (modeled by a trivial fluid) is free to move and experiences capillary forces. The three-phase interface where the fluid, air, and solid vessel wall meet is known as a contact point, and the angle formed between the free interface and the vessel is called the contact angle. We consider a model of this problem that allows for fully dynamic contact points and angles. We develop a scheme of a priori estimates for the model, which then allow us to show that for initial data sufficiently close to equilibrium, the model admits global solutions that decay to equilibrium exponentially fast.