Dynamics and stability of sessile drops with contact points

TL;DR

This paper analyzes the stability and dynamics of sessile drops with moving contact points, demonstrating that solutions near equilibrium decay exponentially, advancing understanding of contact line behavior in fluid mechanics.

Contribution

It introduces a model allowing 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 to a shifted 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 a drop of incompressible viscous Stokes fluid evolving above a one-dimensional flat surface under the influence of gravity. This is a free boundary problem: the interface between the fluid on the surface 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 flat surface 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 a shifted equilibrium exponentially fast.