Stability of a Two-Phase Stokes Problem with Surface Tension

TL;DR

This paper proves the stability and exponential decay of perturbations of circular bubbles in a two-dimensional Stokes flow with surface tension, showing solutions become smooth over time.

Contribution

It establishes the global existence, uniqueness, and exponential stability of solutions near circular bubbles in a 2D Stokes flow with surface tension, including regularity results.

Findings

Solutions decay exponentially to a circle

Perturbations become real analytic over time

Circular bubbles are stable steady states

Abstract

In this work, we study the well-posedness of a system of partial differential equations that model the dynamics of a two-dimensional Stokes bubble immersed in two-dimensional ambient Stokes fluid of the same viscosity that extends to infinity under the effect of surface tension. We assume that the two fluids are immiscible and incompressible and that there is no interfacial jump in the fluid velocity. For this PDE system, a circular fluid bubble is a steady-state solution. Given an initial contour for the fluid bubble which is sufficiently close to a circle, we show that there exists a unique, global-in-time solution. This unique solution decays to a circle exponentially fast, which means that circular fluid bubbles are stable steady-state solutions. We also obtain a result concerning the regularity of the unique solution, that although the initial perturbation around a circular contour is assumed to be of low regularity, any later perturbation becomes real analytic, hence smooth.