Well-posedness and Rayleigh-Taylor instability of the two-phase periodic quasistationary Stokes flow

TL;DR

This paper investigates the mathematical properties of two-phase quasistationary Stokes flow, establishing well-posedness, stability conditions, and Rayleigh-Taylor instability phenomena for fluid interfaces under gravity and surface tension effects.

Contribution

It introduces a nonlinear nonlocal evolution equation for the interface and analyzes stability, providing new insights into the dynamics of two-phase Stokes flows with different viscosities and densities.

Findings

Well-posedness and smoothing properties of the flow model

Rayleigh-Taylor instability for small finger-shaped equilibria

Stability depends on the sign of a specific parameter

Abstract

We study the two-phase, horizontally periodic, quasistationary Stokes flow in two dimensions driven by surface tension and gravity effects in the general context of fluids with (possibly) different viscosities and densities. The sharp interface which separates the fluids is assumed to be the graph of a periodic function. The mathematical model is then recast as a fully nonlinear and nonlocal evolution equation involving only the function parametrizing the interface. Our main results include well-posedness and a parabolic smoothing property, as well as a study of equilibrium solutions in subcritical Sobolev spaces. In particular, we establish the Rayleigh-Taylor instability of small, finger-shaped equilibria and prove that the stability properties of flat interfaces depend on the sign of a certain parameter.