The compressible viscous surface-internal wave problem: nonlinear Rayleigh-Taylor instability

TL;DR

This paper investigates the nonlinear Rayleigh-Taylor instability in a two-layer compressible viscous fluid system with free boundaries, demonstrating instability when surface tension is below a critical threshold.

Contribution

It establishes the nonlinear instability of the free boundary problem for compressible viscous fluids under certain surface tension conditions, extending understanding of fluid interface dynamics.

Findings

Proves nonlinear instability when surface tension is below critical value.

Analyzes free boundary problem for compressible viscous fluids.

Considers effects of gravity and surface tension on interface stability.

Abstract

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We are concerned with the Rayleigh-Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.