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

TL;DR

This paper analyzes the nonlinear Rayleigh-Taylor instability in a two-layer viscous fluid system with free boundaries, identifying conditions under which the system becomes nonlinearly unstable, with or without surface tension effects.

Contribution

It establishes the nonlinear instability of the two-layer viscous fluid problem under certain conditions, including the effect of surface tension and critical thresholds.

Findings

Nonlinear instability occurs when surface tension is below a critical value.

The effect of surface tension can stabilize or destabilize the interface.

The problem is analyzed in a three-dimensional periodic setting.

Abstract

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting. The effect of surface tension is either taken into account at both free boundaries or neglected at both. We are concerned with the Rayleigh-Taylor instability, so we assume that the upper fluid is heavier than the lower fluid. When the surface tension at the free internal interface is below a critical value, which we identify, we establish that the problem under consideration is nonlinearly unstable.