A Model for Rayleigh--Taylor Mixing and Interface Turnover

Abstract

We first develop a new mathematical model for two-fluid interface motion, subjected to the Rayleigh--Taylor (RT) instability in two-dimensional fluid flow, which in its simplest form, is given by $ h_{tt}( \alpha ,t) = A g\, \Lambda h - \frac{ \sigma }{ \rho^++\rho^-} \Lambda^3 h - A \text{\bf\emph{p}}_\alpha(H h_t h_t) $, where $\Lambda = H \text{\bf\emph{p}}_ \alpha $ and $H$ denotes the Hilbert transform. In this so-called $h$-model, $A$ is the Atwood number, $g$ is the acceleration, $ \sigma $ is surface tension, and $\rho^\pm$ denotes the densities of the two fluids. We derive our $h$-model using asymptotic expansions in the Birkhoff--Rott integral-kernel formulation for the evolution of an interface separating two incompressible and irrotational fluids. The resulting $h$-model equation is shown to be locally and globally well-posed in Sobolev spaces when a certain stability condition is satisfied; this stability condition requires that the product of the Atwood number and the initial velocity field ...