Solutions of the buoyancy-drag equation with a time-dependent acceleration

TL;DR

This paper analytically studies the buoyancy-drag equation with a time-dependent acceleration, exploring its symmetries, equivalence classes, and asymptotic behaviors to understand fluid instabilities.

Contribution

It introduces a classification of the buoyancy-drag equation under point transformations and identifies conditions for its reducibility and symmetries.

Findings

Identification of equivalence classes under point transformations.

Derivation of a time-dependent Hamiltonian for certain acceleration functions.

Description of two asymptotic regimes related to fluid instabilities.

Abstract

We perform the analytic study of the the buoyancy-drag equation with a time-dependent acceleration $\gamma(t)$ by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of $\gamma(t)$ define a time-dependent Hamiltonian from which the buoyancy-drag equation can be derived. We then determine the Lie point symmetries of the buoyancy-drag equation, which only exist for values of $\gamma(t)$ including the previous ones, plus additional classes of accelerations for which the equation is reducible to an Abel equation. This allows us to exhibit two r\'egimes for the asymptotic (large time $t$) solution of the buoyancy-drag equation. %\textbf{ It is shown that they describe a mixing zone driven by the Rayleigh--Taylor instability and the Richtmyer--Meshkov instability, respectively.