The lifetime of shape oscillations of a bubble in an unbounded, inviscid and compressible fluid with surface tension

TL;DR

This paper proves a conjecture about the exponential decay rate of shape oscillations of a bubble in a compressible, inviscid fluid with surface tension, providing explicit formulas for decay constants.

Contribution

It rigorously confirms the decay rate conjecture and computes the leading order constants A and B in the asymptotic expression.

Findings

Decay rate $ o e^{- ext{constant} imes t}$ confirmed

Explicit formulas for constants A and B derived

Asymptotic behavior matches numerical and formal evidence

Abstract

General perturbations of a spherical gas bubble in a compressible and inviscid fluid with surface tension were proved in Shapiro and Weinstein (2011), in the linearized approximation, to decay exponentially, $\sim e^{-\Gamma t}, \Gamma>0$, as time advances. Formal asymptotic and numerical evidence led to the conjecture that $\Gamma \approx \frac{A}{\epsilon} \frac{We}{\epsilon^{2}} \exp(-B \frac{We}{\epsilon^2})$, where $0<\epsilon\ll1$ is the Mach number, We is the Weber number, and $A$ and $B$ are positive constants. In this paper, we prove this conjecture and calculate $A$ and $B$ to leading order in $\epsilon$.