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

Ovidiu Costin, Saleh Tanveer, Michael I. Weinstein

Published 2012-10-01Version 1

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 $\epsilon$.