The Lifetime of Shape Oscillations of a Bubble in an Unbounded, Inviscid, and Compressible Fluid with Surface Tension

Abstract

General perturbations of a spherical gas bubble in a compressible and inviscid fluid with surface tension were proved in [A, M. Shapiro and M. I. Weinstein, SIAM J. Math. Anal., 43 (2011), pp. 828--876], 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 and $A$ and $B$ are positive constants. In this paper, we prove this conjecture and calculate $A$ and $B$ to leading order in $\epsilon$.