Asymptotic theory for damped dynamics of gas-filled bubbles

Abstract

The small-amplitude damped oscillations of a gas-filled spherical bubble surrounded by a viscous liquid is studied analytically using a symbolically computable version of the two-time scale expansion (“two-timing”) technique. The developed uniformly valid asymptotic expansions have closed analytical forms under arbitrary specifications for the ideal gas polytropic exponent and the damping coefficient. Heretofore, a comparable level of analytical tractability has been afforded only by a linearized mass-spring-damper model; the corresponding regular perturbation expansion is, however, nonuniform for weak damping. Predictions from the two-timing and the mass-spring-damper models are compared with the full-scale numerical solution of the Rayleigh-Plesset equation. The two-scale analysis resolves several characteristics of the numerical data, such as the disparately large/small acceleration of the bubble wall near the most compressed/expanded states of the bubble, which the linearized model does not. The predictions of the two-scale model are shown to perform favorably against experimental data from the literature. New asymptotic predictions, which are very accurate for overdamped settings, are also developed by extending the mass-spring-damper model.