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Radiative decay of the nonlinear oscillations of an adiabatic spherical bubble at small Mach number

Published online by Cambridge University Press:  19 December 2017

Warren R. Smith Open the ORCID record for Warren R. Smith [Opens in a new window]  and
Qianxi Wang
Warren R. Smith*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Qianxi Wang*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

A theoretical study is carried out for bubble oscillation in a compressible liquid with significant acoustic radiation based on the Keller–Miksis equation using a multi-scaled perturbation method. The leading-order analytical solution of the bubble radius history is obtained to the Keller–Miksis equation in a closed form including both compressible and surface tension effects. Some important formulae are derived including: the average energy loss rate of the bubble system for each cycle of oscillation, an explicit formula for the dependence of the oscillation frequency on the energy, and an implicit formula for the amplitude envelope of the bubble radius as a function of the energy. Our theory shows that the frequency of oscillation does not change on the inertial time scale at leading order, the energy loss rate on the long compressible time scale being proportional to the Mach number. These asymptotic predictions have excellent agreement with experimental results and the numerical solutions of the Keller–Miksis equation over very long times. A parametric analysis is undertaken using the above formula for the energy of the bubble system, frequency of oscillation and minimum/maximum bubble radii in terms of the dimensionless initial pressure of the bubble gases (or, equivalently, the dimensionless equilibrium radius), Weber number and polytropic index of the bubble gas.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 1 - 18
Copyright
© 2017 Cambridge University Press 

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