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Dynamics of the Rayleigh–Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid

Published online by Cambridge University Press:  20 October 2016

Robert A. Van Gorder
Robert A. Van Gorder*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

Temporal dynamics of gas-filled spherical bubbles is often described using the Rayleigh–Plesset equation, a special case of the Navier–Stokes equations that describes the oscillations of a spherical cavity in an infinite incompressible fluid. While analytical approximations and numerical simulations have previously been given in some parameter regimes, we are able to completely classify all possible dynamics exactly, in terms of only the model parameters. We present an analytical study of the solutions to the Rayleigh–Plesset equation in any number of spatial dimensions, and we demonstrate that the possible behaviours of solutions include bubbles of constant radius, bubbles with temporally oscillating radius and bubbles with finite time collapse. Each of these behaviours can be predicted solely in terms of the spatial dimension, pressures acting on the bubble and initial strain. In the case of oscillating bubbles, we give the amplitude and period of these oscillations in terms of an integral which is a function of the aforementioned parameters, while when the bubble collapses, we can similarly give the time of collapse in terms of these parameters. We give a systematic study of all possible behaviours, and capture special case solutions presented numerically or asymptotically in the literature. We also discuss the influence of both surface tension and viscosity when these terms are included in the Rayleigh–Plesset dynamics.

JFM classification

Drops and Bubbles: Breakup/coalescence Drops and Bubbles: Bubble dynamics Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 807 , 25 November 2016 , pp. 478 - 508
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Copyright
© 2016 Cambridge University Press 

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