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Bubble shape oscillations of finite amplitude

Published online by Cambridge University Press:  25 October 2018

Matthieu Guédra Open the ORCID record for Matthieu Guédra [Opens in a new window]  and
Claude Inserra
Matthieu Guédra*
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, Centre Léon Bérard, INSERM, UMR 1032, LabTAU, F-69003, Lyon, France
Claude Inserra
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, Centre Léon Bérard, INSERM, UMR 1032, LabTAU, F-69003, Lyon, France
*
†Email address for correspondence: [email protected]
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Abstract

Shape oscillations arising from the spherical instability of an oscillating bubble can be sustained in a stationary acoustic field. Describing such a steady state requires that nonlinear saturation effects are accounted for to counteract the natural exponential growth of the instability. In this paper, we analyse the establishment of finite-amplitude bubble shape oscillations as a consequence of nonlinear interactions between spherical and non-spherical modes. The set of coupled dynamical equations describing the volume pulsation and the shape oscillations is solved using a perturbation technique based on the Krylov–Bogoliubov method of averaging. A set of first-order differential equations governing the slowly varying amplitudes and phases of the different modes allows us to reproduce the exponential growth and subsequent nonlinear saturation of the most unstable, parametrically excited, shape mode. Solving these equations for steady-state conditions leads to analytical expressions of the modal amplitudes and derivations of the conditionally stable and absolutely stable thresholds for shape oscillations. The analysis of the solutions reveals the existence of a hysteretic behaviour, indicating that bubble shape oscillations could be sustained for acoustic pressures below the classical parametric threshold.

JFM classification

Drops and Bubbles: Bubble dynamics Instability: Parametric instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , pp. 681 - 703
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Copyright
© 2018 Cambridge University Press 

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