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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: matthieu.guedra@inserm.fr
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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

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , pp. 681 - 703
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© 2018 Cambridge University Press 

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References

Ahmed, D., Mao, X., Juluri, B. K. & Huang, T. J. 2009 A fast microfluidic mixer based on acoustically driven sidewall-trapped microbubbles. Microfluid Nanofluidics 7 (5), 727–731.Google Scholar
Benjamin, T. B. & Ellis, A. T. 1990 Self-propulsion of asymmetrically vibrating bubbles. J. Fluid Mech. 212, 65–80.Google Scholar
Benjamin, T. B. & Strasberg, M. 1958 Excitation of oscillations in the shape of pulsating gas bubbles; theoretical work. J. Acoust. Soc. Am. 30 (7), 697–697.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Brenner, M. P., Lohse, D. & Dupont, T. F. 1995 Bubble shape oscillations and the onset of sonoluminescence. Phys. Rev. Lett. 75, 954–957.Google Scholar
Ceschia, M. & Nabergoj, R. 1978 On the motion of a nearly spherical bubble in a viscous liquid. Phys. Fluids 21 (1), 140–142.Google Scholar
Chomas, J. E., Dayton, P., Allen, J., Morgan, K. & Ferrara, K. W. 2001 Mechanisms of contrast agent destruction. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48 (1), 232–248.Google Scholar
Doinikov, A. A. 2004 Translational motion of a bubble undergoing shape oscillations. J. Fluid Mech. 501, 1–24.Google Scholar
Dollet, B., van der Meer, S. M., Garbin, V., de Jong, N., Lohse, D. & Versluis, M. 2008 Nonspherical oscillations of ultrasound contrast agent microbubbles. Ultrasound Med. Biol. 34 (9), 1465–1473.Google Scholar
Eller, A. I. & Crum, L. A. 1970 Instability of the motion of a pulsating bubble in a sound field. J. Acoust. Soc. Am. 47 (3), 762–767.Google Scholar
Feng, Z. C. & Leal, L. G. 1993 On energy transfer in resonant bubble oscillations. Phys. Fluids A 5 (4), 826–836.Google Scholar
Feng, Z. C. & Leal, L. G. 1994 Bifurcation and chaos in shape and volume oscillations of a periodically driven bubble with two-to-one internal resonance. J. Fluid Mech. 226, 209–242.Google Scholar
Francescutto, A. & Nabergoj, R. 1978 Pulsation amplitude threshold for surface waves on oscillating bubbles. Acta Acust. 41 (3), 215–220.Google Scholar
Guédra, M., Cleve, S., Mauger, C., Blanc-Benon, P. & Inserra, C. 2017 Dynamics of nonspherical microbubble oscillations above instability threshold. Phys. Rev. E 96 (6), 063104.Google Scholar
Guédra, M., Inserra, C., Gilles, B. & Mauger, C. 2016a Periodic onset of bubble shape instabilities and their influence on the spherical mode. In 2016 IEEE International Ultrasonics Symposium (IUS), pp. 1–4. IEEE.Google Scholar
Guédra, M., Inserra, C., Mauger, C. & Gilles, B. 2016b Experimental evidence of nonlinear mode coupling between spherical and nonspherical oscillations of microbubbles. Phys. Rev. E 94 (5), 053115.Google Scholar
Hall, P. & Seminara, G. 1980 Nonlinear oscillations of non-spherical cavitation bubbles in acoustic fields. J. Fluid Mech. 101 (2), 423–444.Google Scholar
Harkin, A. A., Kaper, T. J. & Nadim, A. 2013 Energy transfer between the shape and volume modes of a nonspherical gas bubble. Phys. Fluids 25 (6), 062101.Google Scholar
Hilgenfeldt, S., Lohse, D. & Brenner, M. P. 1996 Phase diagrams for sonoluminescing bubbles. Phys. Fluids 8 (11), 2808–2826.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1989a Monopole emission of sound by asymmetric bubble oscillations. Part 1. Normal modes. J. Fluid Mech. 201, 525–541.Google Scholar
Longuet-Higgins, M. S. 1989b Monopole emission of sound by asymmetric bubble oscillations. Part 2. An initial-value problem. J. Fluid Mech. 201, 543–565.Google Scholar
Longuet-Higgins, M. S. 1992 Nonlinear damping of bubble oscillations by resonant interaction. J. Acoust. Soc. Am. 91 (3), 1414–1422.Google Scholar
Lundgren, T. S. & Mansour, N. N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.Google Scholar
Mao, Y., Crum, L. A. & Roy, R. A. 1995 Nonlinear coupling between the surface and volume modes of an oscillating bubble. J. Acoust. Soc. Am. 98 (5), 2764–2771.Google Scholar
Marmottant, P., Biben, T. & Hilgenfeldt, S. 2008 Deformation and rupture of lipid vesicles in the strong shear flow generated by ultrasound-driven microbubbles. Proc. R. Soc. Lond. A 464, 1781–1800.Google Scholar
Mei, C. C. & Zhou, X. 1991 Parametric resonance of a spherical bubble. J. Fluid Mech. 229, 29–50.Google Scholar
Nayfeh, A. H. 2008 Perturbation Methods. Wiley.Google Scholar
Plesset, M. S. 1954 On the stability of fluid flows whit spherical symmetry. J. Appl. Phys. 25, 96–98.Google Scholar
Postema, M., Van Wamel, A., Lancée, C. T. & De Jong, N. 2004 Ultrasound-induced encapsulated microbubble phenomena. Ultrasound Med. Biol. 30 (6), 827–840.Google Scholar
Poulichet, V. & Garbin, V. 2015 Ultrafast desorption of colloidal particles from fluid interfaces. Proc. Natl Acad. Sci. USA 112 (19), 5932–5937.Google Scholar
Poulichet, V., Huerre, A. & Garbin, V. 2017 Shape oscillations of particle-coated bubbles and directional particle expulsion. Soft Matt. 13 (1), 125–133.Google Scholar
Prosperetti, A. 1974 Nonlinear oscillations of gas bubbles in liquids: steady-state solutions. J. Acoust. Soc. Am. 56 (3), 878–885.Google Scholar
Prosperetti, A. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Maths 34, 339–352.Google Scholar
Shaw, S. J. 2006 Translation and oscillation of a bubble under axisymmetric deformation. Phys. Fluids 18, 072104.Google Scholar
Shaw, S. J. 2009 The stability of a bubble in a weakly viscous liquid subject to an acoustic traveling wave. Phys. Fluids 21, 022104.Google Scholar
Shaw, S. J. 2017 Nonspherical sub-millimeter gas bubble oscillations: parametric forcing and nonlinear shape mode coupling. Phys. Fluids 29 (12), 122103.Google Scholar
Tho, P., Manasseh, R. & Ooi, A. 2007 Cavitation microstreaming patterns in single and multiple bubble systems. J. Fluid Mech. 576, 191–233.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1984 Resonant oscillations of inviscid charged drops. J. Fluid Mech. 147, 373–395.Google Scholar
Versluis, M., Goertz, D. E., Palanchon, P., Heitman, I. L., van der Meer, S. M., Dollet, B., de Jong, N. & Lohse, D. 2010 Microbubble shape oscillations excited through ultrasonic parametric driving. Phys. Rev. E 82, 026321.Google Scholar
Yang, S. M., Feng, Z. C. & Leal, L. G. 1993 Nonlinear effects in the dynamics of shape and volume oscillations for a gas bubble in an external flow. J. Fluid Mech. 247, 417–454.Google Scholar