Skip to main content
×
Home

Collective oscillations in bubble clouds

  • ZORANA ZERAVCIC (a1), DETLEF LOHSE (a2) and WIM VAN SAARLOOS (a1) (a3)
Abstract

In this paper the collective oscillations of a bubble cloud in an acoustic field are theoretically analysed with concepts and techniques of condensed matter physics. More specifically, we will calculate the eigenmodes and their excitabilities, eigenfrequencies, densities of states, responses, absorption and participation ratios to better understand the collective dynamics of coupled bubbles and address the question of possible localization of acoustic energy in the bubble cloud. The radial oscillations of the individual bubbles in the acoustic field are described by coupled linearized Rayleigh–Plesset equations. We explore the effects of viscous damping, distance between bubbles, polydispersity, geometric disorder, size of the bubbles and size of the cloud. For large enough clusters, the collective response is often very different from that of a typical mode, as the frequency response of each mode is sufficiently wide that many modes are excited when the cloud is driven by ultrasound. The reason is the strong effect of viscosity on the collective mode response, which is surprising, as viscous damping effects are small for single-bubble oscillations in water. Localization of acoustic energy is only found in the case of substantial bubble size polydispersity or geometric disorder. The lack of localization for a weak disorder is traced back to the long-range 1/r interaction potential between the individual bubbles. The results of the present paper are connected to recent experimental observations of collective bubble oscillations in a two-dimensional bubble cloud, where pronounced edge states and a pronounced low-frequency response had been observed, both consistent with the present theoretical findings. Finally, an outlook to future possible experiments is given.

Copyright
Corresponding author
Email address for correspondence: zorana@seas.harvard.edu
References
Hide All
Anderson P. W. 1958 Absence of diffusion in certain random lattices. Phys. Rev. 109 (5), 14921505.
Arora M., Junge L. & Ohl C. D. 2005 Cavitation cluster dynamics in shock-wave lithotripsy. Part 1. Free field. Ultrasound Med. Biol. 31 (6), 827839.
Arora M., Ohl C. D. & Lohse D. 2007 Effect of nuclei concentration on cavitation cluster dynamics. J. Acoust. Soc. Am. 121 (6), 34323436.
Ashcroft N. W. & Mermin D. N. 1976 Solid State Physics. Saunders College Publishing.
Bremond N., Arora M., Dammer S. & Lohse D. 2006 a Interaction of cavitation bubbles on a wall. Phys. Fluids 18, 121505.
Bremond N., Arora M., Ohl C. D. & Lohse D. 2005 Cavitation on surfaces. J. Phys.: Condens. Matter 17, S3603S3608.
Bremond N., Arora M., Ohl C. D. & Lohse D. 2006 b Controlled multi-bubble surface cavitation. Phys. Rev. Lett. 96, 224501.
Brennen C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.
Brennen C. E. 2002 Fission of collapsing cavitation bubbles. J. Fluid Mech. 472, 153166.
Brenner M. P., Hilgenfeldt S. & Lohse D. 2002 Single bubble sonoluminescence. Rev. Mod. Phys. 74, 425484.
Burns P. N. 1996 Harmonic imaging with ultrasound contrast agents. Clin. Radiol. 51 (Suppl. 1), 5055.
Busnaina A. A., Kashkoush I. I. & Gale G. W. 1995 An experimental study of megasonic cleaning of silicon wafers. J. Electrochem. Soc. 142 (8), 28122817.
Calflisch R. E., Miksis M. J., Papanicolaou C. & Ting L. 1985 Effective equations for wave propagation in bubbly liquids. J. Fluid Mech. 153, 259.
Colonius T., Hagmeijer R., Ando K. & Brennen C. E. 2008 Statistical equilibrium of bubble oscillations in dilute bubbly flow. Phys. Fluids 20, 040902.
Crum L. A. 1983 The polytropic exponent of gas contained within air bubbles pulsating in a liquid. J. Acoust. Soc. Am. 73, 116120.
d'Agostino L. & Brennen C. E. 1988 Acoustical absorption and scattering cross sections of spherical bubble clouds. J. Acoust. Soc. Am. 84 (6), 21262134.
d'Agostino L. & Brennen C. E. 1989 Linearized dynamics of spherical bubble clouds. J. Fluid Mech. 199, 155176.
Dayton P. A., Chomas J. E., Lum A. F. H., Allen J. S., Lindner J. R., Simon S. I. & Ferrara K. W. 2000 Optical and acoustical dynamics of microbubble contrast agents inside neutrophils. Biophys. J. 80, 15471556.
Doinikov A. A. 2004 Mathematical model for collective bubble dynamics in strong ultrasound fields. J. Acoust. Soc. Am. 116, 821.
Doinikov A. A. & Zavtrak T. 1996 On the ‘bubble grapes’ induced by a sound field. J. Acoust. Soc. Am. 99 (6), 38493850.
Garbin V., Cojoc D., Ferrari E., Di Fabrizio E., Overvelde M. L. J., van der Meer S. M., de Jong N., Lohse D. & Versluis M. 2007 Changes in microbubble dynamics near a boundary revealed by combined optical micromanipulation and high-speed imaging. Appl. Phys. Lett. 90 (11) 114103-1–114103-3.
van Hecke M. 2010 Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys.: Condens. Matter 22, 033101.
Hilgenfeldt S., Brenner M. P., Grossmann S. & Lohse D. 1998 Analysis of Rayleigh–Plesset dynamics for sonoluminescing bubbles. J. Fluid Mech. 365, 171204.
Hilgenfeldt S., Lohse D. & Brenner M. P. 1996 Phase diagrams for sonoluminescing bubbles. Phys. Fluids 8 (11), 28082826.
Ida M. 2004 Investigation of transition frequencies of two acoustically coupled bubbles using a direct numerical simulation technique. J. Phys. Soc. Japan 73, 30263033.
Ida M. 2005 Avoided crossings in three coupled oscillators as a model system of acoustic bubbles. Phys. Rev. E 72, 036306.
Kamath V., Prosperetti A. & Egolfopoulos F. N. 1993 A theoretical study of sonoluminescence. J. Acoust. Soc. Am. 94 (1), 248260.
Krefting D., Mettin R. & Lauterborn W. 2004 High-speed observation of acoustic cavitation erosion in multibubble systems. Ultrason. Sonochem. 11 (3–4), 119123.
Kumar S. & Brennen C. E. 1993 Some nonlinear interactive effects in bubbly clouds. J. Fluid Mech. 253, 565591.
Landau L. D. & Lifshitz E. M. 1960 Mechanics. Pergamon.
Leighton T. G. 1994 The Acoustic Bubble. Academic.
Leighton T. G. 2004 From seas to surgeries, from babbling brooks to baby scans: the acoustics of gas bubbles in liquids. Intl J. Mod. Phys. B 18 (25), 32673314.
Liu A. & Nagel S. R. 2010 The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1 347369.
Martinez-Mercado J., Chehata-Gomez D., van Gils D., Sun C. & Lohse D. 2010 On bubble clustering and energy spectra in pseudo-turbulence. J. Fluid Mech. 649, x.
van der Meer S. M., Dollet B., Voormolen M. M., Chin C. T., Bouakaz A., de Jong N., Versluis M. & Lohse D. 2007 Microbubble spectroscopy of ultrasound contrast agents. J. Acoust. Soc. Am. 121, 648656.
Mettin R. 2005 Bubble Structures in Acoustic Cavitation. Research Signpost, Kerala, India.
Mettin R., Akhatov I., Parlitz U., Ohl C. D. & Lauterborn W. 1997 Bjerknes forces between small cavitation bubbles in a strong acoustic field. Phys. Rev. E 56 (3), 29242931.
Mulvagh S. L., DeMaria A. N., Feinstein S. B., Burns P. N., Kaul S., Miller J. G., Monaghan M., Porter T. R., Shaw L. J., Villanueva F. S. & Am Soc Echocardiography Task Force 2000 Contrast echocardiography: current and future applications. J. Am. Soc. Echocardiogr. 13 (4), 331342.
Noordzij L. & van Wijngaarden L. 1974 Relaxation effects, caused by relative motion on shock-waves in gas–bubble–liquid mixtures. J. Fluid Mech. 66 (OCT21), 115143.
Nyborg W. L. & Hughes D. E. 1967 Bubble annihilation in cavitation streamers. J. Acoust. Soc. Am. 42 (4), 891.
Osborn T., Farmer D. M., Vagle S., Thorpe S. A. & Cure M. 1992 Measurements of bubble plumes and turbulence from a submarine. Atmos.-Ocean 30 (3), 419440.
Parlitz U., Mettin R., Luther S., Akhatov I., Voss M. & Lauterborn W. 1999 Spatio-temporal dynamics of acoustic cavitation bubble clouds. Phil. Trans. R. Soc. Cond. Ser. A-Math. Phys. Eng. Sci. A 357 (1751), 313334.
Pelekasis N. A., Gaki A., Doinikov A. A. & Tsamopoulos J. A. 2004 Secondary Bjerknes forces between two bubbles and the phenomenon of acoustic streamers. J. Fluid Mech. 500, 313347.
Plesset M. S. & Prosperetti A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145185.
Porta A. La Voth G. A., Crawford A. M., Alexander J. & Bodenschatz E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409, 10171019.
Prosperetti A. 1977 Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquid. J. Acoust. Soc. Am 61, 1727.
Prosperetti A. & Lezzi A. 1986 Bubble dynamics in a compressible liquid. I. First-order theory. J. Fluid Mech. 168, 457478.
Qin S. P. & Ferrara K. W. 2006 Acoustic response of compliable microvessels containing ultrasound contrast agents. Phys. Med. Biol. 51, 50655088.
Rathgen H., Sugiyama K., Ohl C. D., Lohse D. & Mugele F. 2007 Nanometer-resolved collective micromeniscus oscillations through optical diffraction. Phys. Rev. Lett. 99, 214501.
Sangani A. S. & Didwania A. K. 1993 Dynamic simulations of flows of bubbly liquids at large Reynolds numbers. J. Fluid Mech. 250, 307337.
Sharpe F. A. & Dill L. M. 1997 The behavior of Pacific herring schools in response to artificial humpback whale bubbles. Can. J. Zool.-Rev. Can. Zool. 75 (5), 725730.
Sheng P. 1990 Scattering and Localization of Classical Waves in Random Media. World Scientific.
Sheng P. 1995 Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Academic.
Smereka P. 2002 A Vlasov equation for pressure wave propagation in bubbly fluids. J. Fluid Mech. 454, 287325.
Smereka P. & Banerjee S. 1988 The dynamics of periodically driven bubble clouds. Phys. Fluids 31, 3519.
Sornette D. & Legrend O. 1992 Acoustic wave propagation in one-dimensional stratified gas–liquid media – the different regimes. J. Acoust. Soc. Am. 92 (1), 296308.
Sornette D. & Souillard B. 1988 Strong localization of waves by internal resonances. Europhys. Lett. 7 (3), 269274.
Toschi F. & Bodenschatz E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375404.
Versluis M., v. d. Heydt A., Schmitz B. & Lohse D. 2000 How snapping shrimp snap: through cavitating bubbles. Science 289, 21142117.
Vitelli V., Xu N., Wyart M., Liu A. J. & Nagel S. R. 2010 Heat transport in model jammed solids. Phys. Rev. E 81, 021301021315.
Wang Y. C. & Brennen C. E. 1999 Numerical computation of shock waves in a spherical cloud of cavitation bubbles. J. Fluids Eng. - Trans. ASME 121 (4), 872880.
van Wijngaarden L. 1972 One-dimensional flow of liquids containing small gas bubbles. Ann. Rev. Fluid Mech. 4, 369396.
Wursig B., Greene C. R. & Jefferson T. A. 2000 Development of an air bubble curtain to reduce underwater noise of percussive piling. Mar. Environ. Res. 49 (1), 7993.
Xu Ning, Vitelli Vincenzo, Wyart Matthieu, Liu Andrea J. & Nagel Sidney R. 2009 Energy Transport in Jammed Sphere Packings. Phys. Rev. Lett. 102 (3).
Ye Z. & Alvarez A. 1998 Acoustic localization in bubbly liquid media. Phys. Rev. Lett. 80 (16), 35033506.
Zeravcic Z., van Saarloos W. & Nelson D. R. 2008 Localization of behavior of vibrational modes in granular packing. EPL 83, 44001.
Zijlstra A., Janssens T., Wostyn K., Versluis M., Mertens P. M. & Lohse D. 2009 High speed imaging of 1 mhz driven microbubbles in contact with a rigid wall. Solid State Phenomena 145–146, 710.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 41 *
Loading metrics...

Abstract views

Total abstract views: 164 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th November 2017. This data will be updated every 24 hours.