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Collective oscillations in bubble clouds

Published online by Cambridge University Press:  06 June 2011

ZORANA ZERAVCIC ,
DETLEF LOHSE  and
WIM VAN SAARLOOS
ZORANA ZERAVCIC*
Affiliation:
Instituut Lorentz, University of Leiden, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
DETLEF LOHSE
Affiliation:
Physics of Fluids Group, Mesa+ and Impact Research Institutes and Burgers Center for Fluid Dynamics, Department for Science and Technology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
WIM VAN SAARLOOS
Affiliation:
Instituut Lorentz, University of Leiden, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands FOM Foundation, PO Box 3201, 3502 GA Utrecht, The Netherlands
*
†Email address for correspondence: [email protected]
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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.

JFM classification

Drops and Bubbles: Bubble dynamics Acoustics: Waves in random media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 680 , 10 August 2011 , pp. 114 - 149
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Anderson, P. W. 1958 Absence of diffusion in certain random lattices. Phys. Rev. 109 (5), 1492–1505.CrossRefGoogle Scholar
Arora, M., Junge, L. & Ohl, C. D. 2005 Cavitation cluster dynamics in shock-wave lithotripsy. Part 1. Free field. Ultrasound Med. Biol. 31 (6), 827–839.CrossRefGoogle ScholarPubMed
Arora, M., Ohl, C. D. & Lohse, D. 2007 Effect of nuclei concentration on cavitation cluster dynamics. J. Acoust. Soc. Am. 121 (6), 3432–3436.CrossRefGoogle ScholarPubMed
Ashcroft, N. W. & Mermin, D. N. 1976 Solid State Physics. Saunders College Publishing.Google Scholar
Bremond, N., Arora, M., Dammer, S. & Lohse, D. 2006 a Interaction of cavitation bubbles on a wall. Phys. Fluids 18, 121505.CrossRefGoogle Scholar
Bremond, N., Arora, M., Ohl, C. D. & Lohse, D. 2005 Cavitation on surfaces. J. Phys.: Condens. Matter 17, S3603–S3608.Google Scholar
Bremond, N., Arora, M., Ohl, C. D. & Lohse, D. 2006 b Controlled multi-bubble surface cavitation. Phys. Rev. Lett. 96, 224501.CrossRefGoogle Scholar
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.CrossRefGoogle Scholar
Brennen, C. E. 2002 Fission of collapsing cavitation bubbles. J. Fluid Mech. 472, 153–166.CrossRefGoogle Scholar
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single bubble sonoluminescence. Rev. Mod. Phys. 74, 425–484.CrossRefGoogle Scholar
Burns, P. N. 1996 Harmonic imaging with ultrasound contrast agents. Clin. Radiol. 51 (Suppl. 1), 50–55.Google ScholarPubMed
Busnaina, A. A., Kashkoush, I. I. & Gale, G. W. 1995 An experimental study of megasonic cleaning of silicon wafers. J. Electrochem. Soc. 142 (8), 2812–2817.CrossRefGoogle Scholar
Calflisch, R. E., Miksis, M. J., Papanicolaou, C. & Ting, L. 1985 Effective equations for wave propagation in bubbly liquids. J. Fluid Mech. 153, 259.CrossRefGoogle Scholar
Colonius, T., Hagmeijer, R., Ando, K. & Brennen, C. E. 2008 Statistical equilibrium of bubble oscillations in dilute bubbly flow. Phys. Fluids 20, 040902.CrossRefGoogle Scholar
Crum, L. A. 1983 The polytropic exponent of gas contained within air bubbles pulsating in a liquid. J. Acoust. Soc. Am. 73, 116–120.CrossRefGoogle Scholar
d'Agostino, L. & Brennen, C. E. 1988 Acoustical absorption and scattering cross sections of spherical bubble clouds. J. Acoust. Soc. Am. 84 (6), 2126–2134.CrossRefGoogle Scholar
d'Agostino, L. & Brennen, C. E. 1989 Linearized dynamics of spherical bubble clouds. J. Fluid Mech. 199, 155–176.CrossRefGoogle Scholar
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, 1547–1556.CrossRefGoogle Scholar
Doinikov, A. A. 2004 Mathematical model for collective bubble dynamics in strong ultrasound fields. J. Acoust. Soc. Am. 116, 821.CrossRefGoogle Scholar
Doinikov, A. A. & Zavtrak, T. 1996 On the ‘bubble grapes’ induced by a sound field. J. Acoust. Soc. Am. 99 (6), 3849–3850.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
van Hecke, M. 2010 Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys.: Condens. Matter 22, 033101.Google ScholarPubMed
Hilgenfeldt, S., Brenner, M. P., Grossmann, S. & Lohse, D. 1998 Analysis of Rayleigh–Plesset dynamics for sonoluminescing bubbles. J. Fluid Mech. 365, 171–204.CrossRefGoogle Scholar
Hilgenfeldt, S., Lohse, D. & Brenner, M. P. 1996 Phase diagrams for sonoluminescing bubbles. Phys. Fluids 8 (11), 2808–2826.CrossRefGoogle Scholar
Ida, M. 2004 Investigation of transition frequencies of two acoustically coupled bubbles using a direct numerical simulation technique. J. Phys. Soc. Japan 73, 3026–3033.CrossRefGoogle Scholar
Ida, M. 2005 Avoided crossings in three coupled oscillators as a model system of acoustic bubbles. Phys. Rev. E 72, 036306.Google Scholar
Kamath, V., Prosperetti, A. & Egolfopoulos, F. N. 1993 A theoretical study of sonoluminescence. J. Acoust. Soc. Am. 94 (1), 248–260.CrossRefGoogle Scholar
Krefting, D., Mettin, R. & Lauterborn, W. 2004 High-speed observation of acoustic cavitation erosion in multibubble systems. Ultrason. Sonochem. 11 (3–4), 119–123.CrossRefGoogle ScholarPubMed
Kumar, S. & Brennen, C. E. 1993 Some nonlinear interactive effects in bubbly clouds. J. Fluid Mech. 253, 565–591.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1960 Mechanics. Pergamon.Google Scholar
Leighton, T. G. 1994 The Acoustic Bubble. Academic.Google Scholar
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), 3267–3314.CrossRefGoogle Scholar
Liu, A. & Nagel, S. R. 2010 The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1 347–369.CrossRefGoogle Scholar
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.Google Scholar
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, 648–656.CrossRefGoogle ScholarPubMed
Mettin, R. 2005 Bubble Structures in Acoustic Cavitation. Research Signpost, Kerala, India.Google Scholar
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), 2924–2931.Google Scholar
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), 331–342.CrossRefGoogle ScholarPubMed
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), 115–143.CrossRefGoogle Scholar
Nyborg, W. L. & Hughes, D. E. 1967 Bubble annihilation in cavitation streamers. J. Acoust. Soc. Am. 42 (4), 891.Google Scholar
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), 419–440.CrossRefGoogle Scholar
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), 313–334.CrossRefGoogle Scholar
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, 313–347.CrossRefGoogle Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185.CrossRefGoogle Scholar
Porta, A. La Voth, G. A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409, 1017–1019.CrossRefGoogle ScholarPubMed
Prosperetti, A. 1977 Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquid. J. Acoust. Soc. Am 61, 17–27.CrossRefGoogle Scholar
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. I. First-order theory. J. Fluid Mech. 168, 457–478.CrossRefGoogle Scholar
Qin, S. P. & Ferrara, K. W. 2006 Acoustic response of compliable microvessels containing ultrasound contrast agents. Phys. Med. Biol. 51, 5065–5088.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle ScholarPubMed
Sangani, A. S. & Didwania, A. K. 1993 Dynamic simulations of flows of bubbly liquids at large Reynolds numbers. J. Fluid Mech. 250, 307–337.CrossRefGoogle Scholar
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), 725–730.CrossRefGoogle Scholar
Sheng, P. 1990 Scattering and Localization of Classical Waves in Random Media. World Scientific.CrossRefGoogle Scholar
Sheng, P. 1995 Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Academic.Google Scholar
Smereka, P. 2002 A Vlasov equation for pressure wave propagation in bubbly fluids. J. Fluid Mech. 454, 287–325.CrossRefGoogle Scholar
Smereka, P. & Banerjee, S. 1988 The dynamics of periodically driven bubble clouds. Phys. Fluids 31, 3519.CrossRefGoogle Scholar
Sornette, D. & Legrend, O. 1992 Acoustic wave propagation in one-dimensional stratified gas–liquid media – the different regimes. J. Acoust. Soc. Am. 92 (1), 296–308.CrossRefGoogle Scholar
Sornette, D. & Souillard, B. 1988 Strong localization of waves by internal resonances. Europhys. Lett. 7 (3), 269–274.CrossRefGoogle Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375–404.CrossRefGoogle Scholar
Versluis, M., v. d. Heydt, A., Schmitz, B. & Lohse, D. 2000 How snapping shrimp snap: through cavitating bubbles. Science 289, 2114–2117.CrossRefGoogle ScholarPubMed
Vitelli, V., Xu, N., Wyart, M., Liu, A. J. & Nagel, S. R. 2010 Heat transport in model jammed solids. Phys. Rev. E 81, 021301–021315.Google ScholarPubMed
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), 872–880.CrossRefGoogle Scholar
van Wijngaarden, L. 1972 One-dimensional flow of liquids containing small gas bubbles. Ann. Rev. Fluid Mech. 4, 369–396.CrossRefGoogle Scholar
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), 79–93.CrossRefGoogle ScholarPubMed
Xu, Ning, Vitelli, Vincenzo, Wyart, Matthieu, Liu, Andrea J. & Nagel, Sidney R. 2009 Energy Transport in Jammed Sphere Packings. Phys. Rev. Lett. 102 (3).CrossRefGoogle ScholarPubMed
Ye, Z. & Alvarez, A. 1998 Acoustic localization in bubbly liquid media. Phys. Rev. Lett. 80 (16), 3503–3506.CrossRefGoogle Scholar
Zeravcic, Z., van Saarloos, W. & Nelson, D. R. 2008 Localization of behavior of vibrational modes in granular packing. EPL 83, 44001.CrossRefGoogle Scholar
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, 7–10.CrossRefGoogle Scholar