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Non-spherical bubble dynamics in a compressible liquid. Part 2. Acoustic standing wave

Published online by Cambridge University Press:  24 May 2011

Q. X. WANG  and
J. R. BLAKE
Q. X. WANG*
Affiliation:
School of Mathematics, the University of Birmingham, Birmingham B15 2TT, UK
J. R. BLAKE
Affiliation:
School of Mathematics, the University of Birmingham, Birmingham B15 2TT, UK
*
†Email address for correspondence: [email protected]
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Abstract

This paper investigates the behaviour of a non-spherical cavitation bubble in an acoustic standing wave. The study has important applications to sonochemistry and in understanding features of therapeutic ultrasound in the megahertz range, extending our understanding of bubble behaviour in the highly nonlinear regime where jet and toroidal bubble formation may be important. The theory developed herein represents a further development of the material presented in Part 1 of this paper (Wang & Blake, J. Fluid Mech. vol. 659, 2010, pp. 191–224) to a standing wave, including repeated topological changes from a singly to a multiply connected bubble. The fluid mechanics is assumed to be compressible potential flow. Matched asymptotic expansions for an inner and outer flow are performed to second order in terms of a small parameter, the bubble-wall Mach number, leading to weakly compressible flow formulation of the problem. The method allows the development of a computational model for non-spherical bubbles by using a modified boundary-integral method. The computations show that the bubble remains approximately of a spherical shape when the acoustic pressure is small or is initiated at the node or antinode of the acoustic pressure field. When initiated between the node and antinode at higher acoustic pressures, the bubble loses its spherical shape at the end of the collapse phase after only a few oscillations. A high-speed liquid bubble jet forms and is directed towards the node, impacting the opposite bubble surface and penetrating through the bubble to form a toroidal bubble. The bubble first rebounds in a toroidal form but re-combines to a singly connected bubble, expanding continuously and gradually returning to a near spherical shape. These processes are repeated in the next oscillation.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Cavitation Drops and Bubbles: Sonoluminescence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 679 , 25 July 2011 , pp. 559 - 581
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Batchelor, G. K. 1968 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Best, J. P. 1993 The formation of toroidal bubbles upon collapse of transient cavities. J. Fluid Mech. 251, 79–107.CrossRefGoogle Scholar
Best, J. P. 1994 The rebound of toroidal bubbles. In Bubble Dynamics and Interface Phenomena (ed. Blake, J. R., Boulton-Stone, J. M. & Thomas, N. H.), pp. 405–412. Kluwer.CrossRefGoogle Scholar
Blake, J. R., Keen, G. S., Tong, R. P. & Wilson, M. 1999 Acoustic cavitation: the fluid dynamics of non-spherical bubbles. Phil. Trans. R. Soc. A 357, 251–267.CrossRefGoogle Scholar
Blake, J. R., Taib, B. B. & Doherty, G. 1986 Transient cavities near boundaries. Part 1. Rigid boundary. J. Fluid Mech. 170, 479–497.CrossRefGoogle Scholar
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas bubbles bursting at a free surface. 254, 437–466.Google Scholar
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press (available online).CrossRefGoogle Scholar
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single-bubble sonoluminescence. Rev. Mod. Phys. 74, 425–484.CrossRefGoogle Scholar
Brenner, M. P., Lohse, D. & Dupont, T. F. 1995 Bubble Shape Oscillations and the Onset of Sonoluminescence. Phys. Rev. Lett. 75, 954–957.CrossRefGoogle ScholarPubMed
Calvisi, M. L., Iloreta, J. I. & Szeri, A. J. 2008 Dynamics of bubbles near a rigid surface subjected to a lithotripter shock wave. Part 2. Reflected shock intensifies non-spherical cavitation collapse. J. Fluid Mech. 616, 63–97.CrossRefGoogle Scholar
Calvisi, M. L., Lindau, O., Blake, J. R. & Szeri, A. J. 2007 Shape stability and violent collapse of microbubbles in acoustic traveling waves. Phys. Fluids 19 (4), 047101.CrossRefGoogle Scholar
Chen, H., Brayman, A. A. & Matula, T. J. 2008 Microbubble dynamics in microvessels: Observations of microvessel dilation, invagination and rupture. IEEE Ultrason. Symp. 1–4, 1163–1166.Google Scholar
Cole, R. H. 1948 Underwater Explosions. Princeton University Press.CrossRefGoogle Scholar
Coussis, C. C. & Roy, R. A. 2008 Applications of Acoustics and Cavitation to Noninvasive Therapy and Drug Delivery. Annu. Rev. Fluid Mech. 40, 395–420.CrossRefGoogle Scholar
Crum, L. A. 1975 Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am. 57, 1363–1370.CrossRefGoogle Scholar
Crum, L. A. & Cordry, S. 1994 Single bubble sonoluminescence. In Bubble Dynamics and Interface Phenomena (ed. Blake, J. R., Boulton-Stone, J. M. & Thomas, N. H.), pp. 287–297. Kluwer.CrossRefGoogle Scholar
Dahnke, S., Swamy, K. & Keil, F. J. 1999 Modeling of three dimensional pressure fields in sonochemical reactors with an inhomogeneous density distribution of cavitation bubbles. comparison of theoretical and experimental results. Ultrason. Sonochem. 6, 31–41.CrossRefGoogle ScholarPubMed
Day, C. 2005 Targeted ultrasound mediates the delivery of therapeutic genes to heart muscle. Phys. Today 58 (12).Google Scholar
Feng, Z. C. & Leal, L. G. 1997 Nonlinear bubble dynamics. Annu. Rev. Fluid Mech. 29, 201–243.CrossRefGoogle Scholar
Eller, A. 1968 Force on a bubble in a standing acoustic wave. J. Acoust. Soc. Am. 43 (1), 170–171.CrossRefGoogle Scholar
Epstein, D. & Keller, J. B. 1971 Expansion and contraction of planar, cylindrical, and spherical underwater gas bubbles. J. Acoust. Soc. Am. 52, 977–980.Google Scholar
Gilmore, F. R. 1952 The growth or collapse of a spherical bubble in a viscous compressible liquid. Report No 26-4, Hydrodynamics Laboratory, California Institute of Technology, Pasadena, California, USA.Google Scholar
Goldberg, B. B., Liu, J.-B. & Forsberg, F. 1994 Ultrasound contrast agents – a review. Ultrasound. Med. Biol. 20, 319–333.CrossRefGoogle ScholarPubMed
Herring, C. 1941 The theory of the pulsations of the gas bubbles produced by an underwater explosion. US Nat. Defence Res. Comm. Report.Google Scholar
Hilgenfeldt, S., Brenner, M. P., Grossmann, S. & Lohse, D. 1998 Analysis of Rayleigh–Plesset dynamics for sonoluminescing bubbles. J. Fluid Mech. 365, 171.CrossRefGoogle Scholar
Hua, J. & Lou, J. 2007 Numerical simulation of bubble rising in viscous liquid. J. Comput. Phys. 222 (2), 769–795.CrossRefGoogle Scholar
Johnsen, E. & Colonius, T. 2008 Shock-induced collapse of a gas bubble in shockwave lithotripsy. J. Acoust. Soc. Am. 124, 2011–2020.CrossRefGoogle ScholarPubMed
Johnsen, E. & Colonius, T. 2009 Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231–262.CrossRefGoogle ScholarPubMed
Kamath, V., Prosperetti, A. & Egolfopoulos, F. N. 1993 A theoretical study of sonoluminescence. J. Acoust. Soc. Am. 94, 248–260.CrossRefGoogle Scholar
Kawabata, K. & Umemura, S. 1996 Effect of second-harmonic superimposition on efficient induction of sonochemical effect. Ultrason. Sonochem. 3, 1–5.CrossRefGoogle Scholar
Keller, J. B. & Kolodner, I. I. 1956 Damping of underwater explosion bubble oscillations. J. Appl. Phys. 27 (10), 1152–1161.CrossRefGoogle Scholar
Keller, J. B. & Miksis, M. J. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68, 628–633.CrossRefGoogle Scholar
Klaseboer, E., Turangan, C. K., Khoo, B. C., Szeri, A. J., Calvisi, M. L., Sankin, G. N. & Zhong, P. 2007 Interaction of lithotripter shockwaves with single inertial cavitation bubbles. J. Fluid Mech. 593, 33–56.CrossRefGoogle ScholarPubMed
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73, 106501.CrossRefGoogle Scholar
Lee, M., Klaseboer, E. & Khoo, B. C. 2007 On the boundary integral method for the rebounding bubble. J. Fluid Mech. 570, 407–429.CrossRefGoogle Scholar
Leighton, T. 1994 The Acoustic Bubble. Academic Press.Google Scholar
Lezzi, A. & Prosperetti, A. 1987 Bubble dynamics in a compressible liquid. Part. 2. Second-order theory. J. Fluid Mech. 185, 289–321.CrossRefGoogle Scholar
Lundgren, T. S. & Mansour, N. N. 1991 Vortex ring bubbles. J. Fluid Mech. 72, 391–399.Google Scholar
Matula, T. J. Cordry, S. M., Roy, R. A. & Crum, L. A. 1997 Bjerknes force and bubble levitation under single-bubble sonoluminescence conditions. J. Acoust. Soc. Am. 102 (3), 1522–1527.CrossRefGoogle Scholar
Miksis, J. M., Vanden-Broeck, J. M. & Keller, J. B. 1982 Rising bubbles. J. Fluid Mech. 123, 31–41.CrossRefGoogle Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.CrossRefGoogle Scholar
Pedley, T. J. 1968 The toroidal bubble. J. Fluid Mech. 32, 97–112.CrossRefGoogle Scholar
Popinet, S. & Zaleski, S. 2002 Bubble collapse near a solid boundary: a numerical study of the influence of viscosity. J. Fluid Mech. 464, 137–163.CrossRefGoogle Scholar
Prosperetti, A. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Math. 34, 339–352.CrossRefGoogle Scholar
Prosperetti, A., Crum, L. A. & Commander, K. W. 1988 Nonlinear bubble dynamics. J. Acoust. Soc. Am. 83, 502–514.CrossRefGoogle Scholar
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1. First-order theory. J. Fluid Mech. 168, 457–478.CrossRefGoogle Scholar
Putterman, S. J. & Weninger, K. R. 2000 Sonoluminescence: how bubbles turn sound into light. Annu. Rev. Fluid Mech. 32, 445–476.CrossRefGoogle Scholar
Reddy, A. J. & Szeri, A. J. 2002 Coupled dynamics of translation and collapse of acoustically driven microbubbles. J. Acoust. Soc. Am. 112 (4), 1346–1352.CrossRefGoogle ScholarPubMed
Szeri, A. J., Storey, B. D., Pearson, A. & Blake, J. R. 2003 Heat and mass transfer during the violent collapse of nonspherical bubbles. Phys. Fluids 15, 2576–2586.CrossRefGoogle Scholar
Trinh, E. H. & Hsu, C.-J. 1986 Equilibrium shapes of acoustically levitated drops. J. Acoust. Soc. Am. 79, 1335–1338.CrossRefGoogle Scholar
Turangan, C. K., Jamaluddin, A. R., Ball, G. J. & Leighton, T. G. 2008 Free-Lagrange simulations of the expansion and jetting collapse of air bubbles in water. J. Fluid Mech. 598, 1–25.CrossRefGoogle Scholar
VanDyke, M. D. Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics, 2nd edn. Parabolic Press.Google Scholar
Versluis, M., Goertz, D. E., Palanchon, P., Heitman, I. L., van der Meer, S. M., Dollet, B., Jong, N. D. & Lohse, D. 2010 Microbubble shape oscillations excited through ultrasonic parametric driving. Phys. Rev. E 82, 026321.Google ScholarPubMed
Wang, Q. X. 1998 The numerical analyses of the evolution of a gas bubble near an inclined wall. Theor. Comput. Fluid Dyn. 12, 29–51.CrossRefGoogle Scholar
Wang, Q. X. 2004 Numerical modelling of violent bubble motion. Phys. Fluids 16 (5), 1610–1619.Google Scholar
Wang, Q. X. & Blake, J. R. 2010 Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave. J. Fluid Mech. 659, 191–224.CrossRefGoogle Scholar
Wang, Q. X., Yeo, K. S., Khoo, B. C. & Lam, K. Y. 1996 a Nonlinear interaction between gas bubble and free surface. Comput. Fluids 25 (7), 607.CrossRefGoogle Scholar
Wang, Q. X., Yeo, K. S., Khoo, B. C. & Lam, K. Y. 1996 b Strong interaction between buoyancy bubble and free surface. Theor. Comput. Fluid Dyn. 8, 73.CrossRefGoogle Scholar
Wang, Q. X., Yeo, K. S., Khoo, B. C. & Lam, K. Y. 2005 Vortex ring modelling for toroidal bubbles. Theor. Comput. Fluid Dyn. 19 (5), 303–317.CrossRefGoogle Scholar
Wu, Y., Unger, E. C. McCreery, T. P., Sweitzer, R. H., Shen, D., Wu, G. & Vielhauer, M. D. 1988 Binding and lysing of blood clots using MRX-408. Invest. Radiol. 33, 880–885.CrossRefGoogle Scholar
Young, F. R. 1989 Cavitation. McGraw-Hill.Google Scholar
Yue, P., Feng, J. J., Bertelo, C. A. & Hu, H. H. 2007 An arbitrary Lagrangian–Eulerian method for simulating bubble growth in polymer foaming. J. Comput. Phys. 226 (2), 2229–2249.CrossRefGoogle Scholar
Zhang, S. G. & Duncan, J. H. 1994 On the non-spherical collapse and rebound of a cavitation bubble. Phys. Fluids 6 (7), 2352–2362.CrossRefGoogle Scholar
Zhang, S. G., Duncan, J. H. & Chahine, G. L. 1993 The final stage of the collapse of a cavitation bubble near a rigid wall. J. Fluid Mech. 257, 147–181.CrossRefGoogle Scholar
Zhang, Y. L., Yeo, K. S., Khoo, B. C. & Wang, C. 2001 3D jet impact and toroidal bubbles. J. Comput. Phys. 166, 336–360.CrossRefGoogle Scholar