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Radiative decay of the nonlinear oscillations of an adiabatic spherical bubble at small Mach number

Published online by Cambridge University Press:  19 December 2017

Warren R. Smith Open the ORCID record for Warren R. Smith [Opens in a new window]  and
Qianxi Wang
Warren R. Smith*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Qianxi Wang*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
*
†Email addresses for correspondence: W.Smith@bham.ac.uk, Q.X.Wang@bham.ac.uk
†Email addresses for correspondence: W.Smith@bham.ac.uk, Q.X.Wang@bham.ac.uk
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Abstract

A theoretical study is carried out for bubble oscillation in a compressible liquid with significant acoustic radiation based on the Keller–Miksis equation using a multi-scaled perturbation method. The leading-order analytical solution of the bubble radius history is obtained to the Keller–Miksis equation in a closed form including both compressible and surface tension effects. Some important formulae are derived including: the average energy loss rate of the bubble system for each cycle of oscillation, an explicit formula for the dependence of the oscillation frequency on the energy, and an implicit formula for the amplitude envelope of the bubble radius as a function of the energy. Our theory shows that the frequency of oscillation does not change on the inertial time scale at leading order, the energy loss rate on the long compressible time scale being proportional to the Mach number. These asymptotic predictions have excellent agreement with experimental results and the numerical solutions of the Keller–Miksis equation over very long times. A parametric analysis is undertaken using the above formula for the energy of the bubble system, frequency of oscillation and minimum/maximum bubble radii in terms of the dimensionless initial pressure of the bubble gases (or, equivalently, the dimensionless equilibrium radius), Weber number and polytropic index of the bubble gas.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 1 - 18
Copyright
© 2017 Cambridge University Press 

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References

Amore, P. & Fernández, F. M. 2013 Mathematical analysis of recent analytical approximations to the collapse of an empty spherical bubble. J. Chem. Phys. 138, 084511.Google Scholar
Brennen, C. E. 2013 Cavitation and Bubble Dynamics. Cambridge University Press.Google Scholar
Cole, R. H. 1948 Underwater Explosions. Princeton University Press.Google Scholar
Costin, O., Tanveer, S. & Weinstein, M. I. 2013 The lifetime of shape oscillations of a bubble in an unbounded, inviscid, and compressible fluid with surface tension. SIAM J. Math. Anal. 45, 2924–2936.Google Scholar
Duncan, J. H. & Zhang, S. G. 1991 On the interaction of a collapsing cavity and a compliant wall. J. Fluid Mech. 226, 401–423.Google Scholar
Feng, Z. C. & Leal, L. G. 1997 Nonlinear bubble dynamics. Annu. Rev. Fluid Mech. 29, 201–243.Google Scholar
Fuster, D., Dopazo, C. & Hauke, G. 2011 Liquid compressibility effects during the collapse of a single cavitating bubble. J. Acoust. Soc. Am. 129, 122–131.Google Scholar
Gilmore, F. R.1952 The growth or collapse of a spherical bubble in a viscous compressible liquid. Tech. Rep. 26-4, California Institute of Technology, Pasadena, California, USA.Google Scholar
Hung, C. F. & Hwangfu, J. J. 2010 Experimental study of the behaviour of mini-charge underwater explosion bubbles near different boundaries. J. Fluid Mech. 651, 55–80.Google Scholar
Keller, J. B. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68, 628–633.Google Scholar
Kevorkian, J. & Cole, J. D. 1981 Perturbation Methods in Applied Mathematics. Springer.Google Scholar
Klaseboer, E., Fong, S. W., 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.Google Scholar
Kudryashov, N. A. & Sinelshchikov, D. I. 2014 Analytical solutions of the Rayleigh equation for empty and gas-filled bubble. J. Phys. A: Math. Theor. 47, 405202.Google Scholar
Kuzmak, G. E. 1959 Asymptotic solutions of nonlinear second order differential equations with variable coefficients. Prikl. Mat. Mekh. 23, 515–526 (in Russian); 1959 Z. Angew. Math. Mech. 23, 730–744 (in English).Google Scholar
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73, 106501.Google Scholar
Leighton, T. G. 1994 The Acoustic Bubble. Academic.Google Scholar
Lezzi, A. & Prosperetti, A. 1987 Bubble dynamics in a compressible liquid. Part 2. Second-order theory. J. Fluid Mech. 185, 289–321.Google Scholar
Luke, J. C. 1966 A perturbation method for nonlinear dispersive wave problems. Proc. R. Soc. Lond. A 292, 403–412.Google Scholar
Mancas, S. C. & Rosu, H. C. 2016 Evolution of spherical cavitation bubbles: parametric and closed-form solutions. Phys. Fluids 28, 022009.Google Scholar
Obreschkow, D., Bruderer, M. & Farhat, M. 2012 Analytical approximation for the collapse of an empty spherical bubble. Phys. Rev. E 85, 066303.Google Scholar
Ohl, C. D., Arora, M., Ikink, R., de Jong, N., Versluis, M., Delius, M. & Lohse, D. 2006 Sonoporation from jetting cavitation bubbles. Biophys. J. 91, 4285–4295.Google Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185.Google Scholar
Prosperetti, A. 2004 Bubbles. Phys. Fluids 16, 1852–1865.Google Scholar
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1. First-order theory. J. Fluid Mech. 168, 457–478.Google Scholar
Shapiro, A. M. & Weinstein, M. I. 2011 Radiative decay of bubble oscillations in a compressible fluid. SIAM J. Math. Anal. 43, 828–876.10.1137/100803602Google Scholar
Smith, W. R. 2005 On the sensitivity of strongly nonlinear autonomous oscillators and oscillatory waves to small perturbations. IMA J. Appl. Maths 70, 359–385.Google Scholar
Smith, W. R. 2010 Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop. J. Fluid Mech. 654, 141–159.Google Scholar
Smith, W. R., King, J. R., Tuck, B. & Orton, J. W. 1999 The single-mode rate equations for semiconductor lasers with thermal effects. IMA J. Appl. Maths 63, 1–36.Google Scholar
Smith, W. R. & Wang, Q. X. 2017 Viscous decay of nonlinear oscillations of a spherical bubble at large Reynolds number. Phys. Fluids 29, 082112.Google Scholar
Smith, W. R. & Wissink, J. G. 2015 Travelling waves in two-dimensional plane Poiseuille flow. SIAM J. Appl. Maths 75, 2147–2169.Google Scholar
Smith, W. R. & Wissink, J. G.2017 Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow. Eur. J. Appl. Math. doi:10.1017/S0956792517000213.Google Scholar
Suslick, K. S. & Crum, L. A. 1997 Sonochemistry and sonoluminescence. In Encyclopedia of Acoustics (ed. Crocker, M. J.), pp. 271–282. Wiley.Google Scholar
Tomita, Y. & Shima, A. 1977 On the behavior of a spherical bubble and the impulse pressure in a viscous compressible liquid. Bull. JSME 20, 1453–1460.Google Scholar
Van Gorder, R. A. 2016 Dynamics of the Rayleigh–Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid. J. Fluid Mech. 807, 478–508.Google Scholar
Vokurka, K. 1986 Comparison of Rayleigh’s, Herring’s, and Gilmore’s models of gas bubbles. Acta Acust. United Ac. 59, 214–219.Google Scholar
Wang, Q. X. 2013 Non-spherical bubble dynamics of underwater explosions in a compressible fluid. Phys. Fluids 25, 072104.Google Scholar
Wang, Q. X. 2016 Local energy of a bubble system and its loss due to acoustic radiation. J. Fluid Mech. 797, 201–230.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.Google Scholar
Wang, Q. X. & Blake, J. R. 2011 Non-spherical bubble dynamics in a compressible liquid. Part 2. Acoustic standing wave. J. Fluid Mech. 679, 559–581.Google Scholar
Wang, Q. X. & Manmi, K. 2014 Microbubble dynamics near a wall subjected to a travelling acoustic wave. Phys. Fluids 26, 032104.Google Scholar
Wang, Q. X., Manmi, K. & Liu, K. K. 2015 Cell mechanics in biomedical cavitation. Interface Focus 5, 20150018.Google Scholar
Young, F. R. 1989 Cavitation. McGraw-Hill.Google Scholar
Zhang, A. M., Cui, P., Cui, J. & Wang, X. Q. 2015 Experimental study on bubble dynamics subject to buoyancy. J. Fluid Mech. 776, 137–160.Google 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.Google Scholar
Zhang, Y. N. & Li, S. C. 2012 Effects of liquid compressibility on radial oscillations of gas bubbles in liquids. J. Hydrodyn. B 24, 760–766.Google Scholar