Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T12:36:53.667Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of the Rayleigh–Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid

Published online by Cambridge University Press:  20 October 2016

Robert A. Van Gorder
Robert A. Van Gorder*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: Robert.VanGorder@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Temporal dynamics of gas-filled spherical bubbles is often described using the Rayleigh–Plesset equation, a special case of the Navier–Stokes equations that describes the oscillations of a spherical cavity in an infinite incompressible fluid. While analytical approximations and numerical simulations have previously been given in some parameter regimes, we are able to completely classify all possible dynamics exactly, in terms of only the model parameters. We present an analytical study of the solutions to the Rayleigh–Plesset equation in any number of spatial dimensions, and we demonstrate that the possible behaviours of solutions include bubbles of constant radius, bubbles with temporally oscillating radius and bubbles with finite time collapse. Each of these behaviours can be predicted solely in terms of the spatial dimension, pressures acting on the bubble and initial strain. In the case of oscillating bubbles, we give the amplitude and period of these oscillations in terms of an integral which is a function of the aforementioned parameters, while when the bubble collapses, we can similarly give the time of collapse in terms of these parameters. We give a systematic study of all possible behaviours, and capture special case solutions presented numerically or asymptotically in the literature. We also discuss the influence of both surface tension and viscosity when these terms are included in the Rayleigh–Plesset dynamics.

JFM classification

Drops and Bubbles: Breakup/coalescence Drops and Bubbles: Bubble dynamics Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 807 , 25 November 2016 , pp. 478 - 508
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alikakos, N. D. & Fusco, G. 1998 Slow dynamics for the Cahn–Hilliard equation in higher space dimensions: the motion of bubbles. Arch. Rat. Mech. Anal. 141, 1–61.CrossRefGoogle Scholar
Amore, P. & Fernández, F. M. 2013 Mathematical analysis of recent analytical approximations to the collapse of an empty spherical bubble. J. Chem. Phys. 138 (8), 084511.CrossRefGoogle Scholar
Brennen, C. E. 2013 Cavitation and Bubble Dynamics. Cambridge University Press.Google Scholar
Browne, C., Tabor, R. F., Chan, D. Y., Dagastine, R. R., Ashokkumar, M. & Grieser, F. 2011 Bubble coalescence during acoustic cavitation in aqueous electrolyte solutions. Langmuir 27 (19), 12025–12032.CrossRefGoogle ScholarPubMed
Bogoyavlenskiy, V. 2000 Single-bubble sonoluminescence: shape stability analysis of collapse dynamics in a semianalytical approach. Phys. Rev. E 62, 2158.Google Scholar
Cardoso, V. & Gualtieri, L. 2006 Equilibrium configurations of fluids and their stability in higher dimensions. Class. Quant. Grav. 23 (24), 7151–7198.CrossRefGoogle Scholar
Doinikov, A. A., Novell, A., Escoffre, J.-M. & Bouakaz, A. 2013 Encapsulated bubble dynamics in imaging and therapy. In Bubble Dynamics and Shock Waves (ed. Delale, C. F.), Shock Wave Science and Technology Reference Library, vol. 8, pp. 259–289. Springer.CrossRefGoogle Scholar
Dowker, F., Gauntlett, J. P., Gibbons, G. W. & Horowitz, G. T. 1996 Nucleation of p-branes and fundamental strings. Phys. Rev. D 53 (12), 7115–7128.CrossRefGoogle ScholarPubMed
Elze, H. T., Hama, Y., Kodama, T., Makler, M. & Rafelski, J. 1999 Variational principle for relativistic fluid dynamics. J. Phys. G: Nuclear and Particle Physics 25 (9), 1935–1957.CrossRefGoogle Scholar
Finken, R., Schmidt, M. & Löwen, H. 2001 Freezing transition of hard hyperspheres. Phys. Rev. E 65, 016108.Google Scholar
Fuster, D. & Montel, F. 2015 Mass transfer effects on linear wave propagation in diluted bubbly liquids. J. Fluid Mech. 779, 598–621.CrossRefGoogle Scholar
Herring, C.1941 Theory of the pulsations of the gas bubble produced by an underwater explosion. US NRDC Division 6 Report C4-Sr20.Google Scholar
Khalid, S., Kappus, B., Weninger, K. & Putterman, S. 2012 Opacity and transport measurements reveal that dilute plasma models of sonoluminescence are not valid. Phys. Rev. Lett. 108 (10), 104302.Google Scholar
Klotz, A. R. 2013 Bubble dynamics in N dimensions. Phys. Fluids 25, 082109.Google Scholar
Klotz, A. R. & Hynynen, K. 2010 Simulations of the Devin and Zudin modified Rayleigh–Plesset equations to model bubble dynamics in a tube. Electronic J. Tech. Acoust. 11, http://ejta.org/en/klotz1.Google Scholar
Klotz, A. R., Lindvere, L., Stefanovic, B. & Hynynen, K. 2010 Temperature change near microbubbles within a capillary network during focused ultrasound. Phys. Med. Biol. 55 (6), 1549–1561.Google Scholar
Kudryashov, N. A. & Sinelshchikov, D. I. 2013 An extended equation for the description of nonlinear waves in a liquid with gas bubbles. Wave Motion 50, 351–362.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. Gen. 47, 5202.Google Scholar
Kudryashov, N. A. & Sinelshchikov, D. I. 2015 Analytical solutions for problems of bubble dynamics. Phys. Lett. A 379, 798–802.Google Scholar
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73, 106501.Google Scholar
Leighton, T. G.1994 Derivation of the Rayleigh–Plesset equation in terms of volume. ISVR Tech. Rep. 308. Institute of Sound and Vibration Research.Google Scholar
Lin, H., Storey, B. D. & Szeri, A. J. 2002 Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation. J. Fluid Mech. 452, 145–162.Google Scholar
Lue, L. & Bishop, M. 2006 Molecular dynamics study of the thermodynamics and transport coefficients of hard hyperspheres in six and seven dimensions. Phys. Rev. E 74, 021201.Google Scholar
Obreschkow, D., Bruderer, M. & Farhat, M. 2012 Analytical approximations for the collapse of an empty spherical bubble. Phys. Rev. E 85, 066303.CrossRefGoogle ScholarPubMed
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185.Google Scholar
Prosperetti, A. 1987 The equation of bubble dynamics in a compressible liquid. Phys. Fluids 30 (11), 3626–3628.Google Scholar
Prosperetti, A. 2004 Bubbles. Phys. Fluids 16 (6), 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.CrossRefGoogle 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
Pudovkin, M. I., Meister, C. V., Besser, B. P. & Biernat, H. K. 1997 The effective polytropic index in a magnetized plasma. J. Geophys. Res. 102(A12), 27145–27149.CrossRefGoogle Scholar
Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98.Google Scholar
Trilling, L. 1952 The collapse and rebound of a gas bubble. J. Appl. Phys. 23 (1), 14–17.CrossRefGoogle Scholar
Vokurka, K. 1986 Comparison of Rayleigh’s, Herring’s, and Gilmore’s models of gas bubbles. Acta Acust. United with Acustica 59 (3), 214–219.Google Scholar