Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T21:40:09.026Z Has data issue: false hasContentIssue false
  • English
  • Français

Bubble dynamics in a compressible liquid. Part 2. Second-order theory

Published online by Cambridge University Press:  21 April 2006

A. Lezzi  and
A. Prosperetti
A. Lezzi
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
A. Prosperetti
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA Present address: Dipartimento di Matematica, Universitá degli Studi, Cagliari, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a rigorous singular-perturbation method to second order in the bubble-wall Mach number. The results of Part 1 (Prosperetti & Lezzi, 1986) are recovered at orders zero and one. At second order the ordinary inner and outer structure of the solution proves inadequate to correctly describe the fields and it is necessary to introduce an intermediate region the characteristic length of which is the geometric mean of the inner and outer lengthscales. The degree of indeterminacy for the radial equation of motion found at first order is significantly increased by going to second order. As in Part 1 we examine several of the possible forms of this equation by comparison with results obtained from the numerical integration of the complete partial-differential-equation formulation. Expressions and results for the pressure and velocity fields in the liquid are also reported.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 185 , December 1987 , pp. 289 - 321
Copyright
© 1987 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

Cole, R. H. 1948 Underwater Explosions. Princeton University Press (reprinted by Dover 1965).
Epstein, D. & Keller, J. B. 1971 Expansion and contraction of planar, cylindrical, and spherical underwater gas bubbles. J. Acoust. Soc. Am. 52, 975–980.Google Scholar
Flynn, H. G. 1975 Cavitation dynamics I. A mathematical formulation. J. Acoust. Soc. Am. 57, 1379–1396.Google Scholar
Fujikawa, S. & Akamatsu, T. 1980 Effects of the non-equilibrium condensation of vapor on the pressure wave produced by the collapse of a bubble in a liquid. J. Fluid Mech. 97, 481–512.Google Scholar
Gilmore, F. R. 1952 The collapse and growth of a spherical bubble in a viscous compressible liquid. California Institute of Technology Hydrodynamics Laboratory, Rep. 26–4.
Herring, C. 1941 Theory of the pulsations of the gas bubble produced by an underwater explosion. OSRD Rep. 236.
Hickling, R. & Plesset, M. S. 1964 Collapse and rebound of a spherical bubble in water. Phys. Fluids 7, 7–14.Google Scholar
Kaplun, S. 1967 Fluid Mechanics and Singular Perturbations (ed. P. A. Lagerstrom, L. N. Howard & C. S. Liu), pp. 43–60. Academic.
Keller, J. B. & Kolodner, I. I. 1956 Damping of underwater explosion bubble oscillations. J. Appl. Phys. 27, 1152–1161.Google Scholar
Keller, J. B. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68, 628–633.Google Scholar
Lagerstrom, P. A. & Casten, R. G. 1972 Basic concepts underlying singular perturbation techniques. SIAM Rev. 14, 63–120.Google Scholar
Obermeier, F. 1976 The application of singular perturbation methods to aerodynamic sound generation. In Lecture Notes in Mathematics, vol. 596, pp. 400–418, Springer.
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1. First-order theory. J. Fluid Mech. 168, 457–478.Google Scholar
Rath, H. J. 1980 Free and forced oscillations of spherical gas bubbles and their translational motion in a compressible fluid. In Cavitation and Inhomogeneities in Underwater Acoustics (ed. W. Lauterborn), pp. 64–71. Springer.
Tilmann, P. M. 1980 Nonlinear sound scattering by small bubbles. In Cavitation and Inhomogeneities in Underwater Acoustics (ed. W. Lauterborn), pp. 113–118. Springer.
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
Trilling, L. 1952 The collapse and rebound of a gas bubble. J. Appl. Phys. 23, 14–17.Google Scholar