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A generalization of the impulse and virial theorems with an application to bubble oscillations

Published online by Cambridge University Press:  26 April 2006

Hasan N. Oguz  and
Andrea Prosperetti
Hasan N. Oguz
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
Andrea Prosperetti
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
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Abstract

In the first part of the paper it is shown that the impulse and virial theorems of inviscid incompressible fluid mechanics are special cases of a more general theorem from which an infinity of relations can be obtained. Depending on the problem, only a finite number of these relations may be independent. An application of these results is in the approximate study of the hydrodynamic interaction of bodies. As an example, in the second part of the paper, the case of two freely translating, nonlinearly pulsating bubbles is considered. It is found that in certain parameter ranges the force between the bubbles has a sign opposite to what would be expected on the basis of the linear theory of Bjerknes forces.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 218 , September 1990 , pp. 143 - 162
Copyright
© 1990 Cambridge University Press

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References

Apfel, R. E.: 1976 Technique for measuring the adiabatic compressibility, density and sound speed of sub-microliter liquid samples. J. Acoust. Soc. Am. 59, 339–343.Google Scholar
Aris, R.: 1962 Vectors, Tensors and the Basic Equations of Fluid Mechanics, ch. 9. Prentice Hall.
Bassett, A. B.: 1888 A Treatise on Hydrodynamics, vol. I, p. 248. London: Deighton, Bell and Co. (Reprinted by Dover, New York, 1961.)
Benjamin, T. B.: 1987 Hamiltonian theory for motions of bubbles in an infinite liquid. J. Fluid Mech. 181, 349–379.Google Scholar
Benjamin, T. B.: 1989 Note on shape oscillations of bubbles. J. Fluid Mech. 203, 419–424.Google Scholar
Benjamin, T. B. & Ellis, A. T., 1966 The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A 260, 221–240.Google Scholar
Benjamin, T. B. & Olver, P. J., 1983 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–85.Google Scholar
Blake, J. R.: 1983 The Kelvin impulse: application to bubble dynamics. In Proc. 8th Austral. Fluid Mech. Conf. Newcastle, pp. 10B1–10B4.Google Scholar
Blake, J. R.: 1988 The Kelvin impulse: application to cavitation bubble dynamics. J. Austral. Math. Soc. B 30, 127–46.Google Scholar
Blake, J. R. & Cerone, P., 1982 A note on the impulse due to a vapour bubble near a boundary. J. Austral. Math. Soc. B 23, 383–393.Google Scholar
Blake, J. R. & Gibson, D. C., 1981 Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech. 111, 123–140.Google Scholar
Blake, J. R. & Gibson, D. C., 1987 Cavitation bubbles near boundaries. Ann. Rev. Fluid Mech. 19, 99–123.Google Scholar
Blake, J. R., Taib, B. B. & Doherty, G., 1986 Transient cavities near boundaries. Part 1. Rigid boundary. J. Fluid Mech. 170, 479–497.Google Scholar
Crum, L. A.: 1975 Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am. 57, 1363–1370.Google Scholar
Crum, L. A. & Nordling, D. A., 1972 Velocity of transient cavities in an acoustic stationary wave. J. Acoust. Soc. Am. 52, 294–301.Google Scholar
Kemmer, N.: 1977 Vector Analysis, ch. 10. Cambridge University Press.
Landau, L. D. & Lifshitz, E. M., 1959 Fluid Mechanics, pp. 26–30. Pergamon.
Longuet-Higgins, M. S.: 1989 Some integral theorems relating to the oscillation of bubbles. J. Fluid Mech. 204, 159–166.Google Scholar
Malvern, L. E.: 1969 Introduction to the Mechanics of a Continuous Medium, pp. 210–213. Prentice-Hall.
Minnaert, M.: 1933 On musical air bubbles and the sounds of running water. Phil. Mag. 16, 235–248.Google Scholar
Oguz, H. N. & Prosperetti, A., 1989 Oscillations of a bubble in the vicinity of a nearly plane pressure-release boundary. J. Acoust. Soc. Am., in press.Google Scholar
Prosperetti, A.: 1979 Boundary conditions at a liquid–vapour interface. Meccanica 14, 34–46.Google Scholar
Prosperetti, A.: 1984 Bubble phenomena in sound fields: part two. Ultrasonics 22, 115–23.Google Scholar
Strasberg, M.: 1953 The pulsation frequency of nonspherical gas bubbles in liquids. J. Acoust. Soc. Am. 25, 536–7.Google Scholar
Weiser, M. A. H., Apfel, R. E. & Neppiras, E. A., 1984 Interparticle forces on red cells in a standing wave field. Acustica 56, 114–119.Google Scholar
Wu, T. Y.: 1976 The momentum theorem for a deformable body in a perfect fluid. Schiffstechnik 23, 229–232.Google Scholar
Zabolotskaya, E. A.: 1984 Interaction of gas bubbles in a sound field. Sov. Phys. Acoust. 30, 365–368.Google Scholar
Zavtrak, S. T.: 1987 Bjerknes interaction force of two gas bubbles in a sound field. Sov. Phys. Acoust. 33, 145–147.Google Scholar