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Singular perturbation theory applied to the collective oscillation of gas bubbles in a liquid

Published online by Cambridge University Press:  20 April 2006

J. F. Scott
J. F. Scott
Affiliation:
Department of Applied Mathematical Studies, The University, Leeds, England Present address: Topexpress Limited, 1 Portugal Place, Cambridge CB5 8AF.
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Abstract

By using the technique of multiple scaling a theory of gas-bubble oscillations in a liquid is developed. Collections of bubbles of arbitrary shape under the action of surface tension, buoyancy and solid surfaces are considered. In the absence of thermal conduction in the bubbles and compressibility of the liquid a conserved ‘action’ is defined for each of the modes of oscillation. The equation governing the decay of the action with time is found by carrying the analysis to second order. The geometrical configuration of the bubbles, in which the oscillations take place, evolves in time under convection by an underlying ‘basic’ flow for which the governing equations are derived. The bubble pulsations influence the development of the basic motion. Later in the work a source of gas bubbles is brought in and its effects on the oscillations discussed. The results of the interaction of pulsating bubbles with the liquid surface are also briefly considered. The determination of the amplitude of oscillations induced by the splitting up of bubbles and by the generation of bubbles from the gas source is described. Finally, several applications of the theory to specific problems are given.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 113 , December 1981 , pp. 487 - 511
Copyright
© 1981 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Washington D.C.: National Bureau of Standards.
Crum, L. A. 1975 Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am. 57, 1363–70.Google Scholar
Devin, C. 1959 Survey of thermal, radiation and viscous damping of pulsating air bubbles in water. J. Acoust. Soc. Am. 31, 1654–1667.Google Scholar
Fitzpatrick, H. M. & Strasberg, M. 1957 Hydrodynamic sources of sound. Naval Hydrodynamics Pub. 515, pp. 241–275. National Academy of Sciences, U.S. National Research Council.
Foldy, L. L. 1945 The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Phys. Rev. 67, 107–119.Google Scholar
Grimshaw, R. & Allen, J. S. 1979 Linearly coupled, slowly varying oscillators. Stud. Appl. Math. 61, 55–71.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.
Nayfeh, A. H. 1973 Perturbation Methods. Wiley.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear oscillations. Wiley.
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Ann. Rev. Fluid Mech. 9, 145–185.Google Scholar
Prosperetti, A. 1977 Thermal effects and damping mechanisms in forced radial oscillations of gas bubbles in liquids. J. Acoust. Soc. Am. 61, 17–27.Google Scholar
Tinkham, M. 1964 Group Theory and Quantum Mechanics. McGraw-Hill.
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Whitfield, O. J. & Howe, M. S. 1976 The generation of sound by two-phase nozzle flows and its relevance to excess noise of jet engines. J. Fluid Mech. 75, 553–576.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Van Wijngaarden, L. 1972 One-dimensional flow of liquids containing small gas bubbles. Ann. Rev. Fluid Mech. 4, 369–396.Google Scholar