General linearized solutions of the equations Of bubble pulsations

Abstract

Previous attempts to find linearized solutions of the equations for vapor bubble pulsations [e.g., see Finch and Neppiras, J. Acoust. Soc. Am. 53, 1402–1410 (1973)] assumed that oscillations would be the result of imposed sonic excitation. It was shown by Nicholas and Finch [12th Int. Cong. Acoust., Toronto, paper 14-2 (1986)] that nonlinear solutions of the equations in fact often showed exponential expansions or collapses. In this paper it is shown that if the governing equations are linearized using a Taylor series expansion, without assuming either oscillatory or exponential behavior, then there are regimes of bubble radius in which one or the other of these two is a necessary result. The simple case of a bubble with an insoluble gas content in a nonconducting liquid is considered. There can then be shown to exist a critical radius, with a value given by 2σ/3γP, where σ is the surface tension, γ is the ratio of specific heats, and P is the gas pressure. Below this size the behavior is exponential, above it, oscillatory. The results can be interpreted in terms of the stability of an open-loop control system.