The growth or collapse of a spherical bubble in a viscous compressible liquid

Abstract

With the help of a hypothesis first proposed by Mirkwood and Betbe, the partial differential equations for the flow of a compressible liquid surrounding a spherical bubble are reduced to a single total differential equation for the bubble-wall velocity. The Kirkwood-Bethe hypothesis represents an extrapolation of acoustic theory and would be expected to be most accurate when all liquid velocities are small compared to the velocity of sound in the liquid. However, the present theory is found to agree quite well with the only available numerical solution of the partial differential equations which extends up to a bubble-wall velocity of 2.2 times the sonic velocity. In the particular case of a bubble with constant (or zero) internal pressure, an analytic solution is obtained for the bubble-wall velocity which is valid over the entire velocity range for which the Mirkwood-Bethe hypothesis holds. In the more general situation, when the internal pressure is not constant, simple solutions are obtained only when the bubble-wall velocity is considerably less than sonic velocity. These approximate integral solutions are obtained by neglecting various powers of the ratio of wall velocity to sonic velocity. The zero-order approximation coincides with the equations for a bubble in an incompressible liquid derived by Rayleigh; the first-order approximation agrees with the solution obtained by Herring using a different method. The second-order approximation is presented here for the first time. The complete effects of surface tension, and the principal effects of viscosity, as long as the density variation in the liquid is not great, can be included in the analysis by suitably modifying the boundary conditions at the bubble wall. These effects are equivalent to a change in the internal bubble pressure. With this change, the same equations for the bubble-wall velocity are applicable to a viscous liquid with surface tension. Conditions under which the effects of surface tension and viscosity can be neglected are also determined. First and second-order approximations to the velocity and pressure fields throughout the liquid are derived. From these expressions, the acoustic energy radiated is calculated.

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