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

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.