On the forced oscillations of a small gas bubble in a spherical liquid-filled flask

A spherically-symmetric problem is considered in which a small gas bubble a t the centre of a spherical flask filled with a compressible liquid is exci ted by small radial displacements of the flask wall. The bubble may be comp ressed, expanded and made to undergo periodic radial oscillations. Two asym ptotic solutions have been found for the low-Mach-number stage. The first o ne is an asymptotic solution for the field far from the bubble, and it corr esponds to the linear wave equation. The second one is an asymptotic soluti on for the field near the bubble, which corresponds to the Rayleigh-Plesset equation for an incompressible fluid. For the analytical solution of the l ow-Mach-number regime, matching of these asymptotic solutions is done, yiel ding a generalization of the Rayleigh-Plesset equation. This generalization takes into account liquid compressibility and includes ordinary differenti al equations (one of which is similar to the well-known Herring equation) a nd a difference equation with both lagging and leading time. These asymptot ic solutions are used as boundary conditions for bubble implosion using num erical codes which are based on partial differential conservation equations . Both inverse and direct problems are considered in this study. The invers e problem is when the bubble radial motion is given and the evolution of th e flask wall pressure and velocity is to be calculated. The inverse solutio n is important if one is to achieve superhigh gas temperatures using nonper iodic forcing (Nigmatulin et al. 1996). In contrast, the direct problem is when the evolution of the flask wall pressure or velocity is given, and one wants to calculate the evolution of the bubble radius. Linear and nonlinea r periodic bubble oscillations are analysed analytically. Nonlinear resonan t and near-resonant periodic solutions for the bubble non-harmonic oscillat ions, which are excited by harmonic pressure oscillations on the flask wall , are obtained. The applicability of this approach bubble oscillations in e xperiments on single-bubble sonoluminescence is discussed.