Acoustical correction to the bubble dynamics equation for elastic media

Abstract

Nonlinear oscillations of microbubbles are being investigated extensively for innovative medical and industrial applications. The Rayleigh–Plesset equation successfully describes the dynamics of gas bubbles in liquids and viscoelastic media. To model bubble dynamics in weakly compressible elastic media such as soft tissue, effects of shear stress should be taken into account. These effects were included recently by the authors in a model for bubble oscillations that differs from the Rayleigh–Plesset equation by a single term associated with nonlinear elasticity of the medium surrounding the bubble. Here, we augment this dynamical equation to account for acoustic radiation and thus describe nonlinear bubble oscillations in slightly compressible elastic media. The bubble is assumed to be spherical and the surrounding elastic medium to be isotropic. The analysis is performed in Eulerian coordinates, and attention is devoted to terms associated with radiation loss. Our analysis builds on results obtained by Prosperetti and Lezzi [J. Fluid Mech. 168, 457 (1986)] for a bubble in liquid. The stress tensor for the elastic medium is expressed through invariants of the Green deformation tensor in spherical coordinates. Explicit expressions for components of the stress tensor are obtained using Mooney’s potential function. [Work supported by IR&D at ARL:UT.]