Three-dimensional dynamical equations of interacting bubbles

Abstract

It is known that bubble cavitation plays important role in kidney stone fragmentation in the shock wave lithotripsy and other medical applications of shock wave. The behavior of such bubbles, however, considerably complicated because of its nonlinearity and mutual interactions among the bubbles. For weakly nonlinear oscillations, dynamics of interacting bubble can be approximately expressed by the method of multi-pole expansion with spherical harmonics. In the previous study, the author derived a set of dynamical equations of two interacting aspherical bubbles with Lagrangian mechanics. The axisymmetrical system of two interacting bubbles can be described in two-dimensional coordinate system, and then shape oscillation of the bubbles is expressed with Legendre polynomials. The bubble behavior described by the derived equations qualitatively agreed with experimental results by high-speed photographs. In the dynamics of three or more bubbles, however, the behavior of bubbles is essentially three dimensional, and thus the system of these bubbles should be represented by three-dimensional spherical harmonics (associated Legendre functions). In this study dynamical equations for three interacting aspherical bubbles are derived by multi-pole expansion in the framework of Lagrangian formalism. [Work supported by Grants-in-Aid for Scientific Research 23760142 and a research grant from The Mazda Foundation.]