The dynamics of an ultrasonic cavitation bubble

Abstract

The ultrasonic cavitation bubble is considered to be a small spherical gas bubble set in motion by an ultrasonic wave. The dynamics of this bubble are represented by a problem in partial differential equations, the governing equations of which are the continuity equation and the Navier-Stokes equation for a compressible liquid. I christened this problem the Rayleigh-Noltingk-Neppiras (RANN) problem. The RANN problem has not been solved exactly (numerically or otherwise) but some approximations have been used to reduce it to various initial-value ordinary differential equation problems, namely, the Noltingk-Neppiras model, the Herring- Flynn model, and the Kirkwood-Bethe model. These approximate models are presently the only tools employed in the mathematical analysis of the dynamics of the ultrasonic cavitation bubble. However, the solutions of these models indicate that the behavior of certain bubbles are such that the approximate models are no longer valid. When the predicted velocity of the bubble wall is comparable to the speed of sound, all the three approximate models (the Noltingk-Neppiras model, the Herring-Flynn model, and the Kirkwood-Bethe model) break down. In the periods of large bubble wall velocities, the simulation of the dynamics of the bubble can be accomplished satisfactorily only by the exact solution of the original problem--the RANN problem. The purpose of this investigation is to obtain this exact solution (the solution of the RANN problem).

A finite difference scheme was developed for generating the exact solution of the RANN problem. No actual solutions were obtained because the scheme indicated that the problem has no solution. It was not possible in this investigation to obtain a solution that would satisfy both the governing equations and all the boundary conditions. The problem has more boundary conditions in pressure than can be satisfied; it is an overconstrained problem, Consequently, the RANN problem is not a valid mathematical statement; it is not a legitimate representation of the dynamic behavior of the cavitation bubble.

The Noltingk-Neppiras model, the Herring-Flynn model, and the Kirkwood-Bethe model are all various approximations of the RANN problem. Therefore, they are not valid mathematical statements. In particular, the Noltingk-Neppiras model was singled out and it was shown explicitly that this model is mathematically deficient.

Furthermore, an attempt to reproduce the Herring-Flynn model and the Kirkwood-Bethe model from the RANN problem failed. The application of the acoustic approximations due to Herring, and Kirkwood and Bethe to the RANN problem resulted in equations slightly different from the Herring-Flynn model and the Kirkwood-Bethe model.