Mapping Driven Oscillations in the Size of a Bubble to the Dynamics of a Newtonian Particle in a Potential

TL;DR

This paper presents a novel framework mapping bubble oscillations to a Newtonian particle in a potential, providing analytical tools to understand bubble dynamics, rebound, and energy dissipation during collapse.

Contribution

It introduces a new approach that distinguishes time scales in bubble oscillations and derives analytical approximations for key bubble behaviors.

Findings

Sharp bubble rebound explained by the model

Analytical tensile strength law incorporating gas behavior

Criterion for Bjerknes force reversal based on physical parameters

Abstract

The non-linear dynamics of driven oscillations in the size of a spherical bubble are mapped to the dynamics of a Newtonian particle in a potential within the incompressible liquid regime. The compressible liquid regime, which is important during the bubble's sonic collapse, is approached adiabatically. This new framework naturally distinguishes between the two time scales involved in the non-linear oscillations of a bubble. It also explains the experimentally observed sharp rebound of the bubble upon collapse. Guided by this new vantage point, we develop analytical approximations for several key aspects of bubble motion. First, we formulate a tensile strength law that integrates the bubble's ideal gas behavior with a general polytropic index. Next, we establish a straightforward physical criterion for Bjerknes force reversal, governed by the driving pressure, ambient pressure and tensile strength. Finally, we derive an acoustic energy dissipation formula for the bubble's sonic collapse, dependent solely on the bubble's collapse radii and velocity.