The Rayleigh collapse of two spherical bubbles

TL;DR

This paper derives an exact analytic solution for the nonlinear collapse dynamics of two interacting spherical bubbles, extending Rayleigh's classical single bubble collapse model, with an improved convergent approximation.

Contribution

It provides the first exact analytic solution for two-bubble collapse and introduces an asymptotic approximant to enhance convergence and accuracy.

Findings

Derived an exact solution for two-bubble collapse dynamics

Constructed an asymptotic approximant for better convergence

Generalized Rayleigh's single bubble collapse model

Abstract

The inertial collapse of two interacting and non-translating spherical bubbles of equal size is considered. The exact analytic solution to the nonlinear ordinary differential equation that governs the bubble radii during collapse is first obtained via a slowly converging power series. An asymptotic approximant is then constructed that accelerates convergence of the series and imposes the asymptotic collapse behavior when the radii are small. The solution generalizes the classical 1917 Rayleigh problem of single bubble collapse, as this configuration is recovered when the distance between the bubble centers far exceeds that of their radii.