Evolution of spherical cavitation bubbles: parametric and closed-form solutions

TL;DR

This paper analyzes the Rayleigh-Plesset equation for spherical cavitation bubbles, deriving parametric solutions with and without surface tension, and discusses the effects of viscosity through numerical methods.

Contribution

It introduces a connection between the Rayleigh-Plesset equation with surface tension and Abel's equation, providing new parametric solutions including rational Weierstrass solutions.

Findings

Derived parametric solutions for bubble dynamics with surface tension.

Connected Rayleigh-Plesset equation to Abel's equation for analytical solutions.

Performed numerical analysis for nonzero viscosity case.

Abstract

We present an analysis of the Rayleigh-Plesset equation for a three dimensional vacuous bubble in water. In the simplest case when the effects of surface tension are neglected, the known parametric solutions for the radius and time evolution of the bubble in terms of a hypergeometric function are briefly reviewed. By including the surface tension, we show the connection between the Rayleigh-Plesset equation and Abel's equation, and obtain the parametric rational Weierstrass periodic solutions following the Abel route. In the same Abel approach, we also provide a discussion of the nonintegrable case of nonzero viscosity for which we perform a numerical integration