Analytical Approximations for the Collapse of an Empty Spherical Bubble

TL;DR

This paper develops accurate analytical approximations for the collapse dynamics of an empty spherical bubble modeled by the Rayleigh equation, validated against microgravity cavitation data.

Contribution

It introduces simple and highly accurate analytical formulas for the bubble collapse, improving understanding and prediction of bubble dynamics without closed-form solutions.

Findings

Approximate solution r(t) closely matches numerical results with less than 1% error.

The refined approximation r*(t) achieves 0.001% accuracy using polylogarithm functions.

Validated approximations against microgravity cavitation data, demonstrating practical applicability.

Abstract

The Rayleigh equation 3/2 R'+RR"+p/rho=0 with initial conditions R(0)=Rmax, R'(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density rho. The solution for r=R/Rmax as a function of time t=T/Tcollapse, where R(Tcollapse)=0, is independent of Rmax, p, and rho. While no closed-form expression for r(t) is known we find that s(t)=(1-t^2)^(2/5) approximates r(t) with an error below 1%. A systematic development in orders of t^2 further yields the 0.001%-approximation r*(t)=s(t)[1-a Li(2.21,t^2)], where a=-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these approximations is demonstrated by comparison to high-precision cavitation data obtained in microgravity.