Analytical approximations for the collapse of an empty spherical bubble

Abstract

The Rayleigh equation 3/2Ṙ+RR̈+pρ−1=0 with initial conditions R(0)=R0, Ṙ(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 ρ. The solution for r≡R/R0 as a function of time t≡T/Tc, where R(Tc)≡0, is independent of R0, p, and ρ. While no closed-form expression for r(t) is known, we find that r0(t)=(1−t2)2/5 approximates r(t) with an error below 1%. A systematic development in orders of t2 further yields the 0.001% approximation r*(t)=r0(t)[1−a1 Li2.21(t2)], where a1≈−0.018 320 99 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.