Analytical approximations for the collapse of an empty spherical bubble.

The Rayleigh equation 3/2R+RR+pρ(-1)=0 with initial conditions R(0)=R(0), 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 ρ. The solution for r≡R/R(0) as a function of time t≡T/T(c), where R(T(c))≡0, is independent of R(0), p, and ρ. While no closed-form expression for r(t) is known, we find that r(0)(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)=r(0)(t)[1-a(1)Li(2.21)(t(2))], where a(1)≈-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.