The Rayleigh equation $\frac{3}{2}\dot{R}+R\ddot{R}+p{\rho }^{-1}=0$ with initial conditions $R\left(0\right)=R_0$, $\dot{R}\left(0\right)=0$ models the collapse of an empty spherical bubble of radius $R\left(T\right)$ in an ideal, infinite liquid with far-field pressure $p$ and density $\rho$. The solution for $r\equiv R/R_0$ as a function of time $t\equiv T/T_{\mathrm{c}}$, where $R\left(T_{\mathrm{c}}\right)\equiv 0$, is independent of $R_0$, $p$, and $\rho$. While no closed-form expression for $r\left(t\right)$ is known, we find that $r_0\left(t\right)={(1-t^2)}^{2/5}$ approximates $r\left(t\right)$ with an error below 1$\%$. A systematic development in orders of $t^2$ further yields the 0.001$\%$ approximation $r_{*}\left(t\right)=r_0\left(t\right)[1-a_1{\mathrm{Li}}_{2.21}\left(t^2\right)]$, where $a_1\approx -0.01832099$ 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.

• Received 2 February 2012

DOI:https://doi.org/10.1103/PhysRevE.85.066303