Analytical approximations for the collapse of an empty spherical bubble
The Rayleigh equation 3/2 (R) over dot + R(R) double over dot + p rho(-1) = 0 with initial conditions R(0) = R-0, (R) over dot(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 equivalent to R/R-0 as a function of time t equivalent to T/T-c, where R(T-c) equivalent to 0, is independent of R-0, p, and rho. 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) approximate to -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.