Theory of the Collapsing Axisymmetric Cavity

Abstract

We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we compute the local scaling exponent $\alpha =d\ln h_0/d\ln t^{\prime }$ of the minimum radius $h_0$ of the cavity, where $t^{\prime }$ is the time from collapse. The exponent $\alpha$ very slowly approaches a universal value according to $\alpha =1/2+1/[4\sqrt{-\ln (t^{\prime })}]$. Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single nontrivial scaling exponent. Our predictions are confirmed by numerical simulations.