Theory of the Collapsing Axisymmetric Cavity

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.

Figure 1

Numerical simulation of the time evolution of bubble pinchoff from initial conditions given by the shape with the largest waist. Pinchoff is initiated by surface tension, but the late stages are dominated by inertia, as observed experimentally [10].

Figure 2

A comparison of the exponent $\alpha$ between full numerical simulations of bubble pinchoff (solid line) and the leading order asymptotic theory with $\Gamma =0$ (dashed line).

Figure 3

A normalized graph of $\ddot{a}={\partial }^2{h}^2(z,t)/\partial t^2$ as given by the full numerical simulations, for two different initial conditions, and at $t^{\prime }=3.8\times {10}^{-10}$ (black line) and $t^{\prime }=2.1\times {10}^{-10}$ (grey line, green online). Similar collapse is found if profiles are evaluated at different times for the same initial conditions.

Figure 4

A normalized graph of $h_0/h_{0,\mathrm{pred}}$, where (i) $h_{0,\mathrm{pred}}\propto t^{\prime 1/2}/(-\ln h_0^2{)}^{1/4}$ (dashed line), (ii) $h_{0,\mathrm{pred}}\propto t^{\overline{\prime \alpha }}$ (dotted line), and (iii) $h_{0,\mathrm{pred}}\propto t^{\prime 1/2}\sqrt{e^{-\sqrt{-\ln t^{\prime }}}}$ (solid line).