The bursting of hollow droplets

Abstract

A bubble bursting from a drop of finite size (a hollow droplet) is the generic bursting event in a breaking wave, yet it has been studied almost exclusively at unbounded baths. We follow it from the puncture of the film to the spectrum of a spray, in axisymmetric simulations over liquid-to-gas volume ratios $\Lambda$ from $1/16$ to $512$ and Ohnesorge numbers from $0.005$ to $0.11$. Ejection ceases at ${Oh}_1={Oh}_c(1+2\beta /\lambda )$, with $\beta \simeq 0.83$ and $\lambda =(1+\Lambda {)}^{1/3}$: confinement extends ejection to bubbles too small to eject at a flat surface, and the distance $\delta =1-Oh/{Oh}_1$ to that boundary organizes everything that follows. The collapse, driven by capillary waves, is that of a gas thread pinching next to its mouth, whether the liquid shell is punctured once or twice. The ejected volume is $M_e=C\delta V_{\mathrm{red}}$, with $C\simeq 0.013$ and $V_{\mathrm{red}}=V_{\mathrm{gas}}{V}_{\mathrm{liq}}/(V_{\mathrm{gas}}+V_{\mathrm{liq}})$ a reduced volume. Whatever the confinement, the first droplet is smallest at $\delta \simeq 0.2$ in units of the viscocapillary length ${\ell }_{\mu }={\mu }^2/(\rho \sigma )$ and at $\delta \simeq 0.3$ in units of the bubble radius $R_0$, where the unbounded bath also places its optimum. The largest reaches a tenth of $R_0$. Between them the census is exponential, with a scale set by $R_0$ and no lower cut-off: the smallest droplet a simulation records is set by its resolution, not by ${\ell }_{\mu }$, which belongs to the pinch singularity rather than to the droplets. Mixed over bubble populations, the census gives spray spectra in closed form, whose shape measures the population. The bursting also produces hollow droplets, each a potential new generator.

31 pages, 17 figures