The mass ejected by a bubble bursting from a free drop

Abstract

A bubble bursting at a flat liquid surface ejects droplets only below a critical Ohnesorge number Oh$_c\simeq0.043$. We ask how much it ejects when the bath is a drop of finite size. We solve the axisymmetric Navier--Stokes equations for a bubble of radius $R_0$ tangent internally to a free drop of radius $\lambda R_0$, punctured at $t=0$, over liquid-to-gas volume ratios $\Lambda=V_{\rm liq}/V_{\rm gas}=\lambda^3-1$ from $1/16$ to $512$ and Oh from 0.005 to 0.11. Ejection ceases at Oh$_1=Oh_c(1+2\beta/\lambda)$ with $\beta\cong 0.83$, so confinement extends ejection to liquids too viscous, or bubbles too small, to eject at a flat surface. Two effects of first order in $1/\lambda$ produce the shift: the added Laplace overpressure of the outer surface, and the reduced inertia of the liquid shell. Our main result concerns the ejected mass $M_e$, which unlike the droplet count converges under mesh refinement. It obeys $M_e=C\,\delta\,V_{\rm gas}V_{\rm liq}/(V_{\rm gas}+V_{\rm liq})$, with $\delta=1-Oh/Oh_1$ and $C\simeq 0.013$, for $\Lambda\gtrsim0.2$: the two volumes combine as a reduced volume. When liquid is abundant this reduces to $M_e=C\,\delta\,V_{\rm gas}$, a fixed fraction of the bubble volume, in agreement with classical jet-drop measurements; when gas is abundant, to $M_e=C\,\delta\,V_{\rm liq}$. The fraction of liquid ejected spans four orders of magnitude, exceeding one third in the thinnest shells, where a distinct twin-jet mechanism takes over. Since hollow drops are generic in breaking waves, confinement includes bubbles that a flat surface would exclude and fixes what each delivers: two essential ingredients of sea-spray source functions.