Self-similar Worthington jets
When a micron-sized bubble bursts, capillary waves deform the cavity into a cone that ejects a Worthington jet. The jet is born by inertial focusing, and the local collapse follows self-similar Euler solutions set by the semiangle $β$. Writing $r_j$ and $v_j$ for the dimensionless jet-base radius and velocity, the local Weber number $We_j=r_j v^2_j$ measures inertia relative to capillarity. The theory, supported by accurate numerical simulations gives $r_j\proptoτ^{α(β)}$ with $α\simeq0.63$ and, hence $We_j\gg1$, with $We_j\to\infty$ as $r_j\to0$, so inertia increasingly overwhelms capillarity. In simulations, the interface collapses onto a universal shape for more than two decades in dimensionless time when lengths are scaled using our prediction for $r_j$. For water, this gives incipient radii of $\mathcal{O}(1)$ nm, predicting nanometric sea-spray aerosols.