The diameters and velocities of the jet droplets produced after bubble bursting

Here we provide a theoretical framework revealing that the radius $R_d$ of the top droplet ejected from a bursting bubble of radius $R_b$ can be expressed as $R_d=0.22\,R_b\,\left(1-\left(\frac{Oh}{Oh'_c}\right)^{1/2}\right)$ for $Oh\leq Oh'_c\simeq 0.03$ and $Bo\leq 0.1$ with with $Oh=\mu/\sqrt{\rho R_b\sigma}\ll 1$ the Ohnesorge number, $Bo=\rho g R_b^2/\sigma$ the Bond number and $\rho$, $\mu$ and $\sigma$ the liquid density, viscosity and surface tension coefficient respectively. This prediction, which agrees very well with both experimental data and numerical simulations for all the values of $Oh$ and $Bo$ investigated, can be particularized to express the diameters of the jet droplets produced from the bursting of sea bubbles with radii $R_b\leq 1$ mm, with implications in marine aerosol production. The velocities of the first drops ejected are also expressed as a function of $Oh$ and $Bo$, being this initial drop velocity largely reduced by air drag at tiny distances $\sim R_b$ above the interface. We find that the ratio between the radius of curvature at the tip of the jet and the jet radius controls the growth of capillary instabilities, a fact explaining why no droplets are ejected from the tip of the fast Worthington jet for values of $Oh$ slightly larger than $Oh'_c$.