Universal scaling laws of top jet drop size and speed in bubble bursting*

The collapse of a bubble of radius $R_o$ at the surface of a liquid generating a liquid jet and a subsequent first drop of radius $R$ follows a universal flow pattern that can be universally scaled using the difference between the parent bubble radius and a critical radius $R^*={\text{Oh}^*}^{-2}\mu^2/(\rho\sigma)$ below which no droplet is ejected for a given Newtonian liquid. Here, $\text{Oh}^*=0.037$ is the critical Ohnesorge number (Walls et al. 2015, Phys. Rev. E 92, 021002(R)), where $\text{Oh}=\mu/(\rho\sigma R_o)^{1/2}$; $\rho$, $\sigma$ and $\mu$ are the liquid density, surface tension and viscosity. Based on a flow singularity occurring for $R_o=R^*$, a scaling analysis of the complex flow structure at the onset of jet ejection for $R_o>R^*$ leads to the diameter of the first emitted droplet and the initial ejection velocity: $D = k_d (R_o-R^*)^{5/4}{R^*}^{-1/4}$ and $V=k_v \sigma \mu^{-1} (R_o-R^*)^{3/4}{R^*}^{-3/4}$, respectively. A remarkable collapse of data taken from available literature since 1954 to 2017 furnishes the universal constants $k_d=0.1$ and $k_v=1.6$, for negligible gravity effects.The role of gravity is subdominant and can be reflected by the exponential dependence of the scaling laws obtained on the Bond number.