On the scaling of jetting from bubble collapse at a liquid surface

We present scaling laws for the jet velocity resulting from bubble collapse at a liquid surface which bring out the effects of gravity and viscosity. The present experiments conducted in the range of Bond numbers $0.004<Bo<2.5$ and Ohnesorge numbers $0.001<Oh<0.1$ were motivated by the discrepancy between previous experimental results and numerical simulations. We show here that the actual dependence of $We$ on $Bo$ is determined by the gravity dependency of the bubble immersion (cavity) depth which has no power-law variation. The power-law variation of the jet Weber number, $We\sim 1/\sqrt{Bo}$, suggested by Ghabache et al. (Phys. Fluids, vol. 26 (12), 2014, 121701) is only a good approximation in a limited range of $Bo$ values ($0.1<Bo<1$). Viscosity enters the jet velocity scaling in two ways: (i) through damping of precursor capillary waves which merge at the bubble base and weaken the pressure impulse, and (ii) through direct viscous damping of the jet formation and dynamics. These damping processes are expressed by a dependence of the jet velocity on Ohnesorge number from which critical values of $Oh$ are obtained for capillary wave damping, the onset of jet weakening, the absence of jetting and the absence of jet breakup into droplets.