Capillary waves control the ejection of bubble bursting jets

Here we provide a theoretical framework describing the generation of the fast jet ejected vertically out of a liquid when a bubble, resting on a liquid-gas interface, bursts. The self-consistent physical mechanism presented here explains the emergence of the liquid jet as a consequence of the collapse of the gas cavity driven by the low capillary pressures that appear suddenly around its base when the cap, the thin film separating the bubble from the ambient gas, pinches. The resulting pressure gradient deforms the bubble which, at the moment of jet ejection, adopts the shape of a truncated cone. The dynamics near the lower base of the cone, and thus the jet ejection process, is determined by the wavelength $\unicode[STIX]{x1D706}<^>{\ast }$ of the smallest capillary wave created during the coalescence of the bubble with the atmosphere which is not attenuated by viscosity. The minimum radius at the lower base of the cone decreases, and hence the capillary suction and the associated radial velocities increase, with the wavelength $\unicode[STIX]{x1D706}<^>{\ast }$ . We show that $\unicode[STIX]{x1D706}<^>{\ast }$ increases with viscosity as $\unicode[STIX]{x1D706}<^>{\ast }\propto Oh<^>{1/2}$ for $Oh\lesssim O(0.01)$ , with $Oh=\unicode[STIX]{x1D707}/\sqrt{\unicode[STIX]{x1D70C}R\unicode[STIX]{x1D70E}}$ the Ohnesorge number, $R$ the bubble radius and $\unicode[STIX]{x1D70C}$ , $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70E}$ indicating respectively the liquid density, viscosity and interfacial tension coefficient. The velocity of the extremely fast and thin jet can be calculated as the flow generated by a continuous line of sinks extending along the axis of symmetry a distance proportional to $\unicode[STIX]{x1D706}<^>{\ast }$ . We find that the jet velocity increases with the Ohnesorge number and reaches a maximum for $Oh=Oh_{c}$ , the value for which the crest of the capillary wave reaches the vertex of the cone, and which depends on the Bond number $Bo=\unicode[STIX]{x1D70C}gR<^>{2}/\unicode[STIX]{x1D70E}$ . For $Oh>Oh_{c}$ , the jet is ejected after a bubble is pinched off; in this regime, viscosity delays the formation of the jet, which is thereafter emitted at a velocity which is inversely proportional to the liquid viscosity.