Capillary waves after bubble coalescence and their role in the generation of bubble bursting jets

Here we provide a theoretical framework describing the generation and subsequent breakup of the fast and vertical Worthington jet ejected out of the liquid when a bubble of radius R, initially at rest at an interface, bursts. The correct scalings for the diameters and the velocities of the drops emitted from the highly stretched jet tip, in the limit of negligible gravitational effects, are deduced taking into account the experimental observation that the dynamics of the jet ejection process is governed by the traveling wave excited during the capillary retraction of the rim bordering the bubble cap. The self-consistent physical mechanism presented here explains the emergence of the vertical jet as a consequence of the breakup of the self-similarity of the inertio-capillary cavity collapse caused by the waves excited during the rim retraction process, which possess a characteristic wavelength proportional to ROh^1/2, with Oh=μ/(ρRσ)^1/2 the Ohnesorge number and where ρ, μ and σ indicate the liquid density, viscosity and interfacial tension coefficient, respectively.