Revision of Bubble Bursting: Universal Scaling Laws of Top Jet Drop Size and Speed

The collapse of a bubble of radius Ro at the surface of a liquid generating a liquid jet and a subsequent first drop of radius R is universally scaled using the Ohnesorge number Oh 1⁄4 μ=ðρσRoÞ1=2 and a critical value Oh below which no droplet is ejected; ρ, σ, and μ are the liquid density, surface tension, and viscosity, respectively. First, a flow field analysis at ejection yields the scaling of R with the jet velocity V as R=lμ ∼ ðV=VμÞ−5=3, where lμ 1⁄4 μ2=ðρσÞ and Vμ 1⁄4 σ=μ. This resolves the scaling problem of curvature reversal, a prelude to jet formation. In addition, the energy necessary for the ejection of a jet with a volume and averaged velocity proportional to RoR2 and V, respectively, comes from the energy excess from the total available surface energy, proportional to σR2 o, minus the one dissipated by viscosity, proportional to μðσR3 o=ρÞ1=2. Using the scaling variable φ 1⁄4 ðOh − OhÞOh−2, it yields V=Vμ 1⁄4 kvφ−3=4 and R=lμ 1⁄4 kdφ5=4, which collapse published data since 1954 and resolve the scaling of R and V with kv 1⁄4 16, kd 1⁄4 0.6, and Oh 1⁄4 0.043 when gravity effects are negligible.