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

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 is universally scaled using the Ohnesorge number Oh=μ/(ρσR_{o})^{1/2} 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/l_{μ}∼(V/V_{μ})^{-5/3}, where l_{μ}=μ^{2}/(ρσ) and V_{μ}=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 R_{o}R^{2} and V, respectively, comes from the energy excess from the total available surface energy, proportional to σR_{o}^{2}, minus the one dissipated by viscosity, proportional to μ(σR_{o}^{3}/ρ)^{1/2}. Using the scaling variable φ=(Oh^{*}-Oh)Oh^{-2}, it yields V/V_{μ}=k_{v}φ^{-3/4} and R/l_{μ}=k_{d}φ^{5/4}, which collapse published data since 1954 and resolve the scaling of R and V with k_{v}=16, k_{d}=0.6, and Oh^{*}=0.043 when gravity effects are negligible.