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

Abstract

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.

Scaling laws of top jet drop size and speed from bubble bursting including gravity and inviscid limit

“…The physics involved has been discussed by several authors who have provided successive insightful approaches [8,[12][13][14][15][16]. A synthesis of the existing arguments was briefly discussed in [17]: those arguments pointed to the existence of an overall speed of the capillary wave front that should be of the order of . In other words, one should have t D > t o , which immediately implies that the Ohnesorge number Oh = µ (ρσRo) 1/2 should be below a critical one (here, Oh 1 ) to have a sufficiently energetic jetting for droplet ejection.…”

“…Among others, Krishnan et al [15] neatly described (see their figure 8) how the different wavelengths λ i of the wave packet produced by the breakup of the liquid film sequentially arrive at the axis segregated by their different wave speeds (σ/(ρλ i )) 1/2 . In [17], I observed that when the front of the main capillary wave producing ejection collapses at the axis, the curvature reversal of the surface involved in the onset of ejection imply that all terms of the momentum equation should be locally comparable. In brief, the collapse of a wave with speed V L and amplitude L leading to the ejection of a mass with characteristic radial size R and axial speed V should obey the dimensional balance:…”

“…that the total available energy in the form of surface energy, proportional to σR 2 o , minus the total viscous dissipation of the complete wave pack µ (σR 3 o /ρ) 1/2 should be proportional to the mechanical energy of the liquid ejected column ρV 2 R 2 R o . Observe that the potential energy of gravity was not accounted for in the balance (6) initially formulated in [17]. One should also be careful at considering what is understood as the ejection speed V since it varies strongly with space and time time.…”

“…Besides, the constant Oh 1 is precisely that critical Ohnesorge number above which the viscous dissipation would overcome the available surface energy, as one may readily observe dividing the whole equation (6) by σR 2 o . Combining equations (1), (2), and (6), one obtains [17]:…”

Bubble impact on a tilted wall: Removing bacteria using bubbles

“…Close to the impact moment at t = 0, the schematic is shown in Fig.11(d), the capillary-driven surface deformation leads to the formation of two dimples. The dimple moves with v d velocity which can be defined as v d ∼ σ/ρR (capillary wave speed with the wave number of R −1 ) [18]. So, we can scale the shear stress as τ ∼ µv d /h.…”