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 $\mathrm{Oh}=\mu /(\rho \sigma R_o{)}^{1/2}$ and a critical value ${\mathrm{Oh}}^{*}$ below which no droplet is ejected; $\rho$, $\sigma$, and $\mu$ 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 $R/l_{\mu }\sim {(V/V_{\mu })}^{-5/3}$, where $l_{\mu }={\mu }^2/(\rho \sigma )$ and $V_{\mu }=\sigma /\mu$. 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 $\sigma R_o^2$, minus the one dissipated by viscosity, proportional to $\mu {(\sigma R_o^3/\rho )}^{1/2}$. Using the scaling variable $\phi =({\mathrm{Oh}}^{*}-\mathrm{Oh}){\mathrm{Oh}}^{-2}$, it yields $V/V_{\mu }=k_v{\phi }^{-3/4}$ and $R/l_{\mu }=k_d{\phi }^{5/4}$, which collapse published data since 1954 and resolve the scaling of $R$ and $V$ with $k_v=16$, $k_d=0.6$, and ${\mathrm{Oh}}^{*}=0.043$ when gravity effects are negligible.

Figure 1

A 3D rendering and composition of three fundamental instants of a bubble collapse taken from Fig. 1 of MacIntyre [13]: $t\sim 0$ (just after film rupture), 1.17 (onset of jet ejection), and 1.67 ms (just before the first droplet detachment). (Courtesy of B. Gañán-Riesco, Ingeniatrics Tec. S.L.)

Figure 2

Onset of ejection. The complex vortical structure was already described [14]. $V$ and $V^{\prime }$ stand for the scales of the jet speed and the radial surface velocity, respectively. $R$ is the radial scale (final droplet radius). Thin continuous lines represent vorticity isolines. A cylindrical surface with characteristic axial and radial scales $L$ and $R$, respectively, is indicated with dashed lines.

Figure 3

Nondimensional droplet radius $R/l_{\mu }$ as a function of the nondimensional axial jet velocity at ejection $V/V_{\mu }$ reported in the literature [6, 13, 14, 19, 20] for $\text{Bo}<0.1$. The more realistic thin rim simulations of Duchemin et al. [14] are used.

Figure 4

Axial jet velocities reported in the literature [5, 6, 14, 19, 20] for $\text{Bo}<0.1$.

Figure 5

Universal scaling of experimental results according to the proposed model. The data are taken from figures in Refs. [6, 11, 13, 14, 19, 20, 23, 24, 25, 26, 27]; in particular, all data from Séon and Liger-Belair [see [21], Figs. 16 and 19(b) and the higher viscosity cases of Fig. 17(a)] have been used. The standard deviation of all data from the model (excluding wide rim from Ref. [14]) is 13.6%. The inset shows the raw data. Note that this figure is not just the composition of Figs. 3 and 4: It shows many more points, since many experimental data sets reported in the literature disclose the ejected droplet size but not the speed of the jet’s tip.