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

Jet droplets from bubble bursting are determined by a limited parametrical space: the liquid properties (surface tension, viscosity, and density), mother bubble size, and acceleration of gravity. Thus, the two resulting parameters from dimensional analysis (usually the Ohnesorge and Bond numbers Oh and Bo) completely define this phenomenon when both the trapped gas in the bubble and the environment gas have negligible density. A detailed physical description of the ejection process to model both the ejected droplet radius and its initial launch speed is provided, leading to a scaling law including both Oh and Bo. Two critical values of Oh determine two limiting situations: One (${\text{Oh}}_1=0.038$) is the critical value for which the ejected droplet size is minimum and the ejection speed maximum, and the other (${\text{Oh}}_2=0.0045$) is a critical value which signals when viscous effects vanish. Gravity effects (Bo) are consistently introduced from energy conservation principles. The proposed scaling laws produce a remarkable collapse of published experimental measurements collected for both the ejected droplet radius and ejection speed.

Figure 1

Schematics of the bubble bursting phenomenon showing three stages where the initial free surface level, initial bubble radius $R_o$, ejected droplet radius $R$, main wave speed $V^{\prime }$, and induced ejection speed $V$ are indicated. The profiles shown correspond to three illustrating but arbitrary times selected from the case $\text{Oh}=0.01$ and $\text{Bo}=0.01$ in a detailed numerical simulation [16].

Figure 2

Nondimensional droplet radius $R/l_{\mu }$ as a function of the scaling variable ${\phi }_{\text{R}}$, from the same data as in Fig. 5 in Ref. [17]. The interested reader can assess a small deviation for large values of $\phi$ in that prior figure which is largely emphasized using the variable $R/R_o$ instead, as shown in the Supplemental Material [18].

Figure 3

Ejection velocity at the level of the original free surface $V$ made dimensionless with $V_{\mu }$, as a function of ${\phi }_{\text{V}}$. All available data, including those with $\text{Bo}>0.1$ (that were not included in the previous study [17]), are represented.