Jet drops from bursting bubbles: How gravity and viscosity couple to inhibit droplet production

When a bubble ruptures at a liquid surface the collapsing cavity produces a central jet that frequently breaks up into a series of droplets. Current experiment and theory predict that the production of jet drops will be limited by either viscous or gravitational effects. However, while there are a number of studies focusing on these two limiting cases, less is understood about the production of jet drops when both gravitational and viscous effects are significant. Here, we uncover the existence of an intermediate region where both gravitational and viscous effects play a critical role in jet-drop formation. We propose that the role of gravity is most important before rupture, and carry out simulations that demonstrate the importance of the equilibrium bubble shape in the production of jet drops.

Figure 1

High-speed series of images showing the evolution of two rupturing air bubbles in glycerol-water mixtures. (a) Under certain conditions (cap radius of curvature $R=1.7\mathrm{mm}$, viscosity ${\mu }_{\ell }=3.5\mathrm{mPa}\mathrm{s}$) a central jet rises and produces one or more jet droplets. (b) Under other conditions $\left(R=2.7\mathrm{mm},{\mu }_{\ell }=9.9\mathrm{mPa}\mathrm{s}\right)$, a central jet rises but no droplets are produced. The absence of jetting in this case cannot be explained by gravitational or viscous effects alone (see Supplemental Material videos [25]).

Figure 2

Compilation of individual bubble bursting experiments that lead to jet drops (open symbols) and no jet drops (closed symbols). The regimes in which viscosity or gravity alone would prevent jet drops from forming are indicated with shading. (a) Jet drops are expected only for a certain range of bubble sizes (as measured by cap radius $R$), and this range decreases as the viscosity increases (unshaded region). Yet, jet drops are not observed in much of this region. Here, the symbols indicate the particular liquid represented by the same symbol in Table 1. (b) The experimental data is replotted in terms of the Ohnesorge number, $\mathrm{Oh}={\mu }_{\ell }/\sqrt{{\rho }_{\ell }\gamma R}$, and the Bond number, $\mathrm{Bo}={\rho }_{\ell }g{R}^2/\gamma$, on a logarithmic scale. The dotted line is a guide for the eye separating the observation of droplets from no droplets. The dash-dotted lines denoting the critical ${\mathrm{Bo}}_{\mathrm{c}}$ and ${\mathrm{Oh}}_{\mathrm{c}}$ are calculated using the average property values in Table 1.

Figure 3

A series of simulation results matching the experimental series shown in Fig. 1. Each series is matched by the Bond and Ohnesorge numbers aligned with the inertio-capillary time $\tau =t/\sqrt{{\rho }_{\ell }{R}^3/\gamma }$. (a) The simulations predict that a jet drop forms when $\mathrm{Oh}=0.01,\mathrm{Bo}=0.5$, as expected. (b) The simulations predict that no jet drops are formed when $\mathrm{Oh}=0.02,\mathrm{Bo}=1.3$, consistent with our experiments.

Figure 4

Simulation boundary of jet droplet production illustrating the extent of the intermediate region where gravity and viscosity are significant. The surrounding gas has a minimal influence in determining the limits on droplet formation in the viscosity ratio range ${\mu }_g/{\mu }_{\ell }={10}^{-3}\mathrm{to}{10}^{-2}$. The predicted boundaries are in agreement with both our experimental results [data from Fig. 2] and those results of previous studies [Bond () [27] and Ohnesorge () 26]. The vertical dotted line at $\mathrm{Oh}=0.02$ indicates the location of the simulations used to investigate the role of gravity in the jetting process (Fig. 5).

Figure 5

Results of a series of simulations designed to probe the role of gravity in the production of jet droplets. The effects of Bond number (gravity) in the jetting process are divided into those occurring before $\left({\mathrm{Bo}}^{-}\right)$ and after $\left({\mathrm{Bo}}^{+}\right)$ the bubble bursts. (a) The shape of a bubble in equilibrium is uniquely determined by the value of ${\mathrm{Bo}}^{-}$, tending to flatten out as gravity becomes increasingly important. (b) When ${\mathrm{Bo}}^{-}={\mathrm{Bo}}^{+}$, the result is equivalent to experimental observation (vertical dashed line in Fig. 4). The simulations show that a $100\times$ increase in ${\mathrm{Bo}}^{+}=0.01\to 1.3$ for the ${\mathrm{Bo}}^{-}=0.01$ case has little effect on droplet production. Likewise, a $100\times$ decrease in ${\mathrm{Bo}}^{+}=1.3\to 0.01$ for the ${\mathrm{Bo}}^{-}=1.3$ case does not encourage jet droplets to form. Yet, a change in ${\mathrm{Bo}}^{-}$ (the shape of the bubble) completely accounts for the change in jet production.