Complex behavior very close to the pinching of a liquid free surface

We determine experimentally, with an unprecedented spatiotemporal resolution, the evolution of the liquid free surface during the pinching of a pendant droplet. We observe the transition from the inertio-capillary regime to the inertio-viscous-capillary final stage for a low viscosity liquid. Experiments with 5-cSt silicone oil show the formation of a micrometer subsatellite droplet in the vicinity of the pinching region. This phenomenon does not occur for a glycerine-water mixture with the same Ohnesorge number, which indicates the existence of complex effects during the breakup of the silicone oil droplet. The comparison between the experimental results and numerical simulations of the full Navier-Stokes equations shows that experiments deviate from the Newtonian theoretical predictions when the subsatellite droplet forms. Viscoelasticity might explain that deviation.

Figure 1

Breakup of a pendant drop of 5-cSt SO. The breakage of the liquid thread connecting the upper (small) and lower (large) parent droplets produces a satellite droplet. Very close to the pinching, a microthread forms and gives rise to a subsatellite droplet.

Figure 2

Pinch-off of a drop of (from top to bottom) 1-cSt SO, 5-cSt SO, 10-cSt, G-W 52/48% (v/v), and G-W 60/40% (v/v). The labels indicate the time to the pinching with an error of $\pm 100$ ns.

Figure 3

$R_{\text{min}}\left({\tau }_{\text{expt}}\right)$ obtained experimentally (symbols) and numerically (solid line) from the breakup of a pendant drop of 1-cSt SO. The dashed line corresponds to Eggers's solution (2).

Figure 4

$R_{\text{min}}\left({\tau }_{\text{expt}}\right)$ obtained experimentally (symbols) and numerically (solid line) from the breakup of a pendant drop of 5-cSt SO. The dotted line corresponds to the viscous solution $R_{\text{min}}\left({\tau }_{\text{expt}}\right)=0.0709\sigma /\mu ({\tau }_{\text{expt}}-\mathrm{\Delta }\tau )$. The arrows indicate the points corresponding to the experimental and numerical contours shown in the last graph of Fig. 5 (${\tau }_{\text{expt}}=11.9\mu \mathrm{s}$).

Figure 5

Free-surface shape $R\left(z\right)$ measured experimentally (symbols) and calculated from the numerical simulations (solid line) during the breakup of a 5-cSt SO droplet. The instants of the numerical simulation were those for which $R_{\text{min}}$ took the same value as that of the experiment.

Figure 6

$R_{\text{min}}\left({\tau }_{\text{expt}}\right)$ obtained experimentally (symbols) and numerically (solid line) from the breakup of a pendant drop of G-W 52/48% (v/v). The dotted line corresponds to (1).

Figure 7

Fit of the exponential function $R_{\text{min}}=\alpha exp[{\tau }_{\text{expt}}/\left(3\lambda \right)]$ to the experimental values for ${\tau }_{\text{expt}}<3\mu \mathrm{s}$ during the breakup of a 5-cSt SO droplet.