Figure 1

(a) Formation and (b) breakup of a curved jet. In this experiment, the rotation speed is insufficient and the torus does not form. Instead, the jet breaks up before the torus is generated. (c) Geometrical parameters of a torus: Tube radius, $a_0$, and overall radius, $R_0$. (d) Diagram for the formation of toroidal droplets in terms of the capillary number of the outer liquid, ${\mathrm{Ca}}_o$, and $R_{\mathrm{tip}}/a_{\mathrm{tip}}$, where $R_{\mathrm{tip}}$ is the distance between the needle and the rotation axis and $a_{\mathrm{tip}}$ is the inner radius of the needle. The different symbols correspond to different values of ${\eta }_o$: (◇, ◆) 30000 cP, (○, ●) 10 000 cP, (△, ▲) 5000 cP. The line corresponds to ${\mathrm{Ca}}_o=5.5R_{\mathrm{tip}}/a_{\mathrm{tip}}$ and separates regions where the torus forms (open symbols) or does not form (closed symbols). We use water for the inner liquid and silicone oil for the outer liquid and always add the surfactant sodium dodecyl sulphate above its critical micelle concentration to decrease the interfacial tension to $\gamma =4\mathrm{mN}/\mathrm{m}$ [22].Reuse & Permissions

Figure 2

Snapshots for the time evolution of a torus (from top to bottom) for different aspect ratios. The scale bar in (a1) corresponds to 5 mm. In these experiments, the injected liquid is glycerin and the liquid in the continuous phase is silicone oil (${\eta }_i/{\eta }_o=1/30$).Reuse & Permissions

Figure 3

Number of breakup points in the torus, $n$, as a function of its initial aspect ratio, $R_0/a_0$. The line corresponds to a linear fit of the leftmost points in the steps. We obtain $n\approx 0.57R_0/a_0$.Reuse & Permissions

Figure 4

Time evolution of a torus made of water immersed in silicone oil (${\eta }_i/{\eta }_o=1/30000$, $\gamma =4\mathrm{mN}/\mathrm{m}$). (a)–(c) Breakup of the torus through classical hydrodynamic instabilities. (d)–(f) Shrinkage and coalescence behavior for aspect ratios $R_0/a_0\lesssim 2$. The scale bar corresponds to 5 mm.Reuse & Permissions

Figure 5

(a) Time evolution of the swell (closed symbols) and neck (open symbols) radii normalized with the initial radius, for different values of the aspect ratio: (★, ☆) $R_0/a_0\approx 11.1$, (▲, △) $R_0/a_0\approx 5.6$, (●, ○) $R_0/a_0\approx 3.6$, (▸, ▹) $R_0/a_0\approx 2.6$, (■, □) $R_0/a_0\approx 2.2$, (◂,◃) $R_0/a_0\approx 2.1$, (◆,◇) $R_0/a_0\approx 1.9$, (▼, ▽) $R_0/a_0\approx 1.4$. (b) Time evolution of the torus dimension, quantified through $R_{\mathrm{in}}(t)$ normalized with $R_{\mathrm{in}}(0)$, for different values of the aspect ratio [symbols as in (a)]. Inset: Initial velocity of the shrinkage process as a function of $a_0$. The velocity is independent of $a_0$ and equal to $(34\pm 6)\mu \mathrm{m}/\mathrm{s}$. In all these experiments, ${\eta }_i/{\eta }_o=1/30000$ and $\gamma =4\mathrm{mN}/\mathrm{m}$. The schematic shows a torus with $R_{\mathrm{in}}\approx a_0$.Reuse & Permissions