Charge-Induced Saffman-Taylor Instabilities in Toroidal Droplets

We show that charged toroidal droplets can develop fingerlike structures as they expand due to Saffman-Taylor instabilities. While these are commonly observed in quasi-two-dimensional geometries when a fluid displaces another fluid of higher viscosity, we show that the toroidal confinement breaks the symmetry of the problem, effectively making it quasi-two-dimensional and enabling the instability to develop in this three-dimensional situation. We control the expansion speed of the torus with the imposed electric stress and show that fingers are observed provided the characteristic time scale associated with this instability is smaller than the characteristic time scale associated with Rayleigh-Plateau break-up. We confirm our interpretation of the results by showing that the number of fingers is consistent with expectations from linear stability analysis in radial Hele-Shaw cells.

Figure 1

(a) Schematic of the circular cross section of a torus. $R_0$ is the radius of the central circle, $a_0$ is the tube radius, and $\theta$ is the polar angle. The torus is obtained by revolving this cross section along the $\widehat{\varphi }$ direction. The velocity at $\theta =\pi$ is ${\overrightarrow{U}}_o$. (b) Schematic of the experimental setup. A bath containing silicone oil rotates with an angular speed $\omega \approx 0.25\mathrm{rad}/\mathrm{s}$ while the inner liquid is injected. The resultant toroidal droplet is charged at a voltage $V\in [0,1800]\mathrm{V}$. The subsequent evolution of the torus is captured using a CCD camera.

Figure 2

Snapshots of the evolution of (a)–(c) a breaking torus with $R_0/a_0=3.5$ and $V=500\mathrm{V}$, (d)–(f) a torus with $R_0/a_0=3.7$ and $V=800\mathrm{V}$ where both break-up and viscous fingering are at play, and (g)–(i) a torus with $R_0/a_0=3.2$ and $V=1500\mathrm{V}$ that clearly exhibits viscous fingering. The dashed circle in (a) represents the outer rim of the torus at $t=0$. The dashed circle in (i) illustrates finger division. The snapshots (j)–(l) illustrate that some of the initial perturbations do not develop into fingers. The torus has $R_0/a_0=2.0$ and the applied voltage is $V=800\mathrm{V}$. In (k) and (l), we identify with numbers the initial perturbations and the number of fingers at a late stage of development. Scale bars: 2 mm.

Figure 3

Viscous stress, ${\tau }_{\mu }$, as a function of the pressure difference $\mathrm{\Delta }p=p(\theta =0)-p(\theta =\pi )$. (square) Expanding tori that break, (diamond) tori that evolve via viscous fingering instabilities, and (downward triangle) tori that exhibit both breaking and viscous fingering. The error bars are calculated by propagating the errors in $a_0$, $\xi$, and $U_o$, which mostly result from the pixel size of our CCD camera. The line corresponds to a linear fit of the data going through the origin: ${\tau }_{\mu }=m\mathrm{\Delta }p$, with $m=0.49\pm 0.10$.

Figure 4

State diagram in terms of the capillary number $Ca$ and the aspect ratio $R_o/a_o$. The symbols represent (square) shrinking tori, (circle) tori with stationary central-circle radius, (upward triangle) expanding tori that break, (diamond) tori evolving via viscous fingering, and (downward triangle) tori that exhibit both breaking and viscous fingering. The dashed line represents the critical capillary number where only viscous fingering is observed.

Figure 5

Number of perturbations, $n_m$, in the early stages of evolution [see Fig. 2], as a function of $U_o[(\xi +1)/2{]}^2$. The continuous and dashed lines are fits to the data (see text). Inset: top-view schematic of the unperturbed circular interface with radius $R$ (continuous line) and a sinusoidally perturbed interface for $n=6$ (dashed line).