Shrinking instability of toroidal droplets.

Toroidal droplets are inherently unstable due to surface tension. They can break up, similar to cylindrical jets, but also exhibit a shrinking instability, which is inherent to the toroidal shape. We investigate the evolution of shrinking toroidal droplets using particle image velocimetry. We obtain the flow field inside the droplets and show that as the torus evolves, its cross-section significantly deviates from circular. We then use the experimentally obtained velocities at the torus interface to theoretically reconstruct the internal flow field. Our calculation correctly describes the experimental results and elucidates the role of those modes that, among the many possible ones, are required to capture all of the relevant experimental features.

Toroidal Droplets: Growth Rates, Dispersion Relations, and Behavior in the Thick-Torus Limit.

Toroidal droplets in a viscous liquid are unstable and transform into single or multiple spherical droplets. For thin tori, this can happen via the Rayleigh-Plateau instability causing the breakup of cylindrical jets. In contrast, for thick tori, this can happpen via the shrinking of the "hole". In this work, we use the thin-torus limit to directly measure the growth rate associated with capillary disturbances. In the case of toroidal droplets inside a much more viscous liquid, we even obtain the full dispersion relation, which is in agreement with theoretical results for cylindrical jets. For thick tori, we employ particle image velocimetry to determine the flow field of a sinking toroidal drop inside another viscous liquid. We find that the presence of the "hole" greatly suppresses one of the circulation loops expected for sinking cylinders. Finally, using the flow field of a shrinking toroidal droplet and the time-reversal symmetry of the Stokes equations, we theoretically predict the expected shape deformation of an expanding torus and confirm the result experimentally using charged toroidal droplets.

Stable nematic droplets with handles.

We stabilize nematic droplets with handles against surface tension-driven instabilities, using a yield-stress material as outer fluid, and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle splay on the elastic free energy. For toroidal droplets, we find that the saddle-splay energy screens the twisting energy, resulting in a spontaneous breaking of mirror symmetry; the chiral twisted state persists for aspect ratios as large as ∼20. For droplets with additional handles, we observe in experiments and computer simulations that there are two additional -1 surface defects per handle; these are located in regions with local saddle geometry to minimize the nematic distortions and hence the corresponding elastic free energy.

Breakup dynamics of toroidal droplets in shear-thinning fluids.

We use toroidal droplets to study the breakup dynamics of a Newtonian liquid jet in the presence of rheologically nonlinear materials. We find that the droplets resist breakup for times that are longer than those in the presence of Newtonian liquids. More importantly, we show that our experiments can be explained by incorporating the nonlinearities into the linear treatment of the problem through the strain-rate-dependent viscosity. Finally, we show that the scaling factor required to relate the viscosity to the growth rate associated to unstable modes is given by the elastic modulus of the outside material, illustrating that both the viscoelastic and shear-thinning nature of the outside material play a crucial role in the dynamics of the problem.