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

Abstract

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.

Thickness-Dependent Phase Transition Drives Nanoridge-to-Mesa Instability in Micellar Freestanding Films

“…These observations (Figure b–d) are contrary to the periodicity and symmetry expected for a Rayleigh-type instability as manifested in both freestanding and supported toroidally shaped liquid rings, ,,− where morphologies are quite sensitive to the radius of the toroid and the fluid flows creates a necklace of linked swells and necks. Freestanding toroidal rings have comparable major and minor diameters (of a few mm) and circular cross-section.…”

“…The observations of aperiodicity and the sequential appearance of mesas, their outward motion as well as their coalescence-assisted growth, and the fact that away from mesas, the ridge maintains its shape indicate that the formation of mesas cannot be attributed to the breakdown of the nanoridge through a capillary-driven Rayleigh-type instability, often cited in the stratification literature − ,, as the mechanistic basis of the nanoridge-to-mesa instability. Thus, the quantitative characterization of the formation and evolution of mesas rules out hydrodynamic instabilities due to capillarity or viscous dissipation effects − ,− ,,− and strongly suggests a thickness-dependent phase transition. The topological instabilities underlying mesa formation and the emergence of thinner, darker domains are quite analogous, even though in the latter case a metastable branch corresponding to a lower thickness gets picked.…”

“…Previous studies including Bergeron et al, Heinig et al, and most recently Lee et al claim that the nanoridge instability and the formation of mesas is akin to the capillarity-driven Rayleigh instability that leads to the breakup of a coherent liquid jet , or a toroidal, donut-shaped drop into a stream of droplets. , In the Rayleigh instability, thickness perturbations lead to sinusoidal undulations, creating a train of necks and swells. The capillary pressure progressively increases within the thinning neck, squeezing out fluid into the adjoining thicker swells, until a finite time singularity within the neck leads to pinch-off, creating isolated drops. ,,− The nanoridge-to-mesa instability could be classified as a Rayleigh instability if a similar sequence of thick and thin undulations is visualized around the expanding domains. However, because of the lack of techniques that can be used for visualizing thickness variations associated with the nanoridge-to-mesa instability, such experimental evidence is entirely lacking.…”

An unbounded approach to microfluidics using the Rayleigh–Plateau instability of viscous threads directly drawn in a bath

“…3B, we show that this result holds over a broad range of flow rates Q and nozzle speeds U0. The order of magnitude of the path wavelength L ∼ A must be similar to that of the natural mode of the instability (30). In this range, it is possible to fully program the instability.…”

Sublimation of isolated toric focal conic domains on micro-patterned surfaces

“…The maintenance of fine droplet structures under conditions such as atomization is primarily characterized by a high Laplace number. [52][53][54] It indicates that the influence of surface tension forces is relatively strong compared to the fluid's momentum, affecting the behavior and dynamics of fluid interfaces, as in the case of the smectic A surface.…”