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Toroidal Droplets

Ekapop Pairam and Alberto Fernandez-Nieves

Liquid droplets are naturally driven into a spherical shape by surface tension. That is why non spherical droplets rarely exist in nature. Our goal is to generate such rare droplets and study their hydrodynamic behavior as a function of geometrical parameters.

Newtonian toroidal droplets in another Newtonian medium are unstable and always transform into spherical droplets. There are two different paths of transformation depending on the aspect ratio of the torus.

For a thin torus, where the curvature associated to the overall size of the torus is insignificant compared that associated to the tube radius, the Rayleigh-Plateau Instability induces the transformation of the toroidal droplet into spherical droplets, similar to what happens for cylindrical jets. However, unlike for cylindrical jets, only integer values of the perturbation wavelength can fit inside the torus. As a result, by controlling the aspect ratio of the torus and the viscosity ratio between two liquid interfaces, the droplet breaks in a precise number of spherical droplets.

Figure 1: Viewing from the bottom of breaking Torus (left) and Viewing from the side of breaking Torus (right).

Unlike cylindrical jets, toroidal droplets have an additional non-zero curvature associated with the overall size of the torus. This results in a shrinkage process that couples to the growth of the Rayleigh-Plateau instability. For a sufficiently fat torus, however, no unstable modes of the Rayleigh-Plateau instability can grow and the torus relies on its shrinkage to become a single spherical drop.

Figure 2: Viewing from the bottom of collapsing Torus (left) and Viewing from the side of collapsing Torus (right).

In the case of cylindrical jets, unstable modes can grow when the length of the jet exceeds the length of circumference of the tube. We find that this criteria still works for toroidal drops provided the inner torus perimeter is taken as the length of the jet.

Figure 3: When tube radius exceed the inner radius, toroidal droplet transformation via growth and collapse process..

More complex Droplets
Our method for generating toroidal drops can be extended to generate drops with an arbitrary number of handles.

Figure 4: Flower shape droplet

Next Challemge The drop evolution becomes more complicated when one or both phases are Non-Newtonian. We find that under the right conditions, the toroidal droplets can be stable for very long times. Studying how these toroidal droplets transform into spheres can provide additional insight into the transformation of drops based on simple liquids and allow determination of certain rheological features of the Non-Newtonian material.

Figure 5: Under the right condition toroidal droplet can be really stable and immune to small perturbation.

Useful Reference: E. Pairam and A. FernaŽndez-Nieves, Generation and Stability of Toroidal Droplets in a Viscous Liquid, Physical Review Letters, 102, 234501 (2009).

Contact Information:
Ekapop Pairam
Office: Boggs Building, Room B-55A
Email address: gth675d [at] mail.gatech.edu
Phone: 404-385-3682

Soft Condensed Matter Laboratory, School of Physics, Georgia Institute of Technology
770 State Street NW, Atlanta, GA, 30332-0430, USA
Phone: 404-385-3667 Fax: 404-894-9958
alberto.fernandez [at] physics.gatech.edu