(149j) Toroidal Drops in Compressional Flow

Toroidal drops in compressional flow

AVINOAM NIR¹, OLGA M. LAVRENTEVA¹, IRINA SMAGIN¹ AND MICHAEL ZABARANKIN²

¹ The Wolfson Department of Chemical Engineering, Technion-Israel Institute of Technology,  Haifa 32000, Israel; avinir@tx.technion.ac.il

² Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA.

ABSTRACT

Non spherical droplets are of interest in various areas, and recently also as potential carriers of drugs (Champion et al., 2007) or building blocks for more complex assemblies (Velev et al., 2000). Toroidal drops are known since the experiments by Plateau (1854) in rotating fluids. Such geometry is obtained also when a drop, falling free in a viscous fluid, experiences a finite surface deformation which develops into a toroidal form (Kojima et al., 1984; Baumann et al., 1992; Sostarecz & Belmonte 2003).

In this presentation we shall revisit the stable deformation of spherical drops in compressional viscous flow, e. g. bi-axial extension, within a finite range of the capillary number, Ca, (Zabarankin et al., 2013), and show that loss of stability can lead to formation of toroidal drops. We demonstrate that there is a similar range of Ca in which toroidal stationary solutions exist and that such drops in this flow are inherently unstable. Thus, the stable flat singly-connected drops and the unstable stationary toroidal drops correspond to two branches of the same problem. The dynamics of developing instability of the toroidal drops and the expected evolution of deformation are demonstrated. 

References

BAUMANN, N., JOSEPH, D. D., MOHR, P. & RENARDY, Y. 1992 Vortex rings of one fluid in another in free fall. Phys. Fluids A 4 (3), 567–580.

CHAMPION, J. A., KATARE, Y. K. & MITRAGOTRI, S. 2007 Particle shape: A new design parameter for micro- and nanoscale drug delivery carriers. J. Contr. Release 121 (1–2), 3–9.

KOJIMA, M., HINCH, E. J. & ACRIVOS, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27 (1), 19–32.

PLATEAU, J. 1857 I. Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity.–Third series. Philosophical Magazine Series 4 14 (90), 1–22.

SOSTARECZ, M. C. & BELMONTE, A. 2003 Motion and shape of a viscoelastic drop falling through a viscous fluid. J. Fluid Mech. 497, 235–252.

VELEV, O. D., LENHOFF, A. M. & KALER, E. W. 2000 A class of microstructured particles through colloidal crystallization. Science 287 (5461), 2240–2243.

ZABARANKIN, M., SMAGIN, I., LAVRENTEVA, O. M. & NIR, A. 2013 Viscous drop in compressional Stokes flow. J. Fluid Mech. 720, 169–191.