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Liquid toroidal drop in compressional Stokes flow

Published online by Cambridge University Press:  23 November 2015

Michael Zabarankin ,
Olga M. Lavrenteva  and
Avinoam Nir
Michael Zabarankin*
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA
Olga M. Lavrenteva
Affiliation:
The Wolfson Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
Avinoam Nir
Affiliation:
The Wolfson Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: mzabaran@stevens.edu
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Abstract

The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number, $Ca$), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value, $Ca_{cr}$, toroidal stationary shapes were not found. Remarkably, $Ca_{cr}$ is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.

JFM classification

Mathematical Foundations: Computational methods Drops and Bubbles: Drops Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 785 , 25 December 2015 , pp. 372 - 400
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Copyright
© 2015 Cambridge University Press 

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