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Viscous drop in compressional Stokes flow

Published online by Cambridge University Press:  27 February 2013

Michael Zabarankin ,
Irina Smagin ,
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
Irina Smagin
Affiliation:
The Wolfson Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
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 dynamics of the deformation of a drop in axisymmetric compressional viscous flow is addressed through analytical and numerical analyses for a variety of capillary numbers, $\mathit{Ca}$ , and viscosity ratios, $\lambda $ . For low $Ca$ , the drop is approximated by an oblate spheroid, and an analytical solution is obtained in terms of spheroidal harmonics; whereas, for the case of equal viscosities ( $\lambda = 1$ ), the velocity field within and outside a drop of a given shape admits an integral representation, and steady shapes are found in the form of Chebyshev series. For arbitrary $Ca$ and $\lambda $ , exact steady shapes are evaluated numerically via an integral equation. The critical $\mathit{Ca}$ , below which a steady drop shape exists, is established for various $\lambda $ . Remarkably, in contrast to the extensional flow case, critical steady shapes, being flat discs with rounded rims, have similar degrees of deformation ( $D\sim 0. 75$ ) for all $\lambda $ studied. It is also shown that for almost the entire range of $\mathit{Ca}$ and $\lambda $ , the steady shapes have accurate two-parameter approximations. The validity and implications of spheroidal and two-parameter shape approximations are examined in comparison to the exact steady shapes.

JFM classification

Mathematical Foundations: Computational methods Drops and Bubbles: Drops Low-Reynolds-number flows: Low-Reynolds-number flows

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 720 , 10 April 2013 , pp. 169 - 191
Copyright
©2013 Cambridge University Press

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