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Motion of a drop along the centreline of a capillary in a pressure-driven flow

Published online by Cambridge University Press:  02 November 2009

ETIENNE LAC  and
J. D. SHERWOOD
ETIENNE LAC*
Affiliation:
Schlumberger-Doll Research, One Hampshire Street, Cambridge MA 02139-1578, USA
J. D. SHERWOOD
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: elac@slb.com
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Abstract

The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R, λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ < 1/2, the drop elongates towards a limiting non-zero cylindrical radius. Low-viscosity drops (λ < 1) break up owing to a re-entrant jet at the rear, whereas a travelling capillary wave instability eventually develops on more viscous drops (λ > 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 27 - 54
Copyright
Copyright © Cambridge University Press 2009

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