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Exchange flow of two immiscible fluids and the principle of maximum flux

Published online by Cambridge University Press:  08 July 2011

R. R. KERSWELL
R. R. KERSWELL*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Email address for correspondence: R.R.Kerswell@bristol.ac.uk
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Abstract

The steady, coaxial flow in which two immiscible, incompressible fluids of differing densities move past each other slowly in a vertical cylindrical tube has a continuum of possibilities due to the arbitrariness of the interface between the fluids. By invoking the presence of surface tension to at least restrict the shape of any interface to that of a circular arc or full circle, we consider the following question: which flow will maximise the exchange when there is only one dividing interface Γ? Surprisingly, the answer differs fundamentally from the better-known co-directional two-phase flow situation where an axisymmetric (concentric) core-annular solution always optimises the flux. Instead, the maximal flux state is invariably asymmetric either being a ‘side-by-side’ configuration where Γ starts and finishes at the tube wall or an eccentric core-annular flow where Γ is an off-centre full circle in which the more viscous fluid is surrounded by the less viscous fluid. The side-by-side solution is the most efficient exchanger for a small viscosity ratio β ≲ 4.60 with an eccentric core-annular solution optimal otherwise. At large β, this eccentric solution provides 51% more flux than the axisymmetric core-annular flow which is always a local minimiser of the flux.

JFM classification

Multiphase and Particle-laden Flows: Core-annular flow Mathematical Foundations: General fluid mechanics Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 132 - 159
Copyright
Copyright © Cambridge University Press 2011

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