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On the liquid lining in fluid-conveying curved tubes

Published online by Cambridge University Press:  29 September 2011

Andrew L. Hazel ,
Matthias Heil ,
Sarah L. Waters  and
James M. Oliver
Andrew L. Hazel*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Matthias Heil
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Sarah L. Waters
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK
James M. Oliver
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider axially uniform, two-phase flow through a rigid curved tube in which a fluid (air) core is surrounded by a film of a second, immiscible fluid (water): a simplified model for flow in a conducting airway of the lung. Jensen (1997) showed that, in the absence of a core flow, surface tension drives the system towards a configuration in which the film thickness tends to zero on the inner wall of the bend. In the present work, we demonstrate that the presence of a core flow, driven by a steady axial pressure gradient, allows the existence of steady states in which the film thickness remains finite, a consequence of the fact that the tangential stresses at the interface, imposed by secondary flows in the core, can oppose the surface-tension-driven flow. For sufficiently strong surface tension, the steady configurations are symmetric about the plane containing the tube’s centreline, but as the surface tension decreases the symmetry is lost through a pitchfork bifurcation, which is closely followed by a limit point on the symmetric solution branch. This solution structure is found both in simulations of the Navier–Stokes equations and a thin-film model appropriate for weakly curved tubes. Analysis of the thin-film model reveals that the bifurcation structure arises from a perturbation of the translational degeneracy of the interface location in a straight tube.

JFM classification

Biological Fluid Dynamics: Pulmonary fluid mechanics Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 705: Special issue dedicated to Professor T. J. Pedley on his 70th birthday , 25 August 2012 , pp. 213 - 233
Copyright
Copyright © Cambridge University Press 2011

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