Papers

On the liquid lining in fluid-conveying curved tubes

Andrew L. Hazel^a1 c1, Matthias Heil^a1, Sarah L. Waters^a2 and James M. Oliver^a2

^a1 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

^a2 OCIAM, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK

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.

(Received May 03 2011)

(Reviewed July 29 2011)

(Accepted August 10 2011)

(Online publication September 29 2011)

Key Words:

• pulmonary fluid mechanics;
• thin films

Correspondence:

^c1 Email address for correspondence: Andrew.Hazel@manchester.ac.uk