Skip to main content

On the liquid lining in fluid-conveying curved tubes

  • Andrew L. Hazel (a1), Matthias Heil (a1), Sarah L. Waters (a2) and James M. Oliver (a2)

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.

Corresponding author
Email address for correspondence:
Hide All
1. Band, L. R., Riley, D. S., Matthews, P. C., Oliver, J. M. & Waters, S. L. 2009 Annular thin-film flows driven by azimuthal variations in interfacial tension. Q. J. Mech. Appl. Math. 62 (4), 403430.
2. Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved tubes. Annu. Rev. Fluid Mech. 15, 461512.
3. Bowen, M. 1998 High order diffusion. PhD thesis, University of Nottingham.
4. Cairncross, R. A., Schunk, P. R., Baer, T. A., Rao, R. R. & Sackinger, P. A. 2000 A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion. Intl J. Numer. Meth. Fluids 33, 375403.
5. Daskopoulos, P. & Lenhoff, A. M. 1989 Flow in curved ducts: bifurcation structure for stationary ducts. J. Fluid Mech. 203, 125148.
6. Dean, W. R. 1928 The stream-line motion of fluid in a curved pipe. Phil. Mag. 5, 673695.
7. Gresho, P. M. & Sani, R. L. 1998 Incompressible Flow and the Finite Element Method, vol. 1, Advection–Diffusion and Isothermal Laminar Flow. Wiley.
8. Halpern, D. & Frenkel, A. L. 2003 Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers. J. Fluid Mech. 485, 191220.
9. Heil, M. & Hazel, A. L. 2006 oomph-lib: an object-oriented multi-physics finite-element library. In Fluid–Structure Interaction (ed. M. Schäfer & H.-J. Bungartz), Lecture Notes on Computational Science and Engineering, pp. 19–49. Springer. oomph-lib is available as open-source software at
10. Ito, H. 1987 Flow in curved pipes. JSME Intl J. 30 (262), 543552.
11. Jensen, O. E. 1997 The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373403.
12. Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 6590.
13. King, J. R. & Bowen, M. 2001 Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Math. 12, 321356.
14. Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory, 2nd edn. Berlin, Springer.
15. Lyne, W. H. 1971 Unsteady viscous flow in a curved pipe. J. Fluid Mech. 45, 1331.
16. Myers, T. G. 1998 Thin films with high surface tension. SIAM Rev. 40, 441462.
17. Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931980.
18. Pedley, T. J. 1977 Pulmonary fluid dynamics. Annu. Rev. Fluid Mech. 9, 229274.
19. Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.
20. Ruschak, K. J. 1980 A method for incorporating free boundaries with surface tension in finite element fluid-flow simulators. Intl J. Numer. Meth. Engng 15, 639648.
21. Russell, J. R. D., Williams, J. F. & Xu, X. 2007 MOVCOL4: a moving mesh code for fourth- order time-dependent partial differential equations. SIAM J. Sci. Comput. 29 (1), 197320.
22. Shewchuk, J. R. 1996 Engineering a 2D quality mesh generator and Delaunay triangulator. In Applied Computational Geometry: Towards Geometric Engineering (ed. Lin, M. C. & Manocha, D. ). Lecture Notes in Computer Science , vol. 1148. pp. 203222. Springer.
23. Siggers, J. H. & Waters, S. L. 2008 Unsteady flows in pipes with finite curvature. J. Fluid Mech. 600, 133165.
24. Wei, H.-H. & Rumschitzki, D. S. 2005 The effects of insoluble surfactants on the linear stability of a core-annular flow. J. Fluid Mech. 541, 115142.
25. Zienkiewicz, O. C. & Zhu, J. Z. 1992 The superconvergent patch recovery and a posteriori error estimates. Part 1. The recovery technique. Intl J. Numer. Meth. Engng 33 (7).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed