Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-30T02:42:53.692Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous flow in collapsible tubes of slowly varying elliptical cross-section

Published online by Cambridge University Press:  12 April 2006

Rosemary Wild ,
T. J. Pedley  and
D. S. Riley
Rosemary Wild
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
D. S. Riley
Affiliation:
Department of Mathematics, University of Southampton, England
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with steady flow in collapsible tubes, such as veins, at fairly low Reynolds number. Lubrication theory is used to calculate the velocity and pressure distribution in an elliptical tube whose cross-sectional area and eccentricity vary slowly and in a given way with longitudinal distance x. The transverse velocity field and the effect of inertia on the primary velocity and pressure distributions are calculated to first order in the relevant small parameter. The results of these calculations are combined with a relationship between transmural pressure and the cross-sectional area at any x which is close to that measured in (large) veins, and are used to predict the pressure and flow in a collapsible tube when a given distribution of external pressure is applied. Different relationships between the tube perimeter and cross-sectional area are examined. The theory is applied to an experiment in which a segment of collapsible tube is supported between two rigid segments, and squeezed; predictions of the relationship between the pressure drop and flow rate are made for various experimental conditions. In particular, when the resistance of the downstream rigid segment is held constant, a range of flow rates is found in which the pressure drop falls as the flow rate is raised; this agrees with experiment.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 81 , Issue 2 , 24 June 1977 , pp. 273 - 294
Copyright
© 1977 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Attinger, E. O. 1969 Wall properties of veins. I.E.E.E. Trans. Bio-Med. Engng 16, 253.Google Scholar
Benedict, J. V., Walker, L. B. & Harris, E. H. 1968 Stress — strain characteristics of unembalmed human tendon. J. Biomech. 1, 53.Google Scholar
Brecher, G. A. 1952 Mechanism of venous flow under different degrees of aspiration. Am. J. Physiol. 169, 423.Google Scholar
Caro, C. G. 1965 Extensibility of blood vessels in isolated rabbit lungs. J. Physiol. 178, 193.Google Scholar
Caro, C. G., Pedley, T. J. & Seed, W. A. 1974 Mechanics of the circulation. In Cardiovascular Physiology (ed. Guyton & Jones), pp. 1–47. MTP International Review of Science, Physiology, series 1, vol. 1.
Carton, T. W., Dainauskas, J. & Clark, J. W. 1962 Elastic properties of single elastin fibres. J. Appl. Physiol. 17, 547.Google Scholar
Conrad, W. A. 1969 Pressure — flow relationships in collapsible tubes. I.E.E.E. Trans. Bio-Med. Engng 16, 284.Google Scholar
Flaherty, J., Keller, J. B. & Rubinow, S. I. 1972 Post buckling behaviour of elastic tubes and rings with opposite sides in contact. SIAM J. Appl. Math. 23, 446.Google Scholar
Fung, Y. C. & Sobin, S. S. 1972 Pulmonary alveolar blood flow. Circulation Res. 30, 470.Google Scholar
Hall, P. 1974 Unsteady viscous flow in a pipe of slowly-varying cross-section. J. Fluid Mech. 64, 209.Google Scholar
Katz, A. I., Chen, Y. & Moreno, A. H. 1969 Flow through a collapsible tube. Biophys. J. 9, 1261.Google Scholar
Lee, J. S. & Fung, Y. C. 1970 Flow in locally constricted tubes at low Reynolds numbers. J. Appl. Mech. E37, 9.
Manton, M. J. 1971 Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451.Google Scholar
Moreno, A. H., Katz, A. I., Gold, L. D. & Reddy, R. V. 1970 Mechanics of distension of dog veins and other thin-walled structures. Circulation Res. 27, 1069.Google Scholar
Rubinow, S. I. & Keller, J. B. 1972 Flow of a viscous fluid through an elastic tube with applications to blood flow. J. Theor. Biol. 35, 299.Google Scholar
Sobey, I. J. 1976 Inviscid secondary motions in a tube of slowly varying ellipticity. J. Fluid Mech. 73, 621.Google Scholar
Whitmore, R. L. 1968 Rheology of the Circulation. Pergamon.
Wiederhielm, C. A. 1972 The interstitial space. In Biomechanics: Its Foundations and Objectives (ed. Fung, Perrone & Anliker), pp. 273–286. Prentice-Hall.