Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T07:19:51.548Z Has data issue: false hasContentIssue false
  • English
  • Français

An experimental test of the theory of waves in fluid-filled deformable tubes

Published online by Cambridge University Press:  20 April 2006

J. H. Gerrard
J. H. Gerrard
Affiliation:
Department of Aeronautical Engineering, University of Manchester
Get access Rights & Permissions [Opens in a new window]

Abstract

It seems that no treatment of pulsating flow in deformable tubes is complete without a reference to the work of Womersley (1955) which for an infinitely long tube deals with the waves of axismmetric transverse motion and of longitudinal motion of the walls. This theory has so far been subjected to experimental test only for tethered tubes in which longitudinal wall motion is absent.

A series of measurements of the longitudinal motion has been made on horizontal water-filled latex tubes suspended by an array of strings so that there is minimal longitudinal constraint except at the ends, which are fixed. One end of the tube is driven by oscillating flow produced by a piston; the other end is closed. Theory and experiment agree when the tube is long provided an entrance length greater than a wavelength is included. Tubes which are short enough for reflection from the closed end to be significant present a more complicated problem. It is found that in the entrance length the theory of Womersley cannot be applied. A more refined theory is required which takes into account a distributed end constraint more completely than as a simple boundary condition.

Experiments on tethered tubes in which longitudinal wall motion is absent are also presented. These serve to demonstrate that the theory for such tubes agrees with measurements without any appreciable end effect and also shows that the small viscoelasticity of the latex rubber is correctly included.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 156 , July 1985 , pp. 321 - 347
Copyright
© 1985 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

Atabek, H. B. 1968 Wave propagation through a viscous fluid contained in a tethered, initially stressed, orthotropic elastic tube. Biophys. J. 8, 626.Google Scholar
Atabek, H. B. & Lew, H. S. 1966 Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys. J. 6, 481.Google Scholar
Cox, R. H. 1969 Comparison of linearized wave propagation models for arterial blood flow analysis. J. Biomech. 2, 251.Google Scholar
Klip, W. 1962 Velocity and Damping of the Pulse Wave. The Hague: M. Nijhoff.
Klip, W., Van Loon, P. & Klip, D. A. 1967 Formulas for phase velocity and damping of longitudinal waves in thick-walled viscoelastic tubes. J. Appl. Phys. 38, 3745.Google Scholar
Kuiken, G. D. C. 1984 Wave propagation in a thin-walled liquid-filled initially stressed tube. J. Fluid Mech. 141, 289.Google Scholar
Mcdonald, D. A. 1974 Blood Flow in Arteries, 2nd edn. Arnold.
Milnor, W. R. & Bertram, C. D. 1978 The relation between arterial viscoelasticity and wave propagation in the canine femoral artery in vivo. Circ. Res. 43, 870.Google Scholar
Mirsky, I. 1967 Wave propagation in a viscous fluid contained in an orthotropic elastic tube. Biophys. J. 7, 165.Google Scholar
Nolle, A. W. 1950 Dynamic mechanical properties of rubberlike materials. J. Polymer Sci. 5, 1.Google Scholar
Noordergraaf, A. 1978 Circulatory System Dynamics. Academic.
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.
Rubinow, S. I. & Keller, J. B. 1978 Wave propagation in a viscoelastic tube containing viscous fluid. J. Fluid Mech. 88, 181.Google Scholar
Talbot, S. A. & Gessner, U. 1973 Systems Physiology. Wiley.
Taylor, M. G. 1959–60 An experimental determination of the propagation of fluid oscillations in a tube with a viscoelastic wall; together with an analysis of the characteristics required in an electrical analogue. Phys. in Med. & Biol. 4, 63.Google Scholar
Taylor, L. A. & Gerrard, J. H. 1977 Pressure-radius relationships for elastic tubes and their application to arteries. Med. & Biol. Eng. & Comput. 15, 11.Google Scholar
Van Loon, P., Klip, W. & Bradley, E. L. 1977 Length-force and pressure-volume relationships of arteries. Biorheol. 14, 181.Google Scholar
Whirlow, D. K. & Rouleau, W. T. 1965 Periodic flow of a viscous liquid in a thick-walled elastic tube. Bull. Math. Biophys. 27, 355.Google Scholar
Witzig, K. 1914 Uber erzwungene Wellenbeivergungen zaher, inkompressibler Flussigkeiten in elastischen Rohren. Inaug. Dissertation, University of Berne, Switzerland.
Womersley, J. R. 1955 Oscillatory motion of a viscous liquid in a thin-walled elastic tube I: The linear approximation for long waves. Phil. Mag. 46, 199.Google Scholar
Womersley, J. R. 1957 Oscillatory flow in arteries: the constrained elastic tube as a model of arterial flow and pulse transmission. Phys. in Med. & Biol. 2, 178.Google Scholar