Abstract
The propagation of perturbations in liquid filled elastic tubes depends on the stream velocity of the basic flow. This phenomenon is currently analyzed with the method of the characteristics which relies upon a basic flow with a rectangular velocity profile. It seems that this one-dimensional flow approximation has not been convincingly validated, which justifies to consider other, more general velocity profiles.
In the present analytical study the velocity profile is a quadratic function of the radial coordinate. Small amplitude perturbations are superposed on this inviscid, basic state in which the mean velocityŪ is arbitrarily large. A normal mode analysis shows that the velocity profile and therefore the vorticity of the basic flow influence the more the phenomenon the larger isŪ. For example, a parabolic profile allows countercurrent wave propagation regardless ofŪ.
This questions the one-dimensional wave propagation theory in compliant tubes and, consequently, the interpretation of several physiological and medical problems mainly in the respiratory and cardio-vascular systems.
Resumé
La propagation de perturbations dans un tube élastique conduisant un écoulement fluide dépend de la vitesse de l'écoulement de base. Ce phénomène est habituellement étudié avec la méthode des caractéristiques, où l'on suppose que le profil de vitesse de l'écoulement est rectangulaire. Comme cette simplification ne semble pas avoir été bien validée, il paraît indiqué d'étudier l'impact d'autres profils.
Dans la présente étude analytique, ce profil de vitesse est une fonction quadratique de la coordonée radiale. A cet écoulement non visqueux, dont la vitesse moyenneŪ est arbitraire, l'on superpose des perturbations de faible amplitude. Une analyse linéarisée montre que le profil de vitesse et donc le rotationel de l'écoulement de base influencent d'autant plus ce phénomène d'ondes queŪ est élévée.
Ceci met en question la théorie uni-dimensionelle de la propagation d'ondes dans des tubes compliants et, par là-même, l'interpretation de divers problèmes physiologiques et médicaux, avant tout des systèmes respiratoires et cardio-vasculaires.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Anliker, R. L. Rockwell and E. Odgen,Nonlinear analysis of flow pulses and shock waves in arteries. Part I: Derivation and properties of mathematical model. J. Appl. Math. Phys. (ZAMP)22, 217–246 (Part I), (1971).
A. C. L. Barnard, W. A. Hunt, W. P. Timelake and E. Varley,A theory of fluid flow in compliant tubes. Biophys. J.6, 717–724 and 735–746 (1966).
C. Cancelli and T. J. Pedley,A separated-flow model for collapsible-tube oscillations. J. Fluid Mech.157, 375–404 (1985).
S. V. Dawson and E. A. Elliott,Wave-speed limitation on expiratory flow limitation-a unifying concept. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol.43, 498–515 (1977).
M. B. Histand and M. Anliker,Influence of flow and pressure on wave propagation in the canine aorta. Circulation Res.32, 524–529 (1973).
R. Holenstein, R. M. Nerem and P. F. Niederer,On the propagation of a wave front in viscoelatic arteries. Trans. ASME K: J. Biomech. Engng.106, 115–122 (1984).
R. E. Hyatt, J. Mead, J. R. Rodarte and T. A. Wilson,Changes in lung mechanics: flow-volume relations, pp. 62–107. In: The Lung in Transition Between Health and Disease. P. T. Macklem and S. Permutt (eds.), Dekker, New York 1979.
D. L. Jan, R. D. Kamm and A. H. Shapiro,Filling of partially collapsed compliant tubes. Trans. ASME K: J. Biomech. Engng.106, 12–19 (1983).
R. T. Jones,Blood Flow. Ann. Rev. Fluid Mech.1, 223–244 (1969).
R. D. Kamm and A. H. Shapiro,Unsteady flow in collapsible tube subjected to an external pressure or body forces. J. Fluid Mech.95, 1–78 (1979).
J. W. Lambert,On the nonlinearities of fluid flow in nonrigid tubes. J. Franklin Inst.266, 83–102 (1958).
D. A. McDonald,Blood Flow in Arteries. Arnold, London 1974.
G. W. Morgan and W. R. Ferrante,Wave propagation in elastic tubes filled with streaming liquid. J. Acoust. Soc. Am.27, 715–725 (1955).
A. Müller,Über die Fortpflanzungsgeschwindigkeit von Druckwellen in dehnbaren Röhren bei ruhender und strömender Flüssigkeit. Helv. Physiol. Acta8, 228–241 (1950).
P. F. Niederer,Damping mechanisms and shock-like transitions in the human arterial tree. J. Appl. Math. Phys. (ZAMP)36, 204–220 (1985).
J. H. Olsen and A. H. Shapiro,Large-amplitude unsteady flow in liquid filled elastic tubes. J. Fluid Mech.29, 513–538 (1967).
T. J. Pedley,The Fluid Mechanics of Large Blood Vessels. Cambridge University Press, Cambridge 1980.
E. Rooz, T. F. Wiesner and R. M. Nerem,Epicardial coronary blood flow including the presence of stenoses and aorto-coronary bypasses-I: Model and numerical methods. Trans. ASME K: J. Biomech. Engng.107, 361–367 (1985).
G. Rudinger,Review of current mathematical methods for the analysis of blood flow, pp. 1–33. In: Biomedical Fluid Mechanics Symposium, ASME, New York 1966.
J. A. Rumberger and R. M. Nerem,A method-of-characteristics calculation of coronary blood flow. J. Fluid Mech.82, 429–448 (1977).
H. Schlichting,Grenzschicht-Theorie. Braun, Karlsruhe 1965.
A. H. Shapiro,Physiological and medical aspects of flow in collapsible tubes. Proc. 6th Canadian Congr. Appl. Mech., pp. 883–906 (1977a).
A. H. Shapiro,Steady flow in collapsible tubes. Trans. ASME K: J. Biomech. Engng.99, 126–147 (1977b).
M. Shimizu,Characteristics of pressure-wave propagation in a compliant tube with a fully collapsed segment. J. Fluid Mech.158, 113–135 (1985).
R. Skalak,Synthesis of a complete circulation, pp. 341–376. In: Cardiovascular Fluid Dynamics. D. H. Bergel (ed.) Academic Press, London and New York 1972.
V. L. Streeter, W. F. Keitzer and F. Bohr,Pulsatile pressure and flow through distensible vessels. Circulation Res.13, 3–20 (1963).
J. R. Womersley,An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. WADC Tech. Rep. TR 56-614, Defense Documentation Center (1957).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dardel, E. On wave propagation in elastic tubes conducting rotational flows. Z. angew. Math. Phys. 38, 366–377 (1987). https://doi.org/10.1007/BF00944956
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00944956