Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-04-30T20:13:36.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave propagation in a viscoelastic tube containing a viscous fluid

Published online by Cambridge University Press:  19 April 2006

Sol I. Rubinow  and
Joseph B. Keller
Sol I. Rubinow
Affiliation:
Graduate School of Medical Sciences, Cornell University, Ithaca, New York 10021
Joseph B. Keller
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York 10012
Get access Rights & Permissions [Opens in a new window]

Abstract

Small amplitude, axially symmetric waves in a thin-walled viscoelastic tube containing a viscous compressible fluid are considered. Previous authors have found two modes of propagation for such waves but have studied them only in the low frequency, long wavelength limit. We show that there are infinitely many modes and study them at all frequencies. The appropriate dispersion equation was derived previously (Rubinow & Keller 1971) and analysed for an inviscid fluid. Now it is analysed for a viscous fluid. Asymptotic formulae for the propagation constant k are obtained for both low and high frequencies and for various ranges of the parameters characterizing the tube and the fluid. Special attention is paid to the case of a rigid tube and to parameter values that characterize the flow of blood in mammalian arteries. In addition, numerical results are obtained which complement the asymptotic formulae. Graphs of the velocity c vs. the frequency ω are presented for various modes and for various ranges of the parameters. Transmission-line equations and formulae for the impedance and compliance of the fluid-tube system are obtained, together with asymptotic and numerical results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 88 , Issue 1 , 13 September 1978 , pp. 181 - 203
Copyright
© 1978 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. & Lew, H. S. 1966 Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys. J. 6, 481–503.Google Scholar
Cox, R. H. 1969 Comparison of linearized wave propagation models for arterial blood flow analysis. J. Biomech. 2, 251–265.Google Scholar
Fitz-Gerald, J. M. 1972 Plasma motions in narrow capillary flow. J. Fluid Mech. 51, 463–476.Google Scholar
Goldman, D. E. & Hueter, T. F. 1956 Tabular data of the velocity and absorption of high-frequency sound in mammalian tissues. J. Acoust. Soc. Am. 28, 35–37.Google Scholar
Kirchhoff, G. 1868 Über den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung. Ann. Phys. Chem. 134, 177–193.Google Scholar
Korteweg, D. J. 1878 Über die Fortpflanzungesgeschwindigkeit des Schalles in elastischen Röhren. Ann. Phys. Chem. Ser. 3, 5, 525–542.Google Scholar
Lamb, H. 1898 On the velocity of sound in a tube, as affected by the elasticity of the walls. Manchester Lit. Phil. Soc. Mem. Proc. 42 (9), 1–16.Google Scholar
Mcdonald, D. A. 1974 Blood Flow in Arteries, 2nd edn, chap. 6. Baltimore: Williams & Wilkins.
Morgan, G. W. & Kiely, J. P. 1954 Wave propagation in a viscous liquid contained in a flexible tube. J. Acoust. Soc. Am. 26, 323–328.Google Scholar
Rayleigh, Lord 1945 Theory of Sound, 2nd edn, vol. 2. Dover.
Redwood, M. 1961 Mechanical Waveguides. Pergamon.
Résal, M. H. 1876 Sur les petits mouvements d'un fluide incompressible dans un tuyau élastique. C. R. Acad. Sci. Paris 82, 698–699.Google Scholar
Rubinow, S. I. & Keller, J. B. 1968 Hydrodynamic aspects of the circulatory system. In Hemorheology. Proc. 1st Int. Conf., Univ. Iceland, Reykjavik, 1966 (ed. A. L. Copley). Pergamon.
Rubinow, S. I. & Keller, J. B. 1971 Wave propagation in a fluid-filled tube. J. Acoust. Soc. Am. 50, 198–223.Google Scholar
Scarton, H. A. & Rouleau, W. T. 1973 Axisymmetric waves in compressible Newtonian liquids contained in rigid tubes: steady-periodic modes, shapes and dispersion by the method of eigenvalleys. J. Fluid Mech. 58, 595–621.CrossRefGoogle Scholar
Skalak, R. 1972 Synthesis of a complete circulation. In Cardiovascular Fluid Dynamics, vol. 2 (ed. D. H. Bergel), chap. 19. Academic Press.
Weber, W. 1866 Theorie der durch Wasser oder andere inkompressible Flüssigkeiten in elastischen Röhren fortgepflanzten Wellen. Ber. Ver. König. Sächsischen Gesellschaft Wiss. Leipzig, Math. Phys. Classe 18, 353–357.Google Scholar
Witzig, K. 1914 Über erzwungene Wellenbewegungen zäher, inkompressibler Flüssigkeiten in elastischen Röhren. Inaugural dissertation, Universitat Bern. Wyss Erben, Bern.
Womersley, J. R. 1957 An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. Wright Air Development Center Tech. Rep. no. 56–614.Google Scholar
Young, T. 1808 Hydraulic investigations, subservient to an intended Croonian lecture on the motion of blood. Phil. Trans. Roy. Soc. 98, 164–186.Google Scholar