Pulsatile Flow in an Elastic Tube

  • M. Zamir
Part of the Biological Physics Series book series (BIOMEDICAL)


In the case of a rigid tube it is possible to postulate a fully developed region away from the tube entrance where the flow is independent of x , thus derivatives of U , v with respect to x are zero. The equation of continuity combined with the boundary condition v = 0 at the tube wall then leads to v being identically zero and the governing equations reduce to
$$\rho \frac{{\partial u}}{{\partial t}} + \frac{{\partial p}}{{\partial x}} = \mu \left( {\frac{{{\partial ^2}u}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial u}}{{\partial r}}} \right)$$
(Eq.3.2 .9) As we saw in Chapter 4, the pressure gradient term in that case is a function of t only, not of x, and the velocity U is then a function of r, t only, not of x. For an oscillatory pressure gradient the velocity U then oscillat es with the same frequency, and since it is not a function of x, it will represent the velocity at every cross section of the tube. The fluid then oscillates in bulk, or “en mass. ” There is no wave motion when the tube is rigid .


Wave Speed Tube Wall Pulsatile Flow Oscillatory Flow Oscillatory Cycle 
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References and Further Reading

  1. 1.
    Rouse H, Ince S, 1957. History of Hydraulics. Dover Publications, New York.MATHGoogle Scholar
  2. 2.
    McDonald DA, 1974. Blood Flow in Arteries. Edward Arnold, London.Google Scholar
  3. 3.
    Caro CG, Pedley TJ, Schroter RC, Seed WA, 1978. The Mechanics of the Circulation. Oxford University Press, Oxford.MATHGoogle Scholar
  4. 4.
    Milnor WR, 1989. Hemodynamics. Williams and Wilkins, Baltimore.Google Scholar
  5. 5.
    Lighthill M, 1975. Mathematical Biofluiddynamics. Society for Industrial and Applied Mathematics, Philadelphia.MATHCrossRefGoogle Scholar
  6. 6.
    Fung YC, 1984. Biodynamics: Circulation. Springer-Verlag, New York.Google Scholar
  7. 7.
    McLachlan NW, 1955. Bessel Functions for Engineers. Clarendon Press, Oxford.Google Scholar
  8. 8.
    Watson GN, 1958. A Treatise on the Theory of Bessel Functions. Cambridge University Press. Cambridge.Google Scholar
  9. 9.
    Sechler EE, 1968. Elasticity in Engineering. Dover Publications, New York.Google Scholar
  10. 10.
    Wempner G, 1973. Mechanics of Solids With Applications to Thin Bodies. McGraw-Hill, New York.MATHGoogle Scholar
  11. 11.
    Shames IH, Cozzarelli FA, 1992. Elastic and Inelastic Stress Analysis. Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  12. 12.
    Bradley GL, 1975. A Primer of Linear Algebra. Prentice Hall, Englewood Cliffs, New Jersey.MATHGoogle Scholar
  13. 13.
    Noble B, Daniel JW, 1977. Applied Linear Algebra. Prentice Hall, Englewood Cliffs, New Jersey.MATHGoogle Scholar
  14. 14.
    Lay DC, 1994. Linear Algebra and its Applications. Addison-Wesley, Reading, Massachusetts.Google Scholar
  15. 15.
    Morgan GW, Kiely JP, 1954. Wave propagation in a viscous liquid contained in a flexible tube. Journal of Acoustical Society of America 26:323–328.MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Womersley JR, 1955. Oscillatory motion of a viscous liquid in a thin-walled elastic tube-I: The linear approximation for long waves. Philosophical Magazine 46:199–221.MathSciNetMATHGoogle Scholar
  17. 17.
    Atabek HB, Lew HS, 1966. Wave propagation through a viscous incompressible fluid contained in an initially elastic tube. Biophysical Journal 6:481–503.ADSCrossRefGoogle Scholar
  18. 18.
    Cox RH, 1969. Comparison of linearized wave propagation models for arterial blood flow analysis. Journal of Biomechanics 2:251–265.CrossRefGoogle Scholar
  19. 19.
    Ling SC, Atabek HB, 1972. A nonlinear analysis of pulsatile flow in arteries. Journal of Fluid Mechanics 55:493–511.ADSMATHCrossRefGoogle Scholar
  20. 20.
    Korteweg DJ, 1878. Über die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Rohren. Annalen der Physik und Chemie 5:525–542.ADSMATHGoogle Scholar
  21. 21.
    Lamb H, 1897. On the velocity of sound in a tube, as affected by the elasticity of the walls. Memoirs and Proceedings, Manchester Literary and Philosophical Society A42:1–16.Google Scholar
  22. 22.
    Witzig K, 1914. Über erzwungene Wellenbewegungen zäher, inkom-pressibler Flüssigkeiten in elastischen Rohren. Inaugural Dissertation, Universität Bern, K.J. Wyss, Bern.Google Scholar
  23. 23.
    Lambossy P, 1950. Apercu historique et critique sur le probleme de la propagation des ondes dans un liquide compressible enferme dans un tube elastique. Helvetica Physiologica et Pharmalogica Acta 8:209–227.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • M. Zamir
    • 1
  1. 1.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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