Pulsatile Flow in a Rigid Tube

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

Abstract

Flow in a tube in which the driving pressure varies in time is governed by Eq.3.2.9, namely,
$$\rho \frac{{\partial u}}{{\partial t}} + \frac{1}{\rho }\frac{{\partial p}}{{\partial x}} = \mu \left( {\frac{{{\partial ^2}u}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial u}}{{\partial r}}} \right)$$
Providing that all the simplifying assumptions on which the equation is based are still valid, the equation provides a forum for a solution in which the pressure p is a function of x and t while the velocity u is a function of r and t. Before obtaining this solution, it is important to reiterate the assumptions on which the equation is based, because these assumptions define the idealized features of the flow that the solution represents.

Keywords

Expense Posite Eter Sine 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Further Reading

  1. 1.
    Lighthill M, 1975. Mathematical Biofluiddynamics. Society for Industrial and Applied Mathematics, Philadelphia.MATHCrossRefGoogle Scholar
  2. 2.
    Walker JS, 1988. Fourier Analysis. Oxford University Press, New York.MATHGoogle Scholar
  3. 3.
    Brigham EO, 1988. The Fast Fourier Transform and its Applications. Prentice Hall, Englewood Cliffs, New J ersey.Google Scholar
  4. 4.
    McLachlan NW, 1955. Bessel Functions for Engineers. Clarendon Press, Oxford.Google Scholar
  5. 5.
    Watson GN, 1958. A Treatise on the Theory of Bessel Functions. Cambridge University Press. Cambridge.Google Scholar
  6. 6.
    Sexl T, 1930. Über den von E.G. Richardson entdeckten “Annulareffekt. ” Zeits chrift für Physik 61:349–362.MATHGoogle Scholar
  7. 7.
    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
  8. 8.
    Uchida S, 1956. The pulsating viscous flow superimposed on the steady laminar motion of incompressible fluid in a circular pipe. Zeitschrift für angewandte Mathematik und Physik 7:403–422.MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    McDonald DA, 1974. Blood flow in arteries. Edward Arnold, London.Google Scholar
  10. 10.
    Milnor WR, 1989. Hemodynamics. Williams and Wilkins, Baltimore.Google Scholar
  11. 11.
    Khamrui SR, 1957. On the flow of a viscous liquid through a tube of elliptic section under the influcnce of a periodic gradient. Bulletin of thc Calcutta Mathematical Society 49:57–60.MathSciNetMATHGoogle Scholar
  12. 12.
    Begum R, Zamir M, 1990. Flow in tubes of non-circular cross sections. In: Rahman M (ed), Ocean Waves Mechanics: Computati onal Fluid Dynamics and Mathematical Modeling. Computational Mechanics Publications, Southampton.Google Scholar
  13. 13.
    Duan B, Zamir M, 1991. Approximate solut ion for pulsatile flowin tubes of slightly noncircular cross-sections. Utilitas Mathematica 40:13–26.MathSciNetMATHGoogle Scholar
  14. 14.
    Quadir R, Zamir M, 1997. Entry length and flow development in tubes of reetangular and elliptic cross sect ions. In: Rahman M (cd), Laminar and TUrbulent Boundary Layers. Computational Mechanics Publications, Southampton.Google Scholar
  15. 15.
    Haslam M, Zamir M, 1998. Pulsatilc flow in tubes of elliptic cross sect ions. Annals of Biomedical Engineering 26:1–8.CrossRefGoogle Scholar
  16. 16.
    Moon PR, Spencer DE, 1961. Field Theory for Engineers. Van Nostrand, Princeton, New J ersey.MATHGoogle Scholar
  17. 17.
    McLachlan NW, 1964. Theory and Application of Mathieu Functions. Dover Publications, New York.MATHGoogle 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

Personalised recommendations