Bulletin of Mathematical Biology

, Volume 58, Issue 5, pp 939–955 | Cite as

Solitary waves in prestressed elastic tubes

  • Hilmi Demiray


In this work, we studied the propagation of non-linear waves in a pre-stressed thin elastic tube filled with an inviscid fluid. In the analysis, analogous to the physiological conditions of the arteries, the tube is assumed to be subject to a uniform pressureP 0 and a constant axial stretch ratio λz. In the course of blood flow it is assumed that a large dynamic displacement is superimposed on this static field. Furthermore, assuming that the displacement gradient in the axial direction is small, the non-linear equation of motion of the tube is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. The result is discussed for some elastic materials existing in the literature.


Solitary Wave Solitary Wave Solution Stretch Ratio Elastic Tube Soft Biological Tissue 


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  1. Atabek, H. B. and H. S. Lew. 1966. Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube.Biophys. J. 6, 481–503.CrossRefGoogle Scholar
  2. Cowley, S. J. 1983. On the waves associated with elastic jumps on fluid filled elastic tubes.Quart. J. Mech. Appl. Math. 36, 289–312.MATHMathSciNetGoogle Scholar
  3. Demiray, H. 1972. On the elasticity of soft biological tissues.J. Biomech. 5, 309–311.CrossRefGoogle Scholar
  4. Demiray, H. 1976. Stresses in ventricular wall.J. Appl. Mech., Trans. ASME 43, 194–197.Google Scholar
  5. Demiray, H. 1992. Wave propagation through a viscous fluid contained in a prestressed thin elastic tube.Int. J. Eng. Sci. 30, 1607–1620.MATHMathSciNetCrossRefGoogle Scholar
  6. Erbay, H. A., S. Erbay, and S. Dost. 1992. Wave propagation in fluid filled nonlinear viscoelastic tubes.Acta Mech. 95, 87–102.MATHMathSciNetCrossRefGoogle Scholar
  7. Fung, Y. C. 1984.Biodynamics: Circulation. New York: Springer.Google Scholar
  8. Hashizume, Y. 1985. Nonlinear pressure waves in a fluid-filled elastic tube.J. Phys. Soc. Japan 54, 3305–3312.CrossRefGoogle Scholar
  9. Ishiara, A., N. Hashitzume, and M. Tabibana. 1951. Statistical theory of rubberlike elasticity. IV. Two dimensional stretching.J. Chem. Phys. 19, 1508–1511.MathSciNetCrossRefGoogle Scholar
  10. Jeffrey, A. and T. Kawahara. 1981.Asymptotic Methods in Nonlinear Wave Theory. Boston: Pitmann.Google Scholar
  11. Johnson, R. S. 1970. A nonlinear equation incorporating damping and dispersion.J. Fluid Mech. 42, 49–60.MATHMathSciNetCrossRefGoogle Scholar
  12. Kuiken, G. D. C. 1984. Wave propagation in a thin walled liquid-filled initially stressed tube.J. Fluid Mech. 141, 289–308.MATHCrossRefGoogle Scholar
  13. Lambossy, P. 1951. A pereu et historique sur le problem de la propagation des ondes dans un liquide compressible enferme dans un tube elastique.Helv Physiol. Acta 9, 145–161.Google Scholar
  14. McDonald, D. A. 1966.Blood Flow in Arteries. Baltimore, MD: Williams and Wilkins.Google Scholar
  15. Morgan, G. W. and J. P. Kiely. 1954. Wave propagation in a viscous liquid contained in a flexible tube.J. Acoust. Soc. Amer. 26, 323–328.MathSciNetCrossRefGoogle Scholar
  16. Rachev, A. I. 1980. Effect of transmural pressure and muscular activity on pulse waves in arteries.J. Biomech. Eng. 102, 119–123.CrossRefGoogle Scholar
  17. Rudinger, G. 1970. Shock waves in a mathematical models in the aorta.J. Appl. Mech. 37, 34–41.CrossRefGoogle Scholar
  18. Simon, B. R., A. S. Kobayashi, D. E. Stradness, and C. A. Wiederhielm. 1972. Re-evaluation of arterial constitutive laws.Circulation Res. 30, 491–500.Google Scholar
  19. Skalak, R. 1966. Wave propagation in blood flow. InBiomechanics Symposium, Y. C. Fung (Ed), pp. 20–46. New York: ASME.Google Scholar
  20. Tait, R. J. and T. B. Moodie. 1984. Waves in nonlinear fluid filled tubes.Wave Motion 6, 197–203.MATHMathSciNetCrossRefGoogle Scholar
  21. Witzig, K. 1914. Über erzwungene Wellenbeweg ungen zahber, incompressibler Flüssigkeiten in elastistishen Rohren. Inaugural dissertation, University of Bern.Google Scholar
  22. 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, 199–219.MATHMathSciNetGoogle Scholar
  23. Yomosa, S. 1987. Solitary waves in large blood vessels.J. Phys. Soc. Japan 56, 506–520.MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1996

Authors and Affiliations

  • Hilmi Demiray
    • 1
  1. 1.Department of MathematicsMarmara Research Center, Research Institute for Basic SciencesKocaeliTurkey

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