Advertisement

Bulletin of Mathematical Biology

, Volume 58, Issue 6, pp 1023–1046 | Cite as

Numerical simulation of collapsible-tube flows with sinusoidal forced oscillations

  • J. She
  • C. D. Bertram
Article

Abstract

Collapsible-tube flow with self-excited oscillations has been extensively investigated. Though physiologically relevant, forced oscillation coupled with self-excited oscillation has received little attention in this context. Based on an ODE model of collapsible-tube flow, the present study applies modern dynamics methods to investigate numerically the responses of forced oscillation to a limit-cycle oscillation which has topological characteristics discovered in previous unforced experiments. A devil's staircase and period-doubling cascades are presented with forcing frequency and amplitude as control parameters. In both cases, details are provided in a bifurcation diagram. Poincaré sections, a frequency spectrum and the largest Lyapunov exponents verify the existence of chaos in some circumstances. The thin fractal structure found in the strange attractors is believed to be a result of high damping and low stiffness in such systems.

Keywords

Bifurcation Diagram Chaotic Oscillation Force Frequency Large Lyapunov Exponent Ordinary Differential Equation Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Armitstead, J., C. D. Bertram and O. E. Jensen. 1996. A study of the bifurcation behaviour of a model of flow through a collapsible tube.Bull. Math. Biol. 58, 611–641.MATHCrossRefGoogle Scholar
  2. Bertram, C. D. and T. J. Pedley. 1982. A mathematical model of unsteady collapsible tube behaviour.J. Biomech. 15, 39–50.CrossRefGoogle Scholar
  3. Bertram, C. D. and C. J. Raymond. 1991. Measurements of wave speed and compliance in a collapsible tube during self-excited oscillations: a test of the choking hypothesis.Med. Biol. Eng. Comput. 29, 493–500.CrossRefGoogle Scholar
  4. Bertram, C. D., C. J. Raymond and T. J. Pedley. 1990. Mapping of instabilities during flow through collapsible tubes of differing length.J. Fluids and Structures 4, 125–153.CrossRefGoogle Scholar
  5. Bertram, C. D., C. J. Raymond and T. J. Pedley. 1991. Application of nonlinear dynamics concepts to the analysis of self-excited oscillations of a collapsible tube conveying a fluid.J. Fluids and Structures 5, 391–426.CrossRefGoogle Scholar
  6. Bertram, C. D. and M. D. Sheppeard. 1991. Interaction of self-excited oscillation and pulsatile upstream forcing in a collapsible tube conveying a flow.Med. Biol. Eng. Comput. 29, 259 (Suppl.).CrossRefGoogle Scholar
  7. Bertram, C. D. and X. Tian. 1992. Correlation of local stretchings as a way of characterising chaotic dynamics and noise.Physica D 58, 469–481.CrossRefGoogle Scholar
  8. Conrad, W. A. 1969. Pressure-flow relationships in collapsible tubes.IEEE Trans. Bio-Med. Eng. 16, 284–295.Google Scholar
  9. Doedel, E. and J. P. Kernévez. 1986. AUTO: software for continuation and bifurcation problems in ordinary differential equations. Applied Mathematics Report, California Institute of Technology.Google Scholar
  10. Feigenbaum, M. J. 1980. Universal behavior in nonlinear systems.Los Alamos Science 1, 4–27.MathSciNetGoogle Scholar
  11. Jensen, O. E. 1990. Instabilities of flow in a collapsible tube.J. Fluid Mech. 220, 623–659.MATHMathSciNetCrossRefGoogle Scholar
  12. Jensen, O. E. 1992. Chaotic oscillations in a simple collapsible-tube model.ASME J. Biomech. Eng. 114, 55–59.Google Scholar
  13. Jensen, O. E. and T. J. Pedley. 1989. The existence of steady flow in a collapsed tube.J. Fluid Mech. 206, 339–374.MATHMathSciNetCrossRefGoogle Scholar
  14. Kamm, R. D. and T. J. Pedley. 1989. Flow in collapsible tubes: a brief review.ASME J. Biomech. Eng. 111, 177–179.CrossRefGoogle Scholar
  15. Kaneko, K. 1986.Collapse of Tori and Genesis of Chaos in Dissipative Systems, Chap. 4. Singapore: World Scientific Publishing.MATHGoogle Scholar
  16. Low, H. T. and Y. T. Chew. 1991. Pressure/flow relationships in collapsible tubes: effects of upstream pressure fluctuations.Med. Biol. Eng. Comput. 29, 217–221.CrossRefGoogle Scholar
  17. Matsuzaki, Y. 1986. Self-excited oscillation of a collapsible tube conveying fluid. InFrontiers in Biomechanics, G. W. Schmid-Schönbeinet al. (Eds.), pp. 342–350. New York: Springer-Verlag.Google Scholar
  18. Parlitz, U. and W. Lauterborn. 1987. Period-doubling cascades and devil's staircases of the driven van der Pol oscillator.Phys. Rev. A 36, 1428–1434.CrossRefGoogle Scholar
  19. Pedley, T. J. 1980.The Fluid Mechanics of Large Blood Vessels, Chap. 6. London: Cambridge University Press.MATHGoogle Scholar
  20. Seydel, R. 1988.From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis, Chap. 9. New York: Elsevier.MATHGoogle Scholar
  21. Shapiro, A. H. 1977. Physiologic and medical aspects of flow in collapsible tubes. InProceedings of the Sixth Canadian Congress of Applied Mechanics, pp. 883–906.Google Scholar
  22. Shimizu, M. 1992. Blood flow in a brachial artery compressed externally by a pneumatic cuff.ASME J. Biomech. Eng. 114, 78–83.MathSciNetGoogle Scholar
  23. Thompson, J. M. T. and H. B. Stewart. 1986.Nonlinear Dynamics and Chaos, Chap. 6. Chichester: Wiley.MATHGoogle Scholar
  24. Thompson, J. M. T. and H. B. Stewart. 1993. A tutorial glossary of geometrical dynamics.Int. J. Bifurcation and Chaos 3, 223–239.MATHMathSciNetCrossRefGoogle Scholar
  25. Tian, X. 1991. Nonlinear dynamics and chaos in flow through a collapsed tube. Ph.D. dissertation, University of New South Wales.Google Scholar
  26. Walsh, C., P. A. Sullivan, J. S. Hansen and L.-W. Chen. 1995. Measurement of wall deformation and flow limitation in a mechanical trachea.ASME J. Biomech. Eng. 117, 146–152.Google Scholar
  27. Wolf, A., J. B. Swift, H. L. Swinney and J. Vastano. 1985. Determining Lyapunov exponents from a time series.Physica D 16, 285–317.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1996

Authors and Affiliations

  • J. She
    • 1
  • C. D. Bertram
    • 1
  1. 1.Graduate School of Biomedical EngineeringUniversity of New South WalesSydneyAustralia

Personalised recommendations