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.
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References
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.
Bertram, C. D. and T. J. Pedley. 1982. A mathematical model of unsteady collapsible tube behaviour.J. Biomech. 15, 39–50.
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.
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.
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.
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.).
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.
Conrad, W. A. 1969. Pressure-flow relationships in collapsible tubes.IEEE Trans. Bio-Med. Eng. 16, 284–295.
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.
Feigenbaum, M. J. 1980. Universal behavior in nonlinear systems.Los Alamos Science 1, 4–27.
Jensen, O. E. 1990. Instabilities of flow in a collapsible tube.J. Fluid Mech. 220, 623–659.
Jensen, O. E. 1992. Chaotic oscillations in a simple collapsible-tube model.ASME J. Biomech. Eng. 114, 55–59.
Jensen, O. E. and T. J. Pedley. 1989. The existence of steady flow in a collapsed tube.J. Fluid Mech. 206, 339–374.
Kamm, R. D. and T. J. Pedley. 1989. Flow in collapsible tubes: a brief review.ASME J. Biomech. Eng. 111, 177–179.
Kaneko, K. 1986.Collapse of Tori and Genesis of Chaos in Dissipative Systems, Chap. 4. Singapore: World Scientific Publishing.
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.
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.
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.
Pedley, T. J. 1980.The Fluid Mechanics of Large Blood Vessels, Chap. 6. London: Cambridge University Press.
Seydel, R. 1988.From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis, Chap. 9. New York: Elsevier.
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.
Shimizu, M. 1992. Blood flow in a brachial artery compressed externally by a pneumatic cuff.ASME J. Biomech. Eng. 114, 78–83.
Thompson, J. M. T. and H. B. Stewart. 1986.Nonlinear Dynamics and Chaos, Chap. 6. Chichester: Wiley.
Thompson, J. M. T. and H. B. Stewart. 1993. A tutorial glossary of geometrical dynamics.Int. J. Bifurcation and Chaos 3, 223–239.
Tian, X. 1991. Nonlinear dynamics and chaos in flow through a collapsed tube. Ph.D. dissertation, University of New South Wales.
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.
Wolf, A., J. B. Swift, H. L. Swinney and J. Vastano. 1985. Determining Lyapunov exponents from a time series.Physica D 16, 285–317.
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She, J., Bertram, C.D. Numerical simulation of collapsible-tube flows with sinusoidal forced oscillations. Bltn Mathcal Biology 58, 1023–1046 (1996). https://doi.org/10.1007/BF02458382
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DOI: https://doi.org/10.1007/BF02458382