Abstract
This chapter gives an overview of the main physiological applications of collapsible tube flows and reviews the major theoretical and computational developments of the past twenty-five years, ranging from lumped-parameter models to threedimensional Navier—Stokes simulations. We also discuss some of the significant questions that, despite substantial progress, still remain open.
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References
Aittokallio, T., M. Gyllenberg, and O. Polo: 2001, ‘A model of a snorer’s upper airway’. Math. Biosci. 170, 79–90.
Armaly, B. F., F. J. Durst, C. F. Pereira, and B. Schonung: 1983, ‘Experimental and theoretical investigation of backward-facing step flow’. Journal of Fluid Mechanics 127, 473–496.
Balint, T. S.: 2001, ‘Dynamics of the upper airway’. Ph.D. thesis, University of Warwick, Warwick.
Benjamin, T.: 1963, ‘The threefold classification for unstable disturbances in flexible surfaces bounding inviscid flows’. Journal of Fluid Mechanics 16, 436–450.
Berke, G. S., D. C. Green, M. E. Smith, D. P. Arnstein, V. Honrubia, M. Natividad, and W. A. Conrad: 1991, ‘Experimental-Evidence in the Invivo Canine for the Collapsible Tube Model of Phonation’. Journal of the Acoustical Society of America 89, 1358–1363.
Bertram, C.: 1986, ‘Unstable equilibrium behavior in collapsible tubes’. Journal of Biomechanics 19, 61–69.
Bertram, C. D. and S. A. Godbole: 1997, ‘LDA measurements of velocities in a simulated collapsed tube’. ASME Journal of Biomechanical Engineering 119, 357–363.
Bertram, C. D. and T. J. Pedley: 1982, ‘A mathematical model of unsteady collapsible tube behaviour’. Journal of Biomechanics 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’. Medical and Biological Engineering and Computing 29, 493–500.
Bertram, C. D., C. J. Raymond, and K. S. A. Butcher: 1989, ‘Oscillations in a collapsed-tube analog of the brachial artery under a sphygmomanometer cuff’. ASME Journal of Biomechanical Engineering 111, 185–191.
Bertram, C. D., C. J. Raymond, and T. J. Pedley: 1990, ‘Mapping of instabilities for flow through collapsible tubes of differering length’. Journal of 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’. Journal of Fluids and Structures 5, 391–426.
Binns, R. L. and D. N. Ku: 1989, ‘Effect of stenosis on wall motion — a possible mechanism of stroke and transient ischemic attack’. Arteriosclerosis 9, 842–847.
Bogdanova, E. V. and O. S. Rhyzov: 1983, ‘Free and induced oscillations in Poiseuille flow’. Quarterly Journal of Mechanics and Applied Mathematics 36, 271–287.
Bonis, M. and C. Ribreau: 1978, ‘Etude de quelques proprietes de l’ecoulement dans une conduite collabable’. La Houille Blanche 3/4, 165–173.
Brook, B. S.: 1997, ‘The effect of gravity on the haemodynamics of the giraffe jugular vein’. Ph.D. thesis, University of Leeds, Leeds.
Brook, B. S., S. A. E. G. Falle, and T. J. Pedley: 1999, ‘Numerical solutions for unsteady gravitydriven flows in collapsible tubes: evolution and roll-wave instability of a steady state’. Journal of Fluid Mechanics 396, 223–256.
Brook, B. S. and T. J. Pedley: 2001, ‘A model for time-dependent flow in (giraffe jugular) veins: Uniform tube properties’. Journal of Biomechanics. (in print).
Brower, R. W. and C. Scholten: 1975, ‘Experimental evidence on the mechanism for the instability of flow in collapsible vessels’. Medical and Biological Engineering 13, 839–845.
Cai, Z. and X. Y. Luo: 2001, ‘A fluid-beam model for flow in collapsible channels’. (Preprint).
Cancelli, C. and T. J. Pedley: 1985, ‘A separated-flow model for collapsible-tube oscillations’. Journal of Fluid Mechanics 157, 375–404.
Carew, E. O. and T. J. Pedley: 1997, ‘An active membrane model for peristaltic pumping .1. Periodic activation waves in an infinite tube’. ASME Journal of Biomechanical Engineering 119, 66–76.
Carpenter, P. W., K. Berkouk, and A. D. Lucey: 1999, ‘A theoretical model of pressure propagation in the human spinal CSF system’. Engineering Mechanics 6, 213–228.
Carpenter, P. W., K. Berkouk, and A. D. Lucey: 2001a, ‘Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: Basic theory’. (Submitted).
Carpenter, P. W., K. Berkouk, and A. D. Lucey: 2001b, ‘Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 2: Mechanisms for the pathogenesis of syringomyelia’. (Submitted).
Chang, H.-C. and E. A. Demekhin: 1996, Solitary wave formation and dynamics on falling films’. Adv. Appl. Mech. 32, 1–58.
Conrad, W. A.: 1969, ‘Pressure-Flow relationship in collapsible tubes’. IEEE Transactions on bio-medical engineering BME-16, 284–295.
Cowley, S. J.: 1981, ‘High Reynolds number flows through channels and tubes’. Ph.D. thesis, University of Cambridge, Cambridge.
Cowley, S. J.: 1982, ‘Elastic jumps on fluid-filled elastic tubes’. Journal of Fluid Mechanics 116, 459–473.
Cowley, S. J.: 1983, ‘On the wavetrains associated with elastic jumps on fluid-filled elastic tubes’. Q. J. Mech. Appl. Math. 36, 289–312.
Dai, G., J. P. Gertler, and R. D. Kamm: 1999, ‘The effects of external compression on venous blood flow and tissue deformation in the lower leg’. ASME Journal of Biomechanical Engineering 121, 557–564.
Danahy, D. T. and J. A. Ronan: 1974, ‘Cervical venous hum in patients on chronic hemodialysis’. New England Journal of Medicine 291, 237–239.
Davies, C. and P. W. Carpenter: 1997a, ‘Instabilities in a plane channel flow between compliant walls’. Journal of Fluid Mechanics 352, 205–243.
Davies, C. and P. W. Carpenter: 1997b, ‘Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels’. Journal of Fluid Mechanics 335, 361–392.
Dawson, S. V. and E. A. Elliot: 1977, ‘Wave-speed limitation on expiratory flow — a unifying concept’. Journal of Applied Physiology 43, 498–515.
Dressler, R. F.: 1949, ‘Mathematical solution of the problem of roll-waves in inclined open channels’. Communications on Pure and Applied Mathematics 2, 149–194.
Elad, D., R. D. Kamm, and A. H. Shapiro: 1987, ‘Choking Phenomena in a lung-like model’. ASME Journal of Biomechanical Engineering 109, 1–9.
Elad, D., M. Sahar, J. M. Avidor, and S. Einav: 1992, ‘Steady flow through collapsible tubes: measurements of flow and geometry’. ASME Journal of Biomechanical Engineering 114, 84–91.
Elliot, E. A. and S. V. Dawson: 1977, ‘Test of wave-speed theory of flow limitation in elastic tubes’. Journal of Applied Physiology 43, 516–522.
F. T. Smith, F. T. Smith and O. R. Burggraf: 1985, ‘On the development of large-sized short-scaled disturbances in boundary layers’. Proceedings of the Royal Society of London A 399, 25–55.
Fee, M. S., B. Shraiman, B. Pesaran, and P. Mitra: 1998, ‘The role of nonlinear dynamics of the syrinx in the vocalizations of a songbird’. Nature 395, 67–71.
Flaherty, J. E., J. B. Keller, and S. I. Rubinow: 1972, ‘Post buckling behavior of elastic tubes and rings with opposite sides in contact’. SIAM Journal of Applied Mathematics 23, 446–455.
Gavriely, N. and O. E. Jensen: 1993, ‘Theory and measurements of snores’. Journal of Applied Physiology 74, 2828–2837.
Gavriely, N., Y. Palti, G. Airoy, and J. B. Grotberg: 1984, ‘Measurement and theory of wheezing breath sounds’. Journal of Applied Physiology 57, 481–492.
Gavriely, N., T. R. Shee, D. W. Cugell, and J. B. Grotberg: 1989, ‘Flutter in flow-limited collapsible tubes: a mechanism for the generation of wheezes’. Journal of Applied Physiology 66, 2251–2261.
Gregg, D. E. and L. C. Fisher: 1963, ‘Blood supply to the heart’. In: W. F. Hamilton and P. Dow (eds.): Handbook of Physiology. Section 2: Circulation, Volume II. Washington D. C.: American Physiological Society.
Griffiths, D. J.: 1969, ‘Urethral elasticity and micturition hydrodynamics in females’. Medical and biological engineering and computing 7, 201–215.
Griffiths, D. J.: 1971, ‘Hydrodynamics of male micturition — I: Theory of steady flow through elastic-walled tubes’. Medical and biological engineering and computing 9, 581–588.
Grotberg, J. B.: 1994, ‘Pulmonary flow and transport phenomena’. Annual Review of Fluid Mechanics 26, 529–571.
Grotberg, J. B. and S. H. Davis: 1980, ‘Fluid-dynamical flapping of a collapsible channel: sound generation and flow limitation’. Journal of Biomechanics 13, 219–230.
Grotberg, J. B. and E. L. Reiss: 1984, ‘Subsonic flapping flutter’. Journal of Sound and Vibration 92, 349–361.
Grotberg, J. B. and T. R. Shee: 1985, ‘Compressible-flow channel flutter’. Journal of Fluid Mechanics 159, 175–193.
Guiot, C., P. G. Pianta, C. Cancelli, and T. J. Pedley: 1990, ‘Prediction of coronary blood-flow with a numerical-model based on collapsible tube’. American Journal of Physiology 258, H 1606–H 1614.
Guneratne, J. C.: 1999, ‘High-Reynolds number flow in a collapsible channel’. Ph.D. thesis, Cambridge University, Cambridge.
Guneratne, J. C. and T. J. Pedley: 2001, ‘High Reynolds number flow in a collapsible channel’. (in preparation).
Hayashi, S., T. Hazase, and H. Kawamura: 1998, ‘Numerical analysis for stability and selfexcited oscillation in collapsible tube flow’. ASME Journal of Biomechanical Engineering 120, 468–475.
Hazel, A. L. and M. Heil: 2002, ‘Steady finite Reynolds number flow in three-dimensional collapsible tubes’. (Submitted).
Heil, M.: 1995, ‘Large deformations of cylindrical shells conveying viscous flow’. Ph.D. thesis, University of Leeds, Leeds.
Heil, M.: 1996, ‘The Stability of Cylindrical Shells Conveying Viscous Flow’. Journal of Fluids and Structures 10, 173–196.
Heil, M.: 1997, ‘Stokes flow in collapsible tubes: computation and experiment’. Journal of Fluid Mechanics 353, 285–312.
Heil, M. and T. J. Pedley: 1995, ‘Large Axisymmetric Deformations of Cylindrical Shells Conveying Viscous Flow’. Journal of Fluids and Structures 9, 237–256.
Heil, M. and T. J. Pedley: 1996, ‘Large Post-Buckling Deformations of Cylindrical Shells Conveying Viscous Flow’. Journal of Fluids and Structures 10, 565–599.
Huang, L.: 1995, ‘Flutter of cantilevered plates in axial flow’. Journal of Fluids and Structures 9, 127–147.
Huang, L.: 1998, ‘Reversal of the Bernoulli effect and channel flutter’. Journal of Fluids and Structures 12, 131–151.
Huang, L.: 2001, ‘Viscous flutter of a finite elastic membrane in Poiseuille flow’. Journal of Fluids and Structures 15, 1060–1088.
Ikeda, T. and Y. Matsuzaki: 1999, ‘A one-dimensional unsteady separable and reattachable flow model for collapsible tube-flow analysis’. ASME Journal of Biomechanical Engineering 121, 153–159.
Jensen, O. E.: 1990, ‘Instabilities of flow in a collapsed tube’. Journal of Fluid Mechanics 220, 623–659.
Jensen, O. E.: 1992, ‘Chaotic oscillations in a simple collapsible-tube model’. ASME Journal of Biomechanical Engineering 114, 55–59.
Jensen, O. E.: 1998, ‘An asymptotic model of viscous flow limitation in a highly collapsed channel’. ASME Journal of Biomechanical Engineering 120, 544–547.
Jensen, O. E.: 2001, ‘Self-excited oscillations in a collapsible channel: insights from a simple asymptotic model’. (Preprint).
Jensen, O. E. and M. Heil: 2002, ‘High-frequency self-excited oscillations in a collapsible-channel flow’. Journal of Fluid Mechanics. (Submitted).
Jensen, O. E. and T. J. Pedley: 1989, ‘The existence of steady flow in a collapsed tube’. Journal of Fluid Mechanics 206, 339–374.
Kamm, R. D.: 1982, ‘Bioengineering studies of periodic external compression as prophylaxis against deep vein thrombosis — Part I: numerical studies’. ASME Journal of Biomechanical Engineering 104, 87–95.
Kamm, R. D. and T. J. Pedley: 1989, ‘Flow in collapsible tubes: a brief review’. ASME Journal of Biomechanical Engineering 111, 177–179.
Kamm, R. D. and A. H. Shapiro: 1979, ‘Unsteady flow in a collapsible tube subjected to external pressure or body forces’. Journal of Fluid Mechanics 95, 1–78.
Katz, A. I., Y. Chen, and A. H. Moreno: 1969, ‘Flow through a collapsible tube — Experimental analysis and mathematical model’. Biophysical Journal 9, 1261–1279.
Kececioglu, I., M. E. McClurken, R. D. Kamm, and A. H. Shapiro: 1981, ‘Steady, supercritical flow in collapsible tubes. Part 1. Experimental observations’. Journal of Fluid Mechanics 109, 367–389.
Kistler, S. F. and L. E. Scriven: 1983, ‘Coating Flows’. In: J. Pearson and S. Richardson (eds.): Computational Analysis of Polymer Processing. London: Applied Science Publishers.
Ku, D. N.: 1997, ‘Blood flow in arteries’. Annual Review of Fluid Mechanics 29, 399–434.
Landahl, M.: 1962, ‘On the stability of a laminar incompressible bounadry layer over a flexible surface’. Journal of Fluid Mechanics 13, 607–632.
Larose, P. G. and J. B. Grotberg: 1997, ‘Flutter and long-wave instabilities in compliant channels conveying developing flows’. Journal of Fluid Mechanics 331, 37–58.
Liang, S.-J., G. P. Neitzel, and C. K. Aidun: 1997, ‘Finite element computations for unsteady fluid and elastic membrane interaction problems’. International Journal for Numerical Methods in Fluids 24, 1091–1110.
Lighthill, J.: 1975, Mathematical biofluiddynamics. Philadelphia: SIAM.
Lowe, T. W. and T. J. Pedley: 1995, ‘Computation of Stokes flow in a channel with a collapsible segment’. Journal of Fluids and Structures 9, 885–905.
Luo, X. Y. and T. J. Pedley: 1995, ‘Numerical Simulation of Steady Flow in a 2-D Collapsible Channel’. Journal of Fluids and Structures 9, 149–197.
Luo, X. Y. and T. J. Pedley: 1996, ‘A numerical simulation of unsteady flow in a two-dimensional collapsible channel’. Journal of Fluids Mechanics 314, 191–225.
Luo, X. Y. and T. J. Pedley: 1998, ‘The effects of wall inertia on flow in a two-dimensional collapsible channel’. Journal of Fluid Mechanics 363, 253–280.
Luo, X. Y. and T. J. Pedley: 2000, ‘Multiple solutions and flow limitation in collapsible channel flows’. Journal of Fluid Mechanics 420, 301–324.
Matsuzaki, Y. and K. Fujimura: 1995, ‘Reexamination of steady solutions of a collapsible channel conveying fluid’. ASME Journal of Biomechanical Engineering 117, 492–494.
Matsuzaki, Y. and Y. Fung: 1977, ‘Stability analysis of straight and buckled two-dimensional channels conveying an incompressible flow’. Trans. ASME E: J. Appl. Mech. 44, 548–552.
Matsuzaki, Y., T. Ikeda, T. Kitagawa, and S. Sakata: 1994, ‘Analysis of flow in a two-dimensional collapsible channel using universal “tube” law’. ASME Journal of Biomechanical Engineering 116, 469–476.
Matsuzaki, Y. and T. Matsumoto: 1989, ‘Flow in two-dimensional collapsible channel with rigid inlet and outlet’. ASME Journal of Biomechanical Engineering 111, 180–184.
McClurken, M. E., I. Kececioglu, R. D. Kamm, and A. H. Shapiro: 1981, ‘Steady, supercritical flow in collapsible tubes. Part 2. Theoretical studies’. Journal of Fluid Mechanics 109, 391–415.
McDonald, D. A.: 1974, Blood flow in arteries. London: Edward Arnold, second edition.
Moore, J. E., N. Stergiopulos, X. Golay, D. N. Ku, and J.-J. Meister: 1995, ‘Flow measurements in collapsed stenotic arterial models’. In: R. M. Hochmuth, N. A. Langrana, and M. S. Hefzy (eds.): Proceedings of the 1995 Bioengineering Conference, Vol. ASME-BED 29. pp. 229–230.
Oates, G. C.: 1975, ‘Fluid flow in soft-walled tubes: I. Steady flow’. Medical and Biological Engineering 13, 773–778.
Olson, D. A., R. D. Kamm, and A. H. Shapiro: 1982, ‘Bioengineering studies of periodic external compression as prophylaxis against deep vein thrombosis — Part II: experimental studies on a simulated leg’. ASME Journal of Biomechanical Engineering 104, 96–104.
Pedley, T. J.: 1980, The fluid mechanics of large blood vessels. Cambridge: Cambridge University Press.
Pedley, T. J.: 1992, ‘Longitudinal tension variation in collapsible channels: A new mechanism for the breakdown of steady flow’. ASME Journal of Biomechanical Engineering 114, 60–67.
Pedley, T. J., B. S. Brook, and R. S. Seymour: 1996, ‘Blood pressure and flow rate in the giraffe jugular vein’. Philosophical Transactions of the Royal Society of London B 351, 855–866.
Pedley, T. J. and X. Y. Luo: 1998, ‘Modelling Flow and Oscillations in Collapsible Tubes’. Theoretical and Computational Fluid Dynamics 10, 277–294.
Pedley, T. J. and K. D. Stephanoff: 1985, ‘Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves’. Journal of Fluid Mechanics 160, 337–367.
Pedrizzetti, G.: 1998, ‘Fluid flow in a tube with an elastic membrane insertion’. Journal of Fluid Mechanics 375, 39–64.
Ralph, M. E. and T. J. Pedley: 1988, ‘Flow in a channel with a moving indentation’. Journal of Fluid Mechanics 190, 87–112.
Ralph, M. E. and T. J. Pedley: 1989, ‘Viscous and inviscid flow in a channel with a moving indentation’. Journal of Fluid Mechanics 209, 543–566.
Ralph, M. E. and T. J. Pedley: 1990, ‘Flow in a channel with a moving indentation in one wall’. ASME Journal of Fluids Engineering 112, 468–475.
Rast, M. P.: 1994, ‘Simultaneous Solution of the Navier-Stokes and Elastic Membrane Equations by a Finite-element Method’. International Journal for Numerical Methods in Fluids 19, 1115 – 1135.
Reyn, J. W.: 1987, ‘Multiple solutions and flow limitation for steady flow through a collapsible tube held open at the ends’. Journal of Fluid Mechanics 174, 467–493.
Rodbard, S.: 1966, ‘A hydrodynamics mechanism for autoregulation of flow’. Cardiologia 48, 532–535.
Rodbard, S. and L. Takacs: 1966, ‘Hydrodynamics of autoregulation’. Cardiologia 48, 433–440.
Rosenfeld, M.: 1995, ‘A numerical study of pulsating flow behind a constriction’. Journal of Fluid Mechanics 301, 203–223.
Shapiro, A. H.: 1977, ‘Steady flow in collapsible tubes’. ASME Journal of Biomechanical Engineering 99, 126–147.
Shimizu, M. and Y. Tanida: 1983, ‘On the mechanism of Korotkoff sound generation at diastole’. Journal of Fluid Mechanics 23, 299–312.
Smith, F. T.: 1976a, ‘Flow through constricted or dilated pipes and channels. Part 1’. Q. J. Mech. Appl. Math. 21, 343–364.
Smith, F. T.: 1976b, ‘Flow through constricted or dilated pipes and channels. Part 2’. Q. J. Mech. A ppl. Math. 21, 365–379.
Sobey, I.: 1985, ‘Observations of waves during oscillatory channel flow’. Journal of Fluid Mechanics 151, 395–426.
Stephanoff, K. D., T. J. Pedley, C. J. Lawrence, and T. W. Secomb: 1983, ‘Fluid flow along a channel with an asymmetric oscillating constriction’. Nature 305, 692–695.
Tobak, M. and D. J. Peake: 1982, ‘Topology of three-dimensional separated flows’. Annual Review of Fluid Mechanics 14, 61–85.
Tsuji, T., K. Nakajim, Y. Takeuchi, K. Inoue, K. Shioma, Y. Koyama, K. Tokuchi, T. Yoshikawa, and K. Suma: 1978, ‘Study on haemodynamics during cardiopulmonary bypass (in Japanese)’. Artificial Organs 7, 435–438.
Tutty, O.: 1984, ‘High-Reynolds-number viscous flow in collapsible tubes’. Journal of Fluid Mechanics 146, 451–469.
Tutty, O.: 1989, ‘The viscous flow through symmetric collapsible channels’. Mathematika 36, 153–181.
Tutty, O. R.: 1992, ‘Pulsatile flow in a constricted channel’. ASME Journal of Biomechanical Engineering 114, 50–54.
Tutty, O. R. and T. J. Pedley: 1993, ‘Oscillatory flow in a stepped channel’. Journal of Fluid Mechanics 247, 179–204.
Ur, A. and M. Gordon: 1970, ‘Origin of Korotkoff sounds’. American Journal of Physiology 218, 524–529.
Walsh, C.: 1995, ‘Flutter in one-dimensional collapsible tubes’. Journal of Fluids and Structures 9, 393–408.
Walsh, C., P. Sullivan, and J. Hanson: 1991, ‘Subcritical flutter in collapsible tube flow: a model of expiratory flow in the trachea’. J. Biomech. Engng. 113, 21–26.
Weaver, D. and M. Paidoussis: 1977, ‘On the collapse and flutter phenomena in thin tubes conveying fluid’. Journal of Sound and Vibration 50, 117–132.
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Heil, M., Jensen, O.E. (2003). Flows in Deformable Tubes and Channels. In: Carpenter, P.W., Pedley, T.J. (eds) Flow Past Highly Compliant Boundaries and in Collapsible Tubes. Fluid Mechanics and Its Applications, vol 72. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0415-1_2
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