Hostname: page-component-6d856f89d9-jhxnr Total loading time: 0 Render date: 2024-07-16T04:45:59.338Z Has data issue: false hasContentIssue false
  • English
  • Français

Unsteady flow in a collapsible tube subjected to external pressure or body forces

Published online by Cambridge University Press:  19 April 2006

Roger D. Kamm  and
Ascher H. Shapiro
Roger D. Kamm
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA. 02139
Ascher H. Shapiro
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA. 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

Flows in thin-walled, collapsible tubes are of fundamental importance to various physiologic phenomena and clinical devices.

A one-dimensional, unsteady theory is developed for flows generated either by externally applied pressures or by body forces. Part 1 deals with small-amplitude, linearized flows, part 2 with large amplitude, nonlinear flows. Experimental results for a tube collapsing under external pressure are given in part 3, together with theoretical interpretations and comparative results of numerical simulations.

Several new and unanticipated phenomena are revealed. These are in part associated with the highly nonlinear ‘equation of state’ (transmural pressure versus area) for a partially collapsed tube, and in part with whether the flow speed is sub- or supercritical relative to the speed of area waves. For instance, a flow produced by a spatially uniform external pressure applied to a limited region becomes choked at a flow-limiting throat at which point the fluid speed reaches the local wave speed. This throat forms at the edge of the pressurized region. The critical velocity can be exceeded with the application of certain types of spatially graded external pressures.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 1 , 14 November 1979 , pp. 1 - 78
Copyright
© 1979 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbott, M. B. 1966 An Introduction to the Method of Characteristics. New York: American Elsevier.
Abbott, M. B. 1975 Weak solutions of the equations of open channel flow. In Unsteady Flow in Open Channels (ed. K. Mahmood & V. Yevjevich), vol. 1, pp. 283–312. Water Resources Publications.
Arnitzenius, A. C., Koops, J., Rodrigo, F. A., Elsbach, H. & Brummelen, A. G. E. 1970 Circulatory effects of body acceleration given synchronously with the heart beat (BASH): Ballistocardiography and cardiovascular therapy. Bibl. Cardiol. vol. 26, no. 180.Google Scholar
Ashrae 1969 ASHRAE Guide and Data Book — Equipment Volume. New York: American Society of Heating, Refrigeration and Air Conditioning Engineers.
Attinger, E. O. 1964 Pulsatile Blood Flow. McGraw-Hill.
Beam, R. M. 1968 Finite amplitude waves in fluid-filled elastic tubes: wave distortion, shock waves, and Korotkoff sounds. N.A.S.A. Tech. Note D-4803.Google Scholar
Brawley, R. K., Oldham, H. N., Vasko, J. S., Henney, R. P. & Morrow, A. G. 1966 Influence of right atrial pressure pulse on instantaneous vena caval blood flow. Am. J. Physiol. 211, 347–353.Google Scholar
Collins, R. E. C. 1976 Physical methods of prophylaxis against deep vein thrombosis. In Prophylactic Therapy of Deep Vein Thrombosis and Pulmonary Embolism (ed. J. Fratantoni & S. Wessler), pp. 158–172. DHEW Publication no. (NIH) 76–866.
Dawson, S. V. & Elliott, E. A. 1977 Wave-speed limitation on expiratory flow — a unifying concept. J. Appl. Physiol. 43, 498–515.Google Scholar
Elliott, E. A. & Dawson, S. V. 1977 Test of wave-speed theory of flow limitation in elastic tubes. J. Appl. Physiol. 43, 516–522.Google Scholar
Flaherty, J. E., Keller, J. B. & Rubinow, S. I. 1972 Post buckling behavior of elastic tubes and rings with opposite sides in contact. SIAM J. of Appl. Math. 23, 446–455.Google Scholar
Griffiths, D. J. 1969 Urethral elasticity and micturition hydrodynamics in females. Med. & Biol. Engng 7, 201–215.Google Scholar
Griffiths, D. J. 1971a Hydrodynamics of male micturition — I: Theory of steady flow through elastic-walled tubes. Med. & Biol. Engng 9, 581–588.Google Scholar
Griffiths, D. J. 1971b Steady fluid flow through veins and collapsible tubes. Med. & Biol. Engng 9, 597–602.Google Scholar
Griffiths, D. J. 1973 The mechanics of the urethra and of micturition. Brit. J. of Urology 45, 497–507.Google Scholar
Guyton, A. C., Langston, J. B. & Carrier, O. 1962 Decrease of venous return caused by right atrial pulsation. Circulation Res. 10, 188–196.Google Scholar
Hartree, D. R. 1952 Some practical methods of using characteristics in the calculation of nonsteady compressible flows. Los Alamos Rep. LA-HU-1.Google Scholar
Kantrowitz, A. 1968 Theoretical and experimental analysis of the intra-aortic balloon pump. Trans. Am. Soc. Artif. Int. Organs 14, 338–343.Google Scholar
Kececioglu, I. 1978 Structure of shock waves in collapsible-tube flow. Proc. 31st Ann. Conf. Engng in Medicine & Biol., Atlanta, Ga.
Kivity, Y. & Collins, R. 1974 Nonlinear wave propagation in visco-elastic tubes: application to aortic rupture. J. Biomech. 7, 67–76.Google Scholar
Lister, M. 1960 The numerical solution of hyperbolic partial differential equations by the method of characteristics. In Mathematical Methods for Digital Computers (ed. A. Ralston & H. S. Wilf), pp. 165–179. Wiley.
Macklem, P. T. & Mead, J. 1967 The physiological basis of common pulmonary function tests. Archives of Environ, Health 14, 5–9.Google Scholar
Mcclurken, M. E. 1978 Shape-independent area measurement in collapsible tubes by an electrical impedance technique. Proc. 31st Ann. Conf. Engng in Medicine & Biol., Atlanta, Ga.
Mcmahon, T. A., Clark, C., Murthy, V. S. & Shapiro, A. H. 1971 Intra-aortic balloon experiments in a lumped-element hydraulic model of the circulation. J. Biomech. 4, 335–350.Google Scholar
Mead, J., Turner, J. M., Macklem, P. T. & Little, J. B. 1967 Significance of the relationship between lung recoil and maximum expiratory flow. J. Appl. Physiol. 22, 95–108.Google Scholar
Milne-Thompson, L. M. 1950 Theoretical Hydrodynamics, pp. 515–517. New York: Macmillan.
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics, ch. 4. McGraw-Hill.
Oates, G. C. 1975 Fluid flow in soft-walled tubes; Part 1: Steady flow. Med. & Biol. Engng 13, 773–778.Google Scholar
Olsen, J. G. & Shapiro, A. H. 1967 Large-amplitude unsteady flow in liquid-filled elastic tubes. J. Fluid Mech. 29, 513–538.Google Scholar
Prandtl, L. 1934 Fundamentals of Hydrodynamics and Aerodynamics (transl. by O. G. Tietjens), p. 243. McGraw-Hill.
Roberts, V. C. & Cotton, L. T. 1975 Mechanical methods for the prevention of venous throm-boembolism. In Venous Thrombo-Embolic Disease (ed. C. V. Ruckley & I. M. C. Macintyre), ch. 2. Edinburgh: Churchill Livingstone.
Ross, B. B., Gramiak, R. & Rahn, H. 1955 Physical dynamics of the cough mechanism. J. Appl. Physiol. 8, 264–268.Google Scholar
Rudinger, G. 1970 Shock waves in mathematical models of the aorta. Trans. A.S.M.E., J. Appl. Mech. 37, 34–37.Google Scholar
Schlichting, H. 1960 Boundary Layer Theory, ch. 20. McGraw-Hill.
Shapiro, A. H. & Hawthorne, W. R. 1947 The mechanics and thermodynamics of steady one-dimensional gas flow. J. Appl. Mech. 14, A317–A336.Google Scholar
Shapiro, A. H. 1977a Physiologic and medical aspects of flow in collapsible tubes. Proc. 6th Canadian Cong. Appl. Mech. Vancouver, pp. 883–906.Google Scholar
Shapiro, A. H. 1977b Steady flow in collapsible tubes. J. Biomech. Engng 99, 126–147.Google Scholar
Soroff, H. S., Giron, F., Ruiz, U., Birtwell, W. C., Hirsch, L. J. & Deterling, R. A. 1969 Physiologic support of heart action. New England J. Med. 280, 693–704.Google Scholar
Weissman, K. & Mockros, L. 1967 The mechanics of a collapsing tube heart pump. Int. J. Mech. Sci. 9, 113–121.Google Scholar
Wexler, L., Bergel, D. H., Gabe, I. T., Makin, G. S. & Mills, C. J. 1968 Velocity of blood flow in normal human venae cavae. Circ. Res. 23, 349–359.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves, pp. 164–167. Wiley.