Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T07:45:41.681Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady analysis of transcritical flows in collapsible tubes with discontinuous mechanical properties: implications for arteries and veins

Published online by Cambridge University Press:  04 November 2013

A. Siviglia  and
M. Toffolon
A. Siviglia*
Affiliation:
Laboratory of Hydraulics, Hydrology and Glaciology VAW, ETH Zurich, Switzerland
M. Toffolon
Affiliation:
Department of Civil, Environmental and Mechanical Engineering, University of Trento, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the conditions under which discontinuous mechanical properties of a collapsible tube can induce transcritical flows, i.e. the transition through the critical state where the speed index (analogous to the Mach or the Froude numbers for compressible and free surface flows, respectively) is one. Such a critical transition may strongly modify the flow properties, cause a significant reduction in the cross-sectional area of the tube, and limit the flow. General relationships are obtained for a short segment using a one-dimensional model under steady flow conditions. Marginal curves delimiting the transcritical regions are identified in terms of the speed index and the cross-sectional area ratio. Since there are many examples of such flows in physiology and medicine, we also analyse the specific case of prosthesis (graft or stent) implantation in blood vessels. We then compute transcritical conditions for the case of stiffness and reference area variations, considering a collapsible tube characterized by physiological parameters representative of both arteries and veins. The results suggest that variations in mechanical properties may induce transcritical flow in veins but is unrealistic in arteries.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Blood flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 195 - 215
Copyright
©2013 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

Abraham, P., Leftheriotis, G., Desvaux, B., Saumet, M. & Saumet, J. L. 1994 Diameter and blood velocity changes in the saphenous vein during thermal stress. Eur. J. Appl. Physiol. 69, 305–308.CrossRefGoogle ScholarPubMed
Alastruey, J., Khir, A. W., Matthys, K. S., Segers, P., Sherwin, S. J., Verdonck, P. R., Parker, K. H. & Peiró, J. 2011 Pulse wave propagation in a model human arterial network: assessment of 1-D visco-elastic simulations against in vitro measurements. J. Biomech. 44, 2250–2258.Google Scholar
Alastruey, J., Parker, K. H., Peiró, J., Byrd, S. M. & Sherwin, S. J. 2007 Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J. Biomech. 40, 1794–1805.Google Scholar
Attinger, E. O. 1969 Wall properties of veins. IEEE Trans. Biomed. Engng 16, 253–261.Google Scholar
Avolio, P. 1980 Multi-branched model of the human arterial system. Med. Biol. Engng Comput. 18, 709–718.Google Scholar
Bassez, S., Flaud, P. & Chauveau, M. 2001 Modelling of the deformation of flexible tubes using a single law: application to veins of the lower limb in man. Trans. ASME: J. Biomech. Engng 123, 58–65.Google Scholar
Brook, B. S., Falle, S. A. E. G. & Pedley, T. J. 1999 Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state. J. Fluid Mech. 396, 223–256.CrossRefGoogle Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated-flow model for collapsible-tube oscillations. J. Fluid Mech. 157, 375–404.Google Scholar
Canić, S. 2002 Blood flow through compliant vessels after endovascular repair: wall deformations induced by discontinuous wall properties. Comput. Vis. Sci. 4, 147–155.Google Scholar
Canić, S., Ravi-Chandar, K., Krajcer, Z., Mirkovic, D. & Lapin, S. 2005 Mathematical model analysis of Wallstent® and AneuRx®: dynamic responses of bare-metal endoprosthesis compared with those of stent-graft. Texas Heart Institute J. 32 (4), 502–506.Google Scholar
Caro, C. G., Pedley, T. J., Schroter, R. C. & Seed, W. A. 2012 The Mechanics of the Circulation, 2nd edn. Cambridge.Google Scholar
Charonko, J. J., Ragab, S. A. & Vlachos, P. P. 2009 A scaling parameter for predicting pressure wave reflection in stented arteries. J. Med. Dev. 3, 011006.Google Scholar
Downing, J. M. & Ku, D. N. 1997 Effects of frictional losses and pulsatile flow on the collapse of stenotic arteries. Trans. ASME: J. Biomech. Engng 119, 317–324.Google ScholarPubMed
Drzawiecki, G., Field, S., Moubarak, I. & Li, J. 1997 Vessel growth and collapsible pressure area relationship. Am. J. Physiol. Heart Circ. Physiol. 273, H2030–H2043.Google Scholar
Elad, D. & Kamm, R. D. 1989 Parametric evaluation of forced expiration using a numerical model. Trans. ASME: J. Biomech. Engng 111, 192–199.Google Scholar
Elad, D., Kamm, R. D. & Shapiro, A. H. 1987 Chocking phenomena in a lung-like model. Trans. ASME: J. Biomech. Engng 109, 1–92.Google Scholar
Elad, D., Kamm, R. D. & Shapiro, A. H. 1988 Mathematical simulation of forced expiration. J. Appl. Physiol. 65, 14–25.CrossRefGoogle ScholarPubMed
Elad, D., Kamm, R. D. & Shapiro, A. H. 1989 Steady compressible flow in collapsible tubes: application to forced expiration. J. Fluid Mech. 203, 401–418.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. Appl. Maths 23, 446–455.Google Scholar
Formaggia, L., Quarteroni, A. & Veneziani, A. 2009 Cardiovascular Mathematics: Modelling and Simulation of the Circulatory System. Springer.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.Google Scholar
Jensen, O. E. & Pedley, T. J. 1989 The existence of steady flow in a collapsed tube. J. Fluid Mech. 206, 339–374.Google Scholar
Kadoglou, N. P. E., Moulakakis, K. G., Papadakis, I., Ikonomidis, I., Alepaki, M., Lekakis, J. & Liapis, C. D. 2012 Changes in aortic pulse wave velocity of patients undergoing endovascular repair of abdominal aortic aneurysms. J. Endovasc. Ther. 19, 661–666.Google Scholar
Ku, D. N., Zeigler, M. N. & Downing, J. M. 1990 One-dimensional steady inviscid flow through a stenotic collapsible tube. Trans. ASME: J. Biomech. Engng 112, 444–450.Google Scholar
Lantelme, P., Dzudie, A., Milon, H., Bricca, G., Legedz, L., Chevalier, J. M. & Feugier, P. 2009 Effect of abdominal aortic grafts on aortic stiffness and central hemodynamics. J. Hypertension 27, 1268–1276.CrossRefGoogle ScholarPubMed
Liang, F. Y., Takagi, S., Himeno, R. & Liu, H. 2009 Biomechanical characterization of ventricular-arterial coupling during aging: a multi-scale model study. J. Biomech. 42, 692–704.Google Scholar
Marchi, E. 1968 Il calcolo del rigurgito provocato dalle pile dei ponti: deflusso con transizione attraverso lo stato critico. L’Energia Elettrica 4, 254–259.Google Scholar
Müller, L. O., Parés, C. & Toro, E. F. 2013 Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties. J. Comput. Phys. 242, 53–85.Google Scholar
Müller, L. O. & Toro, E. F. 2013 A global multi-scale mathematical model for the human circulation with emphasis on the venous system. Preprint NI3007, Isaac Newton Institute for Mathematical Sciences.Google Scholar
Nippa, J. H., Alexander, R. H. & Folse, R. 1971 Pulse wave velocity in human veins. J. Appl. Physiol. 30 (4), 558–563.Google Scholar
Pedley, T. J. 2000 Blood flow in arteries and veins. In Perspective in Fluid Dynamics (ed. Batchelor, G. K., Moffat, H. K. & Worster, M. G.), Cambridge.Google Scholar
Pedley, T. J., Brook, B. S. & Seymour, R. S. 1996 Blood pressure and flow rate in the giraffe jugular vein. Phil. Trans. R. Soc. Lond. B 351, 855–866.Google Scholar
Reymond, P., Merenda, F., Rüfenacht, D. & Stergiopulos, N. 2009 Validation of a one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. Physiol. 297, 208–222.Google Scholar
Sakalihasan, N., Limet, R. & Defawe, O. D. 2005 Abdominal aortic aneurysm. Lancet 365 (9470), 1577–1589.Google Scholar
Shapiro, A. H. 1977 Steady flow in collapsible tubes. Trans. ASME: J. Biomech. Engng 99, 126–147.Google Scholar
Sherwin, S. J., Formaggia, L., Peiro, J. & Franke, V. 2003 Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Intl J. Numer. Meth. Fluids 43 (6–7), 673–700.Google Scholar
Stoquart-ElSankari, S., Lehmann, P., Villette, A., Czosnyka, M., Meyer, M. E., Deramond, H. & Balédent, O. 2009 A phase-contrast MRI study of physiologic cerebral venous flow. J. Cerebr. Blood F. Met. 29, 1208–1215.Google Scholar
Toro, E. F. & Siviglia, A. 2013 Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Commun. Comput. Phys. 13 (2), 361–385.Google Scholar
Vardoulis, O., Coppens, E., Martin, B., Reymond, P., Tozzi, P. & Stergiopulos, N. 2011 Impact of aortic grafts on arterial pressure. Eur. J. Vasc. Endovasc. Surg. 42, 704–710.Google Scholar
Wesly, R. L., Vaishnav, R. N, Fuchs, J. C, Patel, D. J. & Greenfield, J. C. 1975 static linear and nonlinear elastic properties of normal and arterialized venous tissue in dog and man. Circulat. Res 37, 509–520.Google Scholar
Zamboni, P., Galeotti, R., Menegatti, E, Malagoni, A. M., Gianesini, S., Bartolomei, I., Mascoli, F. & Salvi, F. 2009 A prospective open-label study of endovascular treatment of chronic cerebrospinal venous insufficiency. J. Vascu. Surg. 50, 1348–1358.Google Scholar