Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-18T10:05:48.611Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions

Published online by Cambridge University Press:  03 June 2015

Eleuterio F. Toro  and
Annunziato Siviglia
Eleuterio F. Toro*
Affiliation:
Laboratory of applied Mathematics, University of Trento, Via Mesiano 77, 1-38100 Trento, Italy
Annunziato Siviglia*
Affiliation:
Laboratory of applied Mathematics, University of Trento, Via Mesiano 77, 1-38100 Trento, Italy
*
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

We formulate a one-dimensional time-dependent non-linear mathematical model for some types of physiological fluid flow in collapsible tubes with discontinuous material properties. The resulting 6 × 6 hyperbolic system is analysed and the associated Riemann problem is solved exactly. Although the solution algorithm deals with idealised cases, it is nonetheless uniquely well-suited for assessing the performance of numerical methods intended for simulating more general situations. Moreover, our model may be a useful starting point for numerical calculations of realistic flows involving rapid and discontinuous material property variations. One important example in mind is the simulation of blood flow in medium-to-large veins in humans. Finally, we also discuss some peculiarities of the model regarding the loss of strict hyperbolicity and uniqueness. In particular we show an example in which the solution of the Riemann problem is non unique.

Keywords

76Z05Collapsible tubesphysiological flowsblood flowsRiemann problem
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 2 , February 2013 , pp. 361 - 385
Copyright
Copyright © Global Science Press Limited 2013

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

[1]Akesson, H., Lindh, M., Ivancev, K., and Risberg, B.Venous stents in chronic iliac vein occlusions. European Journal of vascular and endovascular surgery, 13(3): 334–336, 1997.CrossRefGoogle ScholarPubMed
[2]Alastruey, J., Parker, J., Peiro, K., Byrd, S., and Sherwin, S.Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. Journal of Biomechanics, 40: 1794–1805, 2007.Google Scholar
[3]Alcrudo, F. and Benkhaldoun, F.Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Computers & Fluids, 30(6): 643–671, 2001.CrossRefGoogle Scholar
[4]Andrianov, N. and Warnecke, G.On the solution to the Riemann problem for the compressible duct flow. SIAM Journal on Applied Mathematics, 64(3): 878–901, 2004.Google Scholar
[5]Asay, B., Son, S., and Bdzil, J.The role of gas permeation in convective burning. International Journal of Multiphase Flow, 23: 923–952, 1996.CrossRefGoogle Scholar
[6]Beiko, D., Knudsen, B., and Denstedt, J.Reviews in endourology – Advances in ureteral stent design. Journal of Endourology, 17(4): 195–199, 2003.Google Scholar
[7]Brook, B., Falle, S., and Pedley, T.Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state. Journal of Fluid Mechanics, 396: 223–256, 1999.Google Scholar
[8]Chinnayya, A., Roux, A. L., and Seguin, N.A well-balanced numerical scheme for shallow-water equations with topography: resonance phenomena. International Journal on Finite Volumes, 2004.Google Scholar
[9]Clain, S. and Rochette, D.First- and second-order finite volume methods for the one-dimensional nonconservative Euler system. Journal of Computational Physics, 228(22): 8214–8248, 2009.Google Scholar
[10]Elad, D., Kamm, R., and Shapiro, A.Chocking phenomena in a lung-like model. Journal of Biomechanical Engineering, 109: 1–92, 1987.Google Scholar
[11]Embid, P. and Baer, M.Mathematical analysis of a 2-phase continuum mixture theory. Continuum Mechanics and Thermodynamics, 4:4279–312, 1992.Google Scholar
[12]Formaggia, L., Lamponi, D., and Quarteroni, A.One-dimensional models for blood flow in arteries. Journal of Engineering Mathematics, 47: 251–276, 2003.Google Scholar
[13]Fullana, M. and Zaleski, S.A branched one-dimensional model of vessel networks. Journal of Fluid Mechanics, 621: 183–204, 2009.Google Scholar
[14]Fung, Y.C.Biomechanics:Circulation, Second Edition. Springer Verlag, 1997.CrossRefGoogle Scholar
[15]Goatin, P. and LeFloch, P.-G.The Riemann problem for a class of resonant hyperbolic systems of balance laws. Annales de l’Institut Henry Poicare-analyse non lineaire, 21(6): 881–902, 2004.CrossRefGoogle Scholar
[16]Gosse, L.A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Computers & Mathematics with Applications, 39(9-10): 135–159, 2000.Google Scholar
[17]Gosse, L.A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Mathematical Models & Methods in Applied Sciences, 11(2): 339–365, 2001.CrossRefGoogle Scholar
[18]Greenberg, J., LeRoux, A., Baraille, R., and Noussair, A.Analysis and Approximation of Con-servation Laws with Source Terms. SIAM Journal of Numerical Analysis, 34: 1980–2007, 1997.CrossRefGoogle Scholar
[19]Isaacson, E. and Temple, B.Nonlinear resonance in systems of conservation-laws. SIAM Journal of Applied Mathematics, 52(5): 1260–1278, 1992.Google Scholar
[20]Jeffrey, A.Quasilinear Hyperbolic Systems and Waves. Pitman, 1976.Google Scholar
[21]Kamm, R. and Shapiro, A.Unsteady flow in a collapsible tube subjected to external pressure or body forces. Journal of Fluid Mechanics, 95: 1–78, 1979.Google Scholar
[22]LeFloch, P.G.Shock waves for nonlinear hyperbolic systems in nonconservative form. Institute for Math. and its Appl., Minneapolis, IMA Preprint Series: 593, 1989.Google Scholar
[23]LeVeque, R.Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods. Journal of Computational Physics, 146: 346–365, 1998.Google Scholar
[24]Liu, T.Nonlinear resonance for quasi-linear hyperbolic equation. Journal of Mathematical Physics, 28(11): 2593–2602, 1987.Google Scholar
[25]Moireau, P., Xiao, N., Astorino, M., Figueroa, C.A., Chapelle, D., Taylor, C.A., Gerbeau, J.F.External tissue support and fluid-structure simulation in blood flows. Biomechanics and Modeling in Mechanobiology, to appear, 2011.Google ScholarPubMed
[26]Noussair, A.Riemann problem with nonlinear resonance effects and well-balanced Godunov scheme for shallow fluid past an obstacle. SIAM Journal on Numerical Analysis, 39(1): 52–72, 2001.Google Scholar
[27]Parés, C.Numerical Methods for Nonconservative Hyperbolic Systems: a Theoretical Framework. SIAM Journal on Numerical Analysis, 44: 300–321, 2006.CrossRefGoogle Scholar
[28]Pedley, T. J.The fluid dynamics of large blood vessels, Third Edition. Cambridge University Press, 1980.Google Scholar
[29]Quarteroni, A., Tuveri, M., and Veneziani, A.Computational vascular fluid dynamics: problems, models and methods. Computing and Visualization in Science, 2(4): 163–97, 2000.CrossRefGoogle Scholar
[30]Schwendeman, D., Wahle, C., and Kapila, A.The Riemann Problem and a High-Resolution Godunov Method for a Model of Compressible Two-Phase Flow. Journal of Computational Physics, 212: 490–526, 2006.Google Scholar
[31]Sherwin, S., Formaggia, L., Peiro, J., and Franke, V.Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. International Journal for Numerical Methods in Fluids, 43: 673–700, 2003.Google Scholar
[32]Toro, E. F.Shock capturing methods for free surface shallow water flows. Wiley and Sons, 2001.Google Scholar
[33]Toro, E. F.Riemann Solvers and Numerical Methods for Fluid Dynamics, Third Edition. Springer-Verlag, 2009.Google Scholar
[34]Toro, E. F. and Siviglia, A.Simplified blood flow model with discontinuous vessel properties: analysis and exact solutions. In: Modelling of Physiological Flows (Ambrosi, D., Quarteroni, A., Rozza, G., Eds). Springer-Verlag, 2011.Google Scholar
[35]Umscheid, T. and Stelter, W.Time-related alterations in shape, position, and structure of self-expanding modular aortic stent grafts: a 4-year single-centre follow up. Journal of En-dovascular Surgery, 6: 17–32, 1999.Google Scholar
[36]Zamboni, P., Lanzara, S., Mascoli, F., Caggiati, A., and Liboni, A.Inflammation in venous disease. International Angiology, ProQuest Science Journals, 27(5): 361–369, 2008.Google ScholarPubMed