Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels

Networks and Heterogeneous Media

Volumn 2, Issue 1, 2007, Pages 99-125

  Amadori, Debora ^a   Ferrari, Stefania ^b   Formaggia, Luca ^b  

^a UNIVERSITY OF L'AQUILA   (Italy)

^b POLITECNICO DI MILANO   (Italy)

Author keywords

Fluid structure interaction; Hyperbolic equations; Physiological flows

Indexed keywords

FLUID STRUCTURE INTERACTION; NONLINEAR EQUATIONS; VISCOUS FLOW;

FLUID DYNAMICAL MODEL; HYPERBOLIC EQUATIONS; INCOMPRESSIBLE NAVIER STOKES EQUATIONS; LONGITUDINAL VELOCITY; MODELLING TECHNIQUES; NON-LINEAR HYPERBOLIC SYSTEMS; ONE-DIMENSIONAL MODEL; PHYSIOLOGICAL FLOW;

NAVIER STOKES EQUATIONS;

EID: 52049122224     PISSN: 15561801     EISSN: 1556181X     Source Type: Journal    
DOI: 10.3934/nhm.2007.2.99     Document Type: Article

References (25)

1. S. Alinhac, "Blowup for Nonlinear Hyperbolic Equations," Birkhäuser, Boston, 1995.
2. R. Aris, "Vectors, Tensors and The Basic Equations of Fluid Mechanics," Prentice-Hall, 1962.
3. P. Bergeret, Classification of smooth solutions to 2 × 2 hyperbolic systems with boundary damping, Math. Meth. Appl. Sci., 20 (1997), 1563-1598.
4. S. Čani,́ E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Meth.Appl.Sci., 26 (2003), 1161-1186.
5. S. Čank, D. Lamponi, A. Mikelić, J. Tambaća, Self-consistent effective equations modelling blood flow in medium-to large compliant arteries, Multiscale Model. Simul., 3 (2005), 559-596.
6. C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics" (1st Ed.), Springer Verlag, Berlin, 2000.
7. A. Di Carlo, P. Nardinocchi, G. Pontrelli, L. Teresi, A heterogeneous approach for modelling blood flow in an arterial segment, in "Simulation in Biomedicine V" (eds. Z. M. Arnez, C.A. Brebbia, F. Solina & V. Stankovski), WIT Press, 2003, 69-78.
8. S. Ferrari, F. Saleri, A new two-dimensional shallow water model including pressure effects and slow varying bottom topography, ESAIM: M2AN, 38 (2004), 211-234.
9. S. Ferrari, Convergence analysis of a space-time approximation to a two-dimensional system of shallow water equations, Applicable Analysis, 83 (2004), 757-785.
10. L. Formaggìa, J.-F. Gerbeau, F. Nobile, A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comp.Methods in Appl.Mech.Engng.,191 (2001), 561-582.
11. L. Formaggia, D. Lamponi, A. Veneziani, M. Tuveri, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart, Computer Methods in Biomechanics and Biomedical Engineering, 2007, (to appear).
12. L. Formaggia, F. Nobile, A. Quarteroni, A. Veneziani, Multiscale modelling of the circulatory system: a preliminary analysis, Comp. Vis. Science, 2 (2000), 163-197.
13. L. Formaggia, A. Veneziani, Geometrical multiscale models for the cardiovascular system, in "Blood Flow Modelling and Diagnostics" (ed. T. A. Kowaleski), Institute of Fundamental Technological Research, Warsaw, (2005), 309-360.
14. Y. C. Fung, "Foundations of Solid Mechanics," Prentice-Hall, Englewood Cliffs N.J, 1972.
15. Y. C. Fung, K. Fronek, P. Patitucci, Pseudoelasticity of arteries and the choice of its mathematical expression, Amer. Physiology Soc., (1979), 620-631.
16. J.-F. Gerbeau, B. Perthame, Derivation of viscous Saint- Venant system for laminar shallow water; numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 89-102.
17. J. M. Greenberg and T.-T. Li, The effect of boundary damping for the quasilinear wave equation, J. Differential Equations, 52 (1984), 66-75.
18. L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations," Springer-Verlag, New York, 1997.
19. D. Lamponi, One dimensional and multiscale models for blood flow circulation, Phd Thesis n. 3006, EPFL, Lausanne, (2004).
20. T.-T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," Masson, Paris, 1994.
21. A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables," Springer-Verlag, New York, 1984.
22. T. Pedley, "The Fluid Mechanics of Large Blood Vessels," Cambridge University Press, Cambridge, 1980.
23. A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in "Handbook of Numerical Analysis" (Vol. XII) , North Holland, Amsterdam, 2004, 3-127.
24. S. J. Sherwin, V. Franke, J. Peiró and K. Parker, One dimensional modelling of a vascular network in space-time variables, J.Eng.Math., 47 (2003), 217-250.
25. N. P. Smith, A. J. Pullan and P. J. Hunter, An anatomically based model of transient coronary blood flow in the heart, SIAM J. Appl. Math., 62 (2002), 990-1018.