Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries

SIAM Journal on Applied Dynamical Systems

Volumn 2, Issue 3, 2003, Pages 431-463

  Čanić, Sunčica ^a   Mikelić, Andro ^b  

^a University of Houston   (United States)

^b UNIVERSITÉ LYON 1   (France)

Author keywords

Asymptotic analysis; Blood flow; Compliant vessels; Fluid structure interaction; Navier Stokes equations

Indexed keywords

EID: 3042571169     PISSN: 15360040     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1111111102411286     Document Type: Article

References (24)

1. M. A. BIOT, Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lower frequency range and II. Higher frequency range, J. Acoust Soc. Amer., 28 (1956), pp. 168-191.
2. A. BOURGEAT, A. MIKELIĆ, AND R. TAPIÉRO, Dérivation des équations moyennées décrivant un écoulement non-newtonien dans un domaine de faible épaisseur, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), pp. 965-970.
3. S. ČANIĆ, Blood flow through compliant vessels after endovascular repair: Wall deformations induced by the discontinuous wall properties, Comput. Vis. Sci., 4 (2002), pp. 147-155.
4. S. ČANIĆ AND E.-H. KIM, Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci., 26 (2003), pp. 1-16.
5. P. G. CIARLET, Plates and Junctions in Elastic Multi-structures, Rech. Math. Appl. 14, Masson, Paris, 1990.
6. U. DINAR, Cardiovascular Fluid Dynamics, CRC Press, Boca Raton, FL, 1981.
7. H. DRIDI, Comportement asymptotique des équations de Navier-Stokes dans des domaines "applatis," Bull. Sci. Math. (2), 106 (1982), pp. 369-385.
8. W. JÄGER AND A. MIKELIĆ, On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness, Comm. Pure Appl. Math., 51 (1998), pp. 1073-1121.
9. G. FICHERA, Existence theorems in elasticity, in Handbook der Physik VIa/2, Springer-Verlag, Berlin, 1972.
10. L. FORMAGGIA, F. NOBILE, AND A. QUARTERONI, A one dimensional model for blood flow: Application to vascular prosthesis, in Mathematical Modeling and Numerical Simulation in Continuum Mechanics, Lect. Notes Comput. Sci. Engrg. 19, I. Babuska, T. Miyoshi, and P. G. Ciarlet, eds., Springer-Verlag, Berlin, 2002, pp. 137-153.
11. Y. C. FUNG, Biomechanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York, 1993.
12. J. KEENER AND J. SNEYD, Mathematical Physiology, Interdiscip. Appl. Math. 8, Springer-Verlag, New York, 1998.
13. L. D. LANDAU AND E. M. LIFSCHITZ, Elasticity Theory, Pergamon Press, Oxford, UK, 1975.
14. H. LE DRET, Problemes variationnels dans les multidomaines, Rech. Math. Appl. 19, Masson, Paris, 1991.
15. J.-L. LIONS, Perturbations singulières dans les problèmes aux limites et en controle optimal, Springer-Verlag, Berlin, 1973.
16. P. LUCHINI, M. LUPO, AND A. POZZI, Unsteady Stokes flow in a distensible pipe, ZAMM Z. Angew. Math. Mech., 71 (1991), pp. 367-378.
17. A. MIKELIĆ AND R. TAPIÉRO, Mathematical derivation of the power law describing polymer flow through a thin slab, M2AN Math. Model. Numer. Anal., 29 (1995), pp. 3-22.
18. F. MURAT AND A. SILI, Problèmes monotones dans des cylindres de faible diamètre, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), pp. 567-572.
19. S. A. NAZAROV, Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid, Siberian Math J., 31 (1990), pp. 296-307.
20. M. S. OLUFSEN, C. S. PESKIN, W. Y. KIM, E. M. PEDERSEN, A. NADIM, AND J. LARSEN, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Annals of Biomedical Engineering, 28 (2000), pp. 1281-1299.
21. M. S. OLUFSEN, A structured tree outflow condition for blood flow in the larger systemic arteries, Amer. J. Physiology, 276 (1999), pp. 257-268.
22. K. PERKTOLD AND G. RAPPITSCH, Mathematical modeling of local arterial flow and vessel mechanics, in Computational Methods for Fluid-Structure Interaction, Pitman Res. Notes Math. Ser. 306, J. Crolet and R. Ohayon, eds., Longman, Harlow, UK, 1994, pp. 230-245.
23. A. QUARTERONI, M. TUVERI, AND A. VENEZIANI, Computational vascular fluid dynamics: Problems, models and methods. Survey article, Comput. Vis. Sci., 2 (2000), pp. 163-197.
24. I. TOLSTOY, ed., Acoustics, Elasticity, and Thermodynamics of Porous Media. Twenty-One Papers by M. A. Biot, Acoustical Society of America, New York, 1992.