Comptes Rendus
Fluid–solid interactions: modeling, simulation, bio-mechanical applications
A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation
Comptes Rendus. Mécanique, Volume 333 (2005) no. 12, pp. 867-883.

We derive a closed system of effective equations describing a time-dependent flow of a viscous incompressible Newtonian fluid through a long and narrow elastic tube. The 3D axially symmetric incompressible Navier–Stokes equations are used to model the flow. Two models are used to describe the tube wall: the linear membrane shell model and the linearly elastic membrane and the curved, linearly elastic Koiter shell model. We study the behavior of the coupled fluid–structure interaction problem in the limit when the ratio between the radius and the length of the tube, ɛ, tends to zero. We obtain the reduced equations that are of Biot type with memory. An interesting feature of the reduced equations is that the memory term explicitly captures the viscoelastic nature of the coupled problem. Our model provides significant improvement over the standard 1D approximations of the fluid–structure interaction problem, all of which assume an ad hoc closure assumption for the velocity profile. We performed experimental validation of the reduced model using a mock circulatory flow loop assembled at the Cardiovascular Research Laboratory at the Texas Heart Institute. Experimental results show excellent agreement with the numerically calculated solution. Major applications include blood flow through large human arteries.

Nous obtenons un système fermé d'équations efficaces, décrivant l'écoulement non-stationnaire d'un fluide newtonien incompressible visqueux à travers un tuyau élastique long et de faible épaisseur. Pour modéliser l'écoulement, nous utilisons le système de Navier–Stokes 3D axisymétrique et incompressible. Deux modèles sont employés pour décrire le comportement élastique de la paroi latérale : les équations de Navier pour une membrane courbe élastique linéaire, et ensuite le modèle de Koiter, d'une coque courbe, élastique linéaire. Nous étudions le comportement du système lorsque le rapport ɛ, entre l'épaisseur caractéristique et la longueur du tube, tend vers zéro. Nous obtenons les équations efficaces, essentiellement 1D, qui sont du type de Biot avec mémoire. Une caractéristique intéressante des équations efficaces est que le terme de mémoire capture explicitement la nature viscoélastique du problème couplé. Notre modèle efficace fournit une amélioration significative par rapport aux modèles 1D standards de l'interaction fluide–structure, qui nécessitent une formule de fermeture pour la vitesse, proposée ad hoc. Nous avons effectué la validation expérimentale du modèle réduit en utilisant la boucle d'écoulement simulé au Cardiovascular Research Laboratory, Texas Heart Institute. Les résultats expérimentaux montrent un accord excellent avec la solution calculée numériquement. L'application principale inclut l'écoulement sanguin à travers les grandes artères du corps humain.

Published online:
DOI: 10.1016/j.crme.2005.10.005
Keywords: Computational fluid mechanics, Blood flow, Asymptotic methods, Fluid–structure interaction, 3D Navier–Stokes equations
Mot clés : Mécanique des fluides numérique, Écoulements sanguin, Méthodes asymptotiques, Interaction fluide–structure, Système de Navier–Stokes 3D

Sunčica Čanić 1; Andro Mikelić 2; Josip Tambača 3

1 Department of Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX 77204-3476, USA
2 Institut Camille Jordan, UFR mathématiques, site de Gerland, bâtiment. A, université Claude–Bernard Lyon 1, 50, avenue Tony Garnier, 69366 Lyon cedex 07, France
3 Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
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Sunčica Čanić; Andro Mikelić; Josip Tambača. A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation. Comptes Rendus. Mécanique, Volume 333 (2005) no. 12, pp. 867-883. doi : 10.1016/j.crme.2005.10.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.10.005/

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