In this paper we construct a model to describe some aspects of the deformation of the central region of the human lung considered as a continuous elastically deformable medium. To achieve this purpose, we study the interaction between the pipes composing the tree and the fluid that goes through it. We use a stationary model to determine the deformed radius of each branch. Then, we solve a constrained minimization problem, so as to minimize the viscous (dissipated) energy in the tree. The key feature of our approach is the use of a fixed point theorem in order to find the optimal flow associated to a deformed tree. We also give some numerical results with interesting consequences on human lung deformation during expiration, particularly concerning the localization of the equal pressure point (EPP).
Mots-clés : fixed point, Poiseuille flow, finite tree, elastic wall, lungs, equal pressure point
@article{M2AN_2008__42_4_507_0,
author = {Mauroy, Benjamin and Meunier, Nicolas},
title = {Optimal {Poiseuille} flow in a finite elastic dyadic tree},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {507--533},
publisher = {EDP-Sciences},
volume = {42},
number = {4},
year = {2008},
doi = {10.1051/m2an:2008015},
mrnumber = {2437772},
zbl = {1203.74033},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2008015/}
}
TY - JOUR AU - Mauroy, Benjamin AU - Meunier, Nicolas TI - Optimal Poiseuille flow in a finite elastic dyadic tree JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 507 EP - 533 VL - 42 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008015/ DO - 10.1051/m2an:2008015 LA - en ID - M2AN_2008__42_4_507_0 ER -
%0 Journal Article %A Mauroy, Benjamin %A Meunier, Nicolas %T Optimal Poiseuille flow in a finite elastic dyadic tree %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 507-533 %V 42 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008015/ %R 10.1051/m2an:2008015 %G en %F M2AN_2008__42_4_507_0
Mauroy, Benjamin; Meunier, Nicolas. Optimal Poiseuille flow in a finite elastic dyadic tree. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 507-533. doi : 10.1051/m2an:2008015. http://www.numdam.org/articles/10.1051/m2an:2008015/
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