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Issue ESAIM: M2AN
Volume 42, Number 4, July-August 2008
Page(s) 507 - 533
DOI 10.1051/m2an:2008015
Published online 27 May 2008

ESAIM: M2AN 42 (2008) 507-533
DOI: 10.1051/m2an:2008015

Optimal Poiseuille flow in a finite elastic dyadic tree

Benjamin Mauroy1 and Nicolas Meunier2

1  Laboratoire MSC, Université Paris 7 (Denis Diderot), 2 place Jussieu, building 33/34, 75251 Paris Cedex 05, France. benjamin.mauroy@paris7.jussieu.fr
2  Laboratoire de Mathématiques MAP5, Université Paris 5 (R. Descartes), 45 rue des Saints Pères, 75006 Paris, France. Nicolas.Meunier@math-info.univ-paris5.fr

Received December 22, 2006. Revised September 27, 2007. Published online May 27, 2008.

Abstract
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).


Mathematics Subject Classification. 74D05, 74Q10, 76S05, 92B05.

Key words: Fixed point, Poiseuille flow, finite tree, elastic wall, lungs, equal pressure point.


© EDP Sciences, SMAI 2008


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