Optimal Poiseuille flow in a finite elastic dyadic tree

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: 22 December 2006
Revised: 27 September 2007

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).