A viscoelastic model with non-local damping application to the human lungs

¹ Université Paris Dauphine, 75775 Paris Cedex 16 & INRIA, France.
² Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France. bertrand.maury@math.u-psud.fr
³ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.

Received: 1 July 2005
Revised: 30 October 2005

Abstract

In this paper we elaborate a model to describe some aspects of the human lung considered as a continuous, deformable, medium. To that purpose, we study the asymptotic behavior of a spring-mass system with dissipation. The key feature of our approach is the nature of this dissipation phenomena, which is related here to the flow of a viscous fluid through a dyadic tree of pipes (the branches), each exit of which being connected to an air pocket (alvelola) delimited by two successive masses. The first part focuses on the relation between fluxes and pressures at the outlets of a dyadic tree, assuming the flow within the tree obeys Poiseuille-like laws. In a second part, which contains the main convergence result, we intertwine the outlets of the tree with a spring-mass array. Letting again the number of generations (and therefore the number of masses) go to infinity, we show that the solutions to the finite dimensional problems converge in a weak sense to the solution of a wave-like partial differential equation with a non-local dissipative term.