Volume 40, Number 1, January-February 2006
|Page(s)||201 - 224|
|Published online||23 February 2006|
A viscoelastic model with non-local damping application to the human lungs
Université Paris Dauphine, 75775 Paris Cedex 16 & INRIA, France.
2 Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France. email@example.com
3 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.
Revised: 30 October 2005
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.
Mathematics Subject Classification: 74D05 / 74Q10 / 76S05 / 92B05
Key words: Poiseuille flow / dyadic tree / kernel operator / damped wave equation / human lungs.
© EDP Sciences, SMAI, 2006
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.