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.

Keywords: Poiseuille flow, dyadic tree, kernel operator, damped wave equation, human lungs

@article{M2AN_2006__40_1_201_0, author = {Grandmont, C\'eline and Maury, Bertrand and Meunier, Nicolas}, title = {A viscoelastic model with non-local damping application to the human lungs}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {201--224}, publisher = {EDP-Sciences}, volume = {40}, number = {1}, year = {2006}, doi = {10.1051/m2an:2006009}, mrnumber = {2223510}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2006009/} }

TY - JOUR AU - Grandmont, Céline AU - Maury, Bertrand AU - Meunier, Nicolas TI - A viscoelastic model with non-local damping application to the human lungs JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 201 EP - 224 VL - 40 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006009/ DO - 10.1051/m2an:2006009 LA - en ID - M2AN_2006__40_1_201_0 ER -

%0 Journal Article %A Grandmont, Céline %A Maury, Bertrand %A Meunier, Nicolas %T A viscoelastic model with non-local damping application to the human lungs %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 201-224 %V 40 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2006009/ %R 10.1051/m2an:2006009 %G en %F M2AN_2006__40_1_201_0

Grandmont, Céline; Maury, Bertrand; Meunier, Nicolas. A viscoelastic model with non-local damping application to the human lungs. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 1, pp. 201-224. doi : 10.1051/m2an:2006009. http://archive.numdam.org/articles/10.1051/m2an:2006009/

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