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A viscoelastic model with non-local damping application to the human lungs. (English) Zbl 1330.74040

Summary: 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.

MSC:

74D05 Linear constitutive equations for materials with memory
74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
74L15 Biomechanical solid mechanics
76S05 Flows in porous media; filtration; seepage
92B05 General biology and biomathematics
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