The respiratory system realizes the transfer of oxygen from the outside air to the
alveolar membrane, through which it diffuses into the blood. As pure diffusion is far from
being sufficient to realize that transfer, most of it is of advective type, and this advection
is triggered by inflation-deflation cycles of the parenchyma. The mechanical part of the
lungs can then be seen as a tree-like domain (conducting airways) embedded in an elastic
medium. The flow in the upper part is inertial (incompressible Navier-Stokes equations),
whereas inertia can be neglected for deeper branches (Stokes equations), which allows to
use Poiseuille’s law for each branch, and consequently Darcy like equations on the corresponding
subtrees.
We address here the delicate issues in terms of theory, numerics, and modeling, raised
by the coupling of those models (Navier-Stokes, Darcy equations on a network, elasticity
equations).
Keywords: Alveolar membrane, Darcy equations on a network, elasticity equations, elastic medium, incompressible
Navier-Stokes equations, inflation-deflation cycle, oxygen transfer, parenchyma, respiratory
system, tree-like domain, ventilation process.
