Biological tissues can be seen as porous media in which elastic fibers (such as the heart muscle, lung parenchyma, gray matter) and incompressible viscous fluids (blood, lymph, cerebrospinal fluid) are in interaction. Recently, a new model based on mixture theory was proposed to simulate biological tissues perfusion. This is a fully dynamical model in which the fluid and solid equations are strongly coupled through the interstitial pressure. As such, it generalizes Darcy, Brinkman and Biot equations of poroelasticity.

The mathematical and numerical analysis of this model was first performed for a compressible porous material.  During this presentation, I will focus on the nearly incompressible case with a semigroup approach that also enables to prove the existence of weak solutions.  I will show the existence and uniqueness of strong and weak solutions in the incompressible limit, for which a non-standard divergence constraint arises. Our study also provides guidelines to propose a stable and robust approximation of the problem with mixed finite elements.

Finally, our theoretical results are corroborated by numerical tests. In particular, I will explain how this poromechanical model can be used to reproduce numerically microfluidics experiments conducted at LadHyX (École Polytechnique) to simulate microvessels perfusion.