Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for multi-scale analysis and simulation. In this lecture, the following processes are studied:
diffusion, transport, and reaction of substances in the fluid and the solid phase,
mechanical interactions of the fluid and solid phase,
change of the mechanical properties of the solid phase by chemical reactions,
volume changes (“growth”) of the solid phase.
These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro scale.
In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.
The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plague-formation) or in porous media.
The lecture is based on recent results obtained in cooperation with A. Mikelic, M. Neuss-Radu, F. Weller and Y. Yang.