Fluids and solids interacting over scales
Mathematical models for interactions of flow, chemical reactions and mechanics and applications in lifesciences
In many applications it becomes necessary to couple reactive flow and mechanics of fluids and solids phases, leading to challenges of mathematical modeling and analysis. This lecture gives a survey on results obtain in the team in Heidelberg on dynamics of membranes and tissue, of processes in infarct and bone remodeling.
10.00-10.45: Prof. Andro Mikelić (Université Lyon)
Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system
We undertake establishing well-posedness of the dynamic Biot-Allard equations. It is obtained using the precise properties of the dynamic permeability matrix following the homogenization derivation of the model. By taking the singular limit of the contrast coefficient, the quasi-static Biot system can be obtained from the dynamic Biot equations. These results can be used to formulate an efficient computational algorithm for solving dynamic Biot-Allard equations for subsurface flows with the characteristic reservoir time scales larger than the intrinsical characteristic time. This result appears to be completely new in the literature on Biot's theory. We conclude by showing that in the case of periodic deformable porous media the dynamic permeability has the required properties.
11.00-11.45: Prof. Florin Radu (University of Bergen)
Simulation of concrete carbonation by mixed finite elements
We discuss a prototypical reaction-diffusion-flow scenario in saturated/unsaturated porous media. The special features of our scenario are: the reaction produces water and therefore the flow and transport are coupled in both directions and moreover, the reaction may alter the microstructure. This means we have a variable porosity in our model. For the spatial discretization we propose a mass conservative scheme based on the mixed finite element method. The scheme is semi-implicit in time. Error estimates are obtained for some particular cases. We apply our finite element methodology for the case of concrete carbonation -- one of the most important physico-chemical processes affecting the durability of concrete.