Research questions: We consider problems appearing as microscopic models from two different fields: soil chemistry and biomechanics. Specifically, we investigate reactive flow in porous media and cancellous bone remodeling.
Assuming a stationary microscopic Stokes (or Navier-Stokes) flow in porous media, Darcy s law can be derived at macroscopic level by means of homogenization techniques. In this way tedious computations in complex geometries can be reduced to a simpler, macroscopic situation. Such models can be endowed by accounting chemical processes, such as diffusion, convection and reaction of solutes within the fluid at the pore scale, and adsorption, diffusion and reaction on the grain surface. Crystal precipitation and dissolution may be added to the above-mentioned situations. The resulting models leads to reaction-diffusion and transport systems that are coupled by reaction rates and isotherms at the interface between pores and grains. Homogenization results are known for linear rates, or nonlinearities of certain types, but realistic models are by far not covered by the situations mentioned. Since microscopic models are much too complicated for being approached by standard methods, this also leads to a major impediment in performing large-scale simulations.
In a similar fashion, at microscopic level bones consists of thin, solid elements (trabeculae) separated by marrow. This structure is subject to continuous changes as a consequence of remodeling processes. Molecular units act directly on the trabecular surface, producing changes in the geometry. Damaged or disused bone parts are removed continuously in time. The resulting surface cavities are refilled under the influence of a stimulus, whose production depends on the mechanical loading of the bone. Such hypotheses are analyzed by finite element simulations performed for the full microscopic model. Computer limitations do not allow studies for large pieces of bones, or investigation of complex hypotheses.
Common mathematical features relate both models: a partial differential equation has to be solved in a complex domain and is coupled to an ordinary differential equation on a lower dimensional manifold (the surface of the grain, or the trabecular surface). In both cases microscopic processes may lead to changes in local geometry. The aim of the proposal is to give a rigorous mathematical derivation of macroscopic models, sustained by analysis and numerical simulation. |
