Transmission problems for reactive flow and transport-multiscale analysis of the interactions of solutes with a solid phase

The modelling of reactive flows and transport 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 multiscale analysis, in particular, at the interfaces.
So far, the interactions of the solvent with the solid phase are too roughly approximated in many applications. In this lecture, we are discussing a more detailed mathematical representation of the processes in the solid on the micro-scale and are going to sketch the analysis of the arising transmission problems.
The following specific transmission conditions on the interface between the solid and the fluid phase are considered:
- the continuity of the fluxes of the solutes;
- the following nonlinear relation of the concentrations: $h(v_\varepsilon)=w_\varepsilon$.
Here $w_\varepsilon$ is the vector of concentrations in the solid phase and $v_\varepsilon$ represents the concentrations in the fluid phase. $\varepsilon$ is the scale parameter of the porous media. The structure of $h$ is determined referring to arguments from statistical physics.
The following two problems have to be solved:
(1) to investigate the existence and uniqueness of solutions for a fixed е;
(2) to derive estimates of the solutions needed to pass to the scale limit $\varepsilon\to 0$ and to formulate effective equations.
Trying a standard weak formulation for the underlying partial differential equations does not work since the nonlinear relation on the interface cannot be integrated in functionals or in function spaces directly.
Whereas a scalar diffusion-reaction equation with this nonlinear transmission condition could be solved, the problem for systems was open up to now. Here, a relaxation approach is used to solve the transmission problem for systems for fixed scale $\varepsilon$, using structural assumptions on $h$. Finally, the scale limit is discussed and an effective system is derived.
The results obtained here are based on arguments used by Jäger and Kacur for relaxation approximations of nonlinear parabolic systems.