Synopsis : Upscaling of reactive flows involving free boundaries

of thesis The thesis deals with the upscaling of reactive flows in complex geometry. The reactions (deposition/dissolution) take place at a part of the boundary and depending on the size of the reaction domain, the changes in the pore structure that are due to the deposition process may or may not be neglected. In mathematical terms, the models are defined in a fixed, respectively variable geometry, when the deposition layer generates a free boundary at the pore scale. Specifically, for the chemical vapor deposition (CVD) process on a trenched geometry, we have developed mathematical models for both situations. We have carried out numerical simulations for both the fixed geometry and variable geometry showing a good agreement with the experimental results obtained for the deposition of Titanium dioxide inside trenches of different diameters. This work has implied determining the physical and chemical parameters. For the multi-scale computations, numerical methods inspired from domain decomposition ideas have been proposed and the convergence of the scheme has been proved. Computing the full solution in a domain with oscillating boundary requires a lot of computational effort, as one has to achieve an accuracy that agrees with the scale of oscillations. To approximate these solutions, one defines equations in a simpler domain, where flat boundaries but modified boundary conditions approximate the rough one. The two situations mentioned before were considered: the fixed geometry case, and the time dependent geometry at the microscale (free boundaries). We have derived an approximating (effective) model where a flat boundary is replacing the oscillatory boundary, but defining an effective boundary condition. In the fixed geometry case, we provide rigorous mathematical proofs for the upscaling procedure. The second case, when we take into account the geometry changes at the microscale, is more involved, and we use formal asymptotic methods to derive these boundary conditions. Further, to understand the flow in a domain with variable geometry, we have considered a thin strip with reactions taking place at the lateral boundaries of the strip under dominant transport conditions. We have derived upscaled equations for the solute concentration, which are similar to the Taylor dispersion and we have performed numerical simulations to compare the upscaled equations with other simpler upscaled equations. We have further studied the rigorous homogenization process for the reactive flows for a periodic array of cells and proved the validity of upscaled equations. These reactive flows model the crystal precipitation and dissolution processes in a porous medium. Our results are in agreement with the macroscale equations proposed in the literature. In addition, numerical methods, for instance, finite element methods and mixed finite element method, to compute the solution have been proposed and the proofs for the convergence of these schemes have been obtained. The applications are in the development of all-solid state rechargeable batteries having a high storage capacity. Such devices have a complex 3D geometry for the electrodes to enhance the surface area. The challenges are in the development of the appropriate technologies for the formation of these electrodes. In particular we focus on chemical vapor deposition processes (CVD), with the aim of getting a deeper understanding of the reactions taking place in a complex geometry. Other applications include flows in porous media, bio-film growth etc.