Upscaling of reactive flows

The thesis deals with the upscaling of reactive flows in complex geometry. The reactions which may include deposition or 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. 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. Our contributions in this respect are in dealing with non-Lipschitz reactive terms on the boundary in the fixed geometry case and the formal asymptotic approach for the moving boundary. Both add to the present literature. Next, 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. Reactions take place at the lateral boundaries of the strip (the walls), where the reaction product can deposit in a layer with a non-negligible thickness compared to the width of the strip. This leads to a free boundary problem, in which the moving interface between the fluid and the deposited (solid) layer is explicitly taken into account. Using asymptotic expansion methods, we derive an upscaled, one-dimensional model by averaging in the transversal direction. The upscaled equations are similar to the Taylor dispersion and we have performed numerical simulations to compare the upscaled equations with other simpler upscaled equations and the transversally averaged, two-dimensional solution. The derivation introduces new terms originating from the changing geometry. The numerical computations also provide an insight into the regimes where such an upscaling is useful. 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 precipitation and dissolution processes in a porous medium. We define a sequence of microscopic solutions u" and obtain the upscaled equations as the limit of e \ 0. We adopt the 2-scale framework to achieve this. The challenges are in dealing with the low regularity of microscopic solutions and particular non-linearities in the reaction term. This rigorous derivation closes the gap of the rigorous transition from a given pore scale model to the heuristically proposed macroscopic model. In addition, numerical methods to compute the solution for an upscaled model have been proposed. The upscaled model describes the reactive flow in a porous medium. The reaction term, especially, the dissolution term has a particular, multi-valued character, which leads to stiff dissolution fronts. We have considered both the conformal and mixed schemes for the analysis including both the semi-discrete (time-discretization) and the fully discrete (both in space and time) cases. The fully discrete schemes correspond to the finite element method and the mixed finite element method for conformal, respectively mixed schemes. The numerical schemes have been analyzed and the convergence to the continuous formulation has been proved. Apart from the proof for the convergence, this also yields an existence proof for the solution of the upscaled model. Numerical experiments are performed to study the convergence behavior. The challenges are in dealing with the specific non-linearities of the reaction term. We deal with them by using the translation estimates which are adapted to the specific numerical scheme. The applications are in the development of all-solid state rechargeable batteries having a high dtorage 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.