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

Convergence analysis of mixed numerical schemes for reactive in a porous medium

This paper deals with the numerical analysis of an upscaled model describing the reactive flow in a porous medium. The solutes are transported by advection and diffusion and undergo precipitation and dissolution. The reaction term and, in particular, the dissolution term has a particular, multi-valued character, which leads to stiff dissolution fronts. We consider the Euler implicit method for the temporal discretization and the mixed finite element for the discretization in time. More precisely, we use the lowest order Raviart-Thomas elements. As an intermediate step we consider also a semi-discrete mixed variational formulation (continuous in space). We analyse the numerical schemes and prove the convergence to the continuous formulation. Apart from the proof for the convergence, this also yields an existence proof for the solution of the model in mixed variational formulation. Numerical experiments are performed to study the convergence behavior

Rigorous upscaling of rough boundaries for reactive flows

We consider a mathematical model for reactive flow in a channel having a rough (periodically oscillating) boundary with both period and amplitude e. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non-linear and multi-valued rates and is posed in a fixed geometry with rough boundaries. We provide a rigorous justification for the upscaling process in which we define an upscaled problem defined in a simpler domain with flat boundaries. To this aim, we use periodic unfolding techniques combined with translation estimates. Numerical experiments confirm the theoretical predictions and illustrate a practical application of this upscaling process. Keywords: Reactive flows; rough boundaries; homogenization

Rigorous upscaling of rough boundaries for reactive flows

We consider a mathematical model for reactive ¿ow in a channel having a rough (periodically oscillating) boundary with both period and amplitude e. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non-linear and multi-valued rates. We provide a rigorous justi¿cation for the upscaling process in which we de¿ne an upscaled problem de¿ned in a simpler domain with ¿at boundaries. To this aim, we use periodic unfolding techniques combined with translation estimates. Numerical experiments con¿rm the theoretical predictions and illustrate a practical application of this upscaling process

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen-Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen-Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term which makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above