Thesis Summary:

This thesis is concerned with the mathematical and numerical analysis of partial differential equations (PDE)-based models for the coupling of flow equations and transport arising from problems related with the simulation of transport phenomena and chemical interactions within saturated porous media. This framework is encountered in a vast variety of engineering applications such as polymer flooding in petroleum extraction, wastewater treatment, food and chemicals processing, chromatography and others. Among the applications mentioned, those that motivated the development of this work are mainly related to the design of equipment used in water treatment. This includes settlers, clarifiers/thickeners, and filtration equipment. However, we point out, that other applications where envision and developed during the work on the general models. In fact, this dissertation also includes some results related to traffic flow, bioconvection and thermohaline circulation. Other extensions which require more extensive modifications or additions, such as fluid-structure interactions are briefly discussed in the current and future works section, at the end of this thesis. As a quick overview, in Chapter 2 we begin by studying the phenomenon of sedimentation, in the first instance, through polydisperse sedimentation models, considering from the numerical point of view a finite volume method with entropy conservation properties. In Chapter 3, we introduce models for the coupling of flow and transport equations motivated by the study of double-diffusive flows. Here we change the approach of the numerical approximation, to focus on the finite element method, with divergence-free approximations for velocity. The analysis and numerical scheme designed for the non- stationary setting is then extended, to develop a second approach for the sedimentation phenomenon, which in turn, motivated Chapter 4. It is a complete three-dimensional model for clarifiers, where we incorporate the one-dimensional Kynch density function describing hindered settling, used in the first approach, in a transport equation coupled with a Navier-Stokes-Brinkman model for the flux. Finally Chapter 5, discuss the application to the modelling of soil-based water filters of a similar scheme adapted to the context of an axisymmetric domain and a non-stationary system.

ISI Publications from the Thesis

Graham BAIRD, Raimund BüRGER, Paul E. MéNDEZ, Ricardo RUIZ-BAIER: Second-order schemes for axisymmetric Navier-Stokes-Brinkman and transport equations modelling water filters. Numerische Mathematik, vol. 147, 2, pp. 431-479, (2021).

Raimund BüRGER, Paul E. MéNDEZ, Ricardo RUIZ-BAIER: Convergence of H(div)-conforming schemes for a new model of sedimentation in circular clarifiers with a rotating rake. Computer Methods in Applied Mechanics and Engineering, vol. 367, article: 113130, (2020).

Raimund BüRGER, Paul E. MéNDEZ, Carlos PARéS: On entropy stable schemes for degenerate parabolic multispecies kinematic flow models. Numerical Methods for Partial Differential Equations, vol. 35, 5, pp. 1847-1872, (2019).

Raimund BüRGER, Paul E. MéNDEZ, Ricardo RUIZ-BAIER: On H(div)-conforming methods for double-diffusion equations in porous media. SIAM Journal on Numerical Analysis, vol. 57, 3, pp. 1318-1343, (2019).