BALIKE, Dieudonné Zirhumanana (2024) Analysis and simulations of free boundary problems arising from biofilm modelling. [Tesi di dottorato]

Documento PDF TesiDieudonneZirhumananaBALIKE.pdf Visibile a [TBR] Amministratori dell'archivio Download (5MB) | Richiedi una copia

Abstract

This Ph.D. dissertation is related to the mathematical modelling of biofilms and their interactions with metals in the context of wastewater treatment. The thesis deals with both modelling and numerical simulations of metal interactions with biofilms but it also addresses the qualitative analysis (existence, uniqueness, and stability of solutions) of the same models. It consists of four chapters besides the introduction and the conclusion which are organized as follows. First, a literature survey of the numerical methods mainly adopted in the resolution of biofilm models is presented. Such work has been motivated by the considerable progress made in recent decades in biofilm modelling and by the variety of numerical techniques adopted. In order to be correctly interpreted, these models need to be solved numerically, as their analytical solutions are very difficult to find in most instances. The choice of an efficient numerical method depends on the mathematical model and varies according to the objectives and the approach followed by the researchers. In this review, we have described and discussed the most used numerical methods adopted in biofilm modelling in the last decades. The review focuses only on deterministic continuous models for biofilms growing in liquid environments and classifies them in two broad categories, namely the one-dimensional and multidimensional models. For each category, the seminal models have been individuated and reviewed with reference to the numerical methods adopted for their resolution. In the third Chapter, a mathematical model describing the precipitation of (trace-) metals in a multispecies biofilm is presented. The model is derived from mass conservation principles. The growth of biomass and the formation of precipitates are governed by a system of nonlinear hyperbolic partial differential equations, while the diffusion of substrates, cations, and anions is modelled by a system of parabolic partial differential equations. A nonlinear ordinary differential equation is defined to take into account the biofilm thickness evolution, which represents the moving boundary of the problem. A supplementary system of nonlinear ordinary differential equations models the biofilm and the bulk liquid interactions within the bioreactor. This model is applied to two real-world problems in wastewater treatment and in anaerobic digestion. The model is general and can be applied to any type of metal involved in a chemical precipitation mechanism. It considers the effect of the precipitates accumulation in reducing the biofilm porosity.\\ The fourth Chapter presents the existence and uniqueness of solutions to the free boundary problem related to biofilm growth, which represents a simplification of the model introduced in the third Chapter. The main novelty consists in studying the original hyperbolic-parabolic systems without the hypothesis of quasi-stationary conditions for the substrate equations. The method of characteristics and fixed point strategies are used to prove the existence and uniqueness theorem in small and all times. All the equations are converted into integral equations, in particular this transformation is made for the parabolic equations by using the Green's functions. Dirichlet-Neumann and Neumann-Robin boundary conditions are considered for the substrate equations and their extension to the case with variable diffusivity. Finally, the fifth Chapter addresses the question of stability for the models studied in the third and fourth Chapters. The existence of the stationary solutions together with the stability analysis of the classical solutions are presented. The conclusion of the thesis and indications for future work are summarized in the sixth Chapter of this dissertation.

Downloads

Actions (login required)

Modifica documento