Modélisation et simulations numériques pour des systèmes de la mécanique des fluides avec contraintes; application à la biologie et au trafic routier

The works presented in this thesis are devoted to the study of partial differential equations systems (PDE). In particular, we are interested in constrained systems coming from the fluid mechanics field which allow to described, in time and space, physical quantities such as density or speed. In this context, we build models for biology which are then numerically tested. We also present an original numerical method for a road traffic model.In the first part, using the theory of mixtures, we model the development of a photosynthetic micro-algae biofilm. The growth of micro-algae is precisely described by taking into account their composition and their access to nutrients dissolved in the surrounding liquid and light. Then, using numerical simulations, we estimate the biofilm productivity.In the second part, using the mixture theory we propose a model describing the rheology of the large intestine and its mucus layer. Thanks to this model we can give an accurate description of the velocity field induced by intestinal flow. This velocity field will then be used to build a modeldescribing precisely interactions between the intestinal microbiota, the gastric broth and the host. For these two models numerical schemes are proposed and allow a first validation of the models.The last part is devoted to developing an asymptotic preserving scheme for the constraint Aw-Rascle system for road traffic. We present an explicit-implicit method based on a splitting technique in order to approximate the solutions of Aw-Rascle system with constraint, while relaxing the stability condition (CFL).Les travaux présentés dans cette thèse sont consacrés à l’étude de systèmes d’équations aux dérivées partielles (EDP). En particulier, nous nous intéressons à des systèmes issus de la mécanique des fluides avec contraintes, qui permettent de décrire de manière continue, en temps et en espace, des quantités physiques telles que la densité ou la vitesse. Dans ce cadre, nous construisons des modèles pour la biologie, qu’ensuite nous testons numériquement. Nous proposons également avec des méthodes similaires une approche numérique originale pour un système de trafic routier.Dans une première partie, à l’aide de la théorie des mélanges, nous modélisons le développement d’un biofilm de micro-algues photosynthétiques. La croissance des micro-algues y est précisément décrite, en tenant compte de leur composition et de l’accès aux nutriments dissouts, contenus dans le liquide environnant ainsi que de la lumière. Puis, à l’aide de simulations numériques, nous estimons la productivité du biofilm.Dans la seconde partie, en utilisant la théorie des mélanges, nous proposons un modèle permettant de décrire la rhéologie du gros intestin et de sa couche de mucus. Grâce à ce modèle, nous donnerons une description précise du champ de vitesse, induit par le flux intestinal. Puis, ce champ de vitesse sera utilisé pour construire un modèle décrivant les interactions entre le microbiote intestinal, le bouillon gastrique et l’hôte. Pour ces deux modèles, un schéma numérique est proposé et permet une première validation.Enfin, la dernière partie est consacrée à l’élaboration d’un schéma asymptotic preserving pour le système de trafic routier d’Aw-Rascle avec contrainte. Nous y présentons une méthode explicite-implicite basée sur une technique de splitting permettant d’approcher les solutions du systèmed’Aw-Rascle avec contrainte, tout en réduisant la contrainte de stabilité (CFL)

Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams

International audienceWe discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicles density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions

A two-dimensional population balance model for cell growth including multiple uptake systems

Cell growth in a chemostat is a well-documented research topic. How cells uptake the avail-able substrate to gain weight and engage cell division is not generally taken into account inthe modelling bioreactors. In fact, the growth rate is related to a population doubling timewhereas the microorganisms’ growth in mass is due to the mass transfer of substrates fromthe liquid phase to the biotic phase. Clearly, growth in mass precedes growth in number.Similarly, the transport of substrates down to the cell scale precedes the mass transfer. Thisarticle’s main feature is a two-dimensional population balance model that allows to uncou-ple growth in mass and growth in number when the equilibrium between a cell populationand its environment is disrupted. The cell length and the rate of anabolism are chosen asinternal variables. It is proved that the hypothesis “growth in number = growth in mass” isvalid at steady-state or in exponential growth only. The glucose uptake is assumed drivenby two transport systems with a different affinity constant for the substrate. This combina-tion of two regulated uptake systems operating in parallel explains a 3-fold increase in theuptake following a glucose pulse, but can also predict substrate uptake rates higher thanthe maximal batch value as observed in some experiments. These features are obtainedby considering carbon fluxes in the formulation of regulation principles for uptake dynam-ics. The population balance’s implementation in a multi-compartment reactor is a naturalprospective work and allows extensions to industrial processes

A multi-scale epidemic model of Salmonella infection with heterogeneous shedding∗

Salmonella strains colonize the digestive tract of farm livestock, such as chickens or pigs, without affecting them, and potentially infect food products, representing a threat for human health ranging from food poisoning to typhoid fever. It has been shown that the ability to excrete the pathogen in the environment and contaminate other animals is variable. This heterogeneity in pathogen carriage and shedding results from interactions between the host’s immune response, the pathogen and the commensal intestinal microbiota. In this paper we propose a novel generic multiscale modeling framework of heterogeneous pathogen transmission in an animal population. At the intra-host level, the model describes the interaction between the commensal microbiota, the pathogen and the inflammatory response. Random fluctuations in the ecological dynamics of the individual microbiota and transmission at between-host scale are added to obtain a drift-diffusion PDE model of the pathogen distribution at the population level. The model is further extended to represent transmission between several populations. The asymptotic behavior as well as the impact of control strategies including cleaning and antimicrobial administration are investigated through numerical simulation

Modeling and numerical simulations for fluid mechanics systems with constraints : application to biology and road traffic

Cette thèse est consacrée à l'étude de systèmes d'équations aux dérivées partielles. En particulier nous nous intéressons à des systèmes issus de la mécanique des fluides avec contraintes qui permettent de décrire de manière continue en temps et en espace des quantités physiques telles que la densité ou la vitesse. Dans ce cadre nous construisons des modèles pour la biologie : modélisation de la croissance d'un biofilm de micro-algues et modélisation du gros intestin et de sa couche de mucus. Ces modèles sont ensuite testés numériquement à l'aide de schémas numériques spécifiquement élaborés pour ces modèles. Cette thèse est complétée par une étude numérique du modèle d'Aw-Rascle avec contrainte pour le trafic routierThis thesis is devoted to the study of partial differential equation systems. In particular, we are interested in constrained systems coming from the fluid mechanics field which allow to describe, in time and space, physical quantities such as density or speed. In this context we build models for biology: modeling of the growth of micro-algae biofilms and modeling of the large intestine and its mucus layer. These models are then tested numerically using numerical schemes specifically developed for these models. This thesis is supplemented with a numerical study of Aw-Rascle model with constraint for road traffi

Modélisation et simulations numériques pour des systèmes de la mécanique des fluides avec contraintes : application à la biologie et au trafic routier

This thesis is devoted to the study of partial differential equation systems. In particular, we are interested in constrained systems coming from the fluid mechanics field which allow to describe, in time and space, physical quantities such as density or speed. In this context we build models for biology: modeling of the growth of micro-algae biofilms and modeling of the large intestine and its mucus layer. These models are then tested numerically using numerical schemes specifically developed for these models. This thesis is supplemented with a numerical study of Aw-Rascle model with constraint for road trafficCette thèse est consacrée à l'étude de systèmes d'équations aux dérivées partielles. En particulier nous nous intéressons à des systèmes issus de la mécanique des fluides avec contraintes qui permettent de décrire de manière continue en temps et en espace des quantités physiques telles que la densité ou la vitesse. Dans ce cadre nous construisons des modèles pour la biologie : modélisation de la croissance d'un biofilm de micro-algues et modélisation du gros intestin et de sa couche de mucus. Ces modèles sont ensuite testés numériquement à l'aide de schémas numériques spécifiquement élaborés pour ces modèles. Cette thèse est complétée par une étude numérique du modèle d'Aw-Rascle avec contrainte pour le trafic routie

: École d’Été 2019 - Feuilletages et géométrie algébrique

We give a summary of the Minimal Model Program (namely, Mori Theory) in the case of surfaces

A time-space model for the growth of microalgae biofilms for biofuel production

International audienceWe present in this paper a spatial model describing the growth of a photosynthetic microalgae biofilm. In this model we consider photosynthesis, extracellular matrix excretion, and mortality. These mechanisms are described precisely using kinetic laws that take into account some saturation effects which limit the reaction rates and involve different components that we treat individually. In particular, to obtain a more detailed description of the microalgae growth, we consider separately the lipids they contain and the functional part of microalgae (proteins, RNA, etc ...), the latter playing a leading role in photosynthesis. We also consider the components dissolved in liquid phase as CO 2. The model is based on mixture theory and the behavior of each component is described on the one hand by mass conservation, which takes into account biological features of the system, and on the other hand by conservation of momentum, which describes the physical properties of the components. Some numerical simulations are displayed in the one-dimensional case and show that the model is able to estimate accurately the biofilm productivity

: École d’Été 2019 - Feuilletages et géométrie algébrique

We give a summary of the Minimal Model Program (namely, Mori Theory) in the case of surfaces

: École d’Été 2019 - Feuilletages et géométrie algébrique

We give a summary of the Minimal Model Program (namely, Mori Theory) in the case of surfaces