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Thesis submitted in fulfilment of the requirements for the degree Doctor technologiae (engineering: chemical) In the faculty of engineering at the cape peninsula university of technology / Since the first uses of hollow-fiber membrane bioreactors (MBR’s) to immobilize whole cells were reported in the early 1970’s, this technology has been used in as wide ranging applications as enzyme production to bone tissue engineering. The potential of these devices in industrial applications is often diminished by the large diffusional resistances of the membranes. Currently, there are no analytical studies on the performance of the MBR which account for both convective and diffusive transport. The purpose of this study was to quantify the efficiency of a biocatalytic membrane reactor used for the production of enzymes. This was done by developing exact solutions of the concentration and velocity profiles in the different regions of the membrane bioreactor (MBR). The emphasis of this study was on the influence of radial convective flows, which have generally been neglected in previous analytical studies. The efficiency of the MBR was measured by means of the effectiveness factor. An analytical model for substrate concentration profiles in the lumen of the MBR was developed. The model was based on the solution of the Navier-Stokes equations and Darcy’s law for velocity profiles, and the convective-diffusion equation for the solute concentration profiles. The model allowed for the evaluation of the influence of both hydrodynamic and mass transfer operating parameters on the performance of the MBR. These parameters include the fraction retentate, the transmembrane pressure, the membrane hydraulic permeability, the Reynolds number, the axial and radial Peclet numbers, and the dimensions of the MBR. The significant findings on the hydrodynamic studies were on the influence of the fraction retentate. In the dead-end mode it was found that there was increased radial convective flow, and hence more solute contact with the enzymes/biofilm immobilised on the surface of the membrane. The improved solute-biofilm contact however was only limited to the entrance half of the MBR. In the closed shell mode there was uniform distribution of solute, however, radial convective flows were significantly reduced. The developed model therefore allowed for the evaluation of an optimum fraction retentate value, where both the distribution of solutes and radial convective flows could be maximised.

Na presente tese é apresentada uma nova solução analítica para a equação de ad-vecção-difusão bidimensional transiente para simular a dispersão de poluentes na atmosfera. Para tanto, a equação de advecção-difusão é resolvida pela combinação da transformada de Laplace e da técnica GILTT (Generalized Integral Laplace Transform Technique). O fechamento da turbulência para os casos Fickiano e não-Fickiano é considerado. É investigado o problema de modelagem da dispersão de poluentes em condições de ventos fortes e fracos considerando, para o caso de ventos fracos, a difusão longitudinal na equação de advecção-difusão. Além disso, foi incluída no modelo a velocidade vertical e avaliada sua influência considerando-se o campo de velocidades constante e também geradas via LES (Large Eddy Simulation), para poder simular uma camada limite turbulenta mais realística. Os resultados obtidos por essa metodologia são validados com resultados experimentais disponíveis na literatura. / In the present thesis it is presented a new analytical solution for the transient two- dimensional advection-diffusion equation to simulate the pollutant dispersion in atmosphere. For that, the advection-diffusion equation is solved combining the Laplace transform and the GILTT (Generalized Integral Laplace Transform Technique) techniques. The turbulence closure for Fickian and non-Fickian cases is considered. It is investigated the problem of modeling the pollutant dispersion in strong and weak winds considering, for the case of low wind conditions, the longitudinal diffusion in the advection-diffusion equation. Moreover, it was considered in the model the vertical velocity and its influence was evaluated considering velocities field constant and also generated by means of LES (Large Eddy Simulation), to simulate a more realistic turbulent boundary layer. The results attained by this methodology are validated with experimental results available in literature.

De nos jours, la modélisation numérique constitue un outil indispensable pour le prototypage de convertisseurs électromagnétiques. Les matériaux magnétiques jouent un rôle essentiel dans la conversion de l’énergie, il est donc nécessaire de maîtriser leur comportement et leur représentation. L’objectif de ce travail s’inscrit dans ce cadre et s’attache à élaborer des lois réalistes de comportement de matériaux afin de les inclure dans des simulateurs de circuits. Concernant le comportement statique, le modèle de Jiles-Atherton a été implémenté puis adapté, simplifié et modifié afin d’en améliorer la précision et l’implémentation. La modélisation dynamique du matériau a été effectuée grâce au modèle DWM élaboré au laboratoire Ampère. Ce modèle intègre les effets dynamiques excédentaires grâce à une loi « dynamique de matériau » implémentée au sein de l’équation de diffusion magnétique. Ce modèle a été ensuite homogénéisé afin d’en améliorer son implémentation future dans un simulateur de circuit. Chacun des différents modèles a été testé et validé sur plusieurs échantillons. / Nowadays, numerical modeling is an indispensable tool for the prototyping of electromagnetic converters. Magnetic materials play an essential role into the energy conversion so it is necessary to control their behavior as well as their modeling. The objective of this work is to develop realistic laws of material behavior for circuit simulators use. Regarding the static behavior, the Jiles-Atherton model has been implemented and adapted, simplified and modified to improve accuracy and implementation. Dynamic modeling of the material was performed using the model DWM developed into the Ampere laboratory. This model incorporates the excedentary dynamic effects thanks to a "dynamical material law" implemented into the magnetic diffusion equation. Then this model was homogenized to improve its future implementation in a circuit simulator. Each of the different models has been tested and validated on several samples.

In this dissertation, we are interested in the dynamics of spatially distributed populations. In particular, we focus on persistence conditions and minimal traveling periodic wave speeds for stage-structured populations in heterogeneous landscapes. The model includes structured populations of two age groups, juveniles and adults, in patchy landscapes. First, we present a stage-structured population model, where we divide the population into pre-reproductive and reproductive stages. We assume that all parameters of the two age groups are piecewise constant functions in space. We derive explicit formulas for population persistence in a single-patch landscape and in heterogeneous habitats. We find the critical size of a single patch surrounded by a non-lethal matrix habitat. We derive the dispersion relation for the juveniles-adults model in homogeneous and heterogeneous landscapes. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest. We apply sensitivity and elasticity formulas to the critical size of a single-patch landscape and to the minimal traveling wave speed in a homogeneous landscape. Secondly, we use asymptotic techniques to find an explicit formula for the traveling periodic wave speed and to calculate the spread rates for structured populations in heterogeneous landscapes. We illustrate the power of the homogenization method by comparing the dispersion relation and the resulting minimal wave speeds for the approximation and the exact expression. We find an excellent agreement between the fully heterogeneous speed and the homogenized speed, even though the landscape period is on the same order as the diffusion coefficients and not as small as the formal derivation requires. We also generalize this work to the case of structured populations of n age groups. Lastly, we use a finite difference method to explore the numerical solutions for the juveniles-adults model. We compare numerical solutions to analytic solutions and explore population dynamics in non-linear models, where the numerical solution for the time-dependent problem converges to a steady state. We apply our theory to study various aspects of marine protected areas (MPAs). We develop a model of two age groups, juveniles and adults, in which only adults can be harvested and only outside MPAs, and recruitment is density dependent and local inside MPAs and fishing grounds. We include diffusion coefficients in density matching conditions at interfaces between MPAs and fishing grounds, and examine the effect of fish mobility and bias movement on yield and fish abundance. We find that when the bias towards MPAs is strong or the difference in diffusion coefficients is large enough, the relative density of adults inside versus outside MPAs increases with adult mobility. This observation agrees with findings from empirical studies.

Cette thèse est consacrée à l’étude de phénomènes de propagation et de concentration dans des modèles d’équations intégro-différentielles venant de la écologie. On étudie certaines équations de réaction-diffusion non locales apparaissant en dynamique de populations, ainsi que des modèles représentant l’évolution Darwinienne avec un mode de reproduction sexué.Dans une première partie, nous étudions la propagation spatiale pour une équation de réaction-diffusion ou la dispersion opère via un noyau de convolution à queue lourde. Nous mesurons de manière précise l’accélération du front de propagation de la solution. Nous proposons également une échelle adaptée pour mesurer les «petites» mutations. Dans les deux cas nous utilisons le formalisme des équations de Hamilton-Jacobi.Dans un second temps nous étudions un modèle de génétique quantitative, avec un mode de reproduction sexuée. Un petit paramètre mesure la déviation entre le trait des descendants est la moyenne des traits des parents. Dans le régime où ce paramètre est petit nous étudions l’existence de solutions stationnaires, puis le problème de Cauchy lié à ce modèle. Les solutions se concentrent autour des optima de sélection, sous la forme de perturbations de distributions Gaussiennes avec petite variance fixée par le paramètre. Notre analyse généralise le cas linéaire de la reproduction asexuée en utilisant des outils d’analyse perturbative. Enfin dans une dernière partie nous fournissons des simulations numériques et des méthodes mathématiques pour étudier la dynamique interne des équilibres dans le régime de petite variance, pour les deux modes de reproduction : asexué et sexué. / This manuscript tackles propagation and concentration phenomena in different integro-differential equations with a background in ecology. We study non local reaction-diffusion equations from population dynamics, and models for Darwinian evolution with a sexual or asexual mode of reproduction, with a preference for the former.In a first part, we study spatial propagation for a reaction diffusion equation where dispersion acts through a fat tailed kernel. We measure accurately the acceleration of the propagation front of the population. We propose as well a scaling well adapted to “small mutations” when we consider the model in the context of adaptative dynamics. This scaling is very natural following the previous spatial investigation. In both cases we look at the long time behavior and we use the Hamilton-Jacobi framework. Then we turn our attention towards a quantitative genetics model, with a sexual mode of reproduction, imposed by the “infinitesimal operator”. In this non-linear setting, a small parameter tunes the deviation between the phenotypic trait of the offspring and the mean of the traits of the parents. In the regime where this parameter is small, we prove existence of stationary solutions, and their local uniqueness. We also provide an example of non-uniqueness in the case where the selection function admits several extrema. We prove that the solution concentrates around the points of minimum of the selection function. The analysis is carried by the small perturbations of special profiles : Gaussian distributions with small variance fixed by the parameter.We then study the stability of the Cauchy problem associated to the previous model. This time we prove that at all times, for a well prepared initial data, the solutions is arbitrary close to a Gaussian distribution with small variance. The proof follows the framework of the previous : we use perturbative analysis tools, but this time an even more precise description of the correctors is needed and we linearize the equation to obtain it. In a final part we show numerical simulations and different mathematical approaches to study inside dynamics of phenotypic lineages in the regime of small variance, with a moving environement.

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We consider a singularly perturbed reaction-diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

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Ce travail de thèse est dédié à la visualisation par microscopie optique des transitions de phases, thermo- et photo-induites dans des monocristaux à transition de spin. L’étude des cristaux du composé [{Fe(NCSe)(py)2}2(m-bpypz)] a permis de montrer la possibilité de contrôler la dynamique de l’interface HS/BS (haut spin/bas spin) par une irradiation lumineuse appliquée sur toute la surface du cristal ou de manière localisée. Les investigations expérimentales menées sur l’effet de l’intensité de la lumière sur la température de transition ont mis en évidence d’une part l’importance du couplage entre le cristal et le bain thermique, et d’autre part le rôle de la diffusion de la chaleur dans le monocristal. En parallèle, un modèle basé sur une description de type Ginzburg-Landau, a permis de mettre sur pied une description de type réaction diffusion des effets spatio-temporels accompagnant la transition de spin dans un monocristal. Celui-ci a permis d’identifier et de comprendre le rôle des paramètres pertinents entrant en jeu dans le contrôle du mouvement de l’interface HS/BS. Les résultats obtenus sont très encourageants et reproduisent avec une grande fidélité les données expérimentales. Cependant l’origine de l’orientation de l’interface HS/BS observée par microscopie optique dans les cristaux du composé [{Fe(NCSe)(py)2}2(m-bpypz)] était restée mystérieuse. Pour résoudre cette question, nous avons développé un modèle électro-élastique qui tient compte du changement de volume au cours de la transition de spin. Ce dernier nous a conduits à analyser l’effet de la symétrie du réseau cristallin et de la forme du cristal sur l’orientation de l’interface élastique. En l’appliquant au composé [{Fe(NCSe)(py)2}2(m-bpypz)], en tenant compte du caractère anisotrope du changement de la maille élémentaire lors du passage HSBS, nous avons réussi à retrouver quantitativement l’orientation du front observée expérimentalement en microscopie optique. Ceci confirme bien le rôle primordial de l’élasticité dans le comportement des matériaux à transition de spin. Des études sous lumière à très basse température nous ont donné la possibilité de suivre en temps réel, l’effet LIESST (Light Induced Excited Spin State Trapping), la re-laxation coopérative du cristal ainsi que l’instabilité photo-induite LITH (Light Induced Thermal Hysteresis). Un monde fascinant est apparu autour de cette dernière, avec la présence de comportements totalement inédits. Ainsi, et pour la première fois, nous avons mis en évidence l’existence de phénomènes d’auto-organisation et de comportements autocatalytiques du front de transition. Cette physique non-linéaire dénote un comportement actif du cristal, par suite d’une subtile préparation autour d’un état instable. Ces comportements rappellent les structures dissipatives de Turing et ouvrent des perspectives fascinantes pour cette thématique, tant sur le plan expérimental que théorique. / This thesis work is devoted to visualization by optical microscopy of thermo- and photo-induced phase transitions, in switchable spin transition single crystals. The study of crystals of the compound [{Fe (NCSe) (py) 2} 2 (m-bpypz)] showed the possibility to control reversibly the dynamics of the HS/LS interface through a photo-thermal effect generated by an irradiation of the whole crystal or using a spatially localized light spot on the crystal surface. The investigations of the effect of the light intensity on the transition temperature have highlighted the importance of the coupling between the crystal and the thermal bath in these experiments. Concomitantly, we developped a reaction diffusion model allowing to describe and iden-tify the relevant physical parameters involved in the control of the movement of HS/LS interface. The obtained results are very encouraging and reproduce the main features of the experimental data. However the origin of the interface orientation observed by the optical microscopy in the crystal of the compound [{Fe (NCSe) (py) 2} 2 (m-bpypz)] re-mained mysterious, and needed an elastic approach to be handled. At this end, an electro-elastic model including the volume change at the spin transition was developed. By taking into account for the anisotropy of the unit cell deformation at the transition, we were able to reproduce quantitatively the experimental HS/LS interface orientation. This result confirms the crucial role of the lattice symmetry and its elastic properties in the emergence of a stable interface orientation. The last part of the thesis is devoted to the investigation of photo-induced effects at very low temperatures (~10K). There, we visualized for the first time the real time transformation of a single crystal under LIESST (Light Induced Excited Spin State Trapping) effect as well as its subsequent relaxation at higher temperatures. We have also studied the light induced instabilities through investigation on the LITH (Light Induced Thermal Hysteresis) loops. Around the latter, a fascinating world made of nonlinear effects, and patterns formation emerged, recalled the well known Turing structures. These results lead to new horizons that will give access to new theories and original experimental observations that will enrich the topics opening the new avenues to study of nonlinear phenomena in spin crossover solids.