Desenvolvemos urna teoria cinética para urna mistura de gases ionizados em presença de campos elétricos e magnéticos. As leis de Ohm, Fourier e Navier-Stokes são obtidas por dois métodos distintos que se baseiam na equação de Boltzmann. Verificamos que o emprego de teoremas de representação torna o método de Chapman-Enskog mais direto. Entretanto o método combinado mostrou-se extremamente simples, onde os coeficientes de transporte são determinados através da inversão de tensores de segunda e quarta ordens. Calculamos também a integral de colisão para as possíveis interações em gases ionizados tais como, entre partículas carregadas, partícula carregada e partícula neutra e entre partículas neutras. Como uma aplicação do método combinado, determinamos os coeficientes de condutividade elétrica, condutividade térmica, coeficiente termo-elétrico e o coeficiente de viscosidade cisalhante para um gás totalmente ionizado. Obtemos seus respectivos gráficos, considerando então um gás ionizado formado a partir do gás de hélio. / We develop a kinetic theory for ionized gases mixtures under the presence of electric and magnetic fields. The laws of Ohm, Fourier and Navier-Stokes are obtained by two different methods based on the Boltzmann equation. We verify that the use of representation theorems makes the Chapman-Enskog method more direct. However the combined method shows up as extremely simple where the transport coefficients are determined through inversion of second-order and fourth order tensors. We calculate also the collision integrals for possible interactions in ionized gases like: between charged particles, between charged particles and neutral particles and between neutral particles. As an application of the combined method, we determine the electrical and thermal conductivity coefficients, thermo-electric and shear viscosity coefficients for a completely ionized gas. We obtain their respective graphics considering an ionized gas of helium.

The subject of the thesis is the derivation of non-local diffusion equations from kinetic models with heavy-tailed equilibrium in velocity. We are particularly interested in confining the kinetic equations and developing methods that allow us, from the confined kinetic models, to derive confined versions of non-local diffusion equations.

Cette thèse porte sur le développement de modèles multi-échelles et de méthodes de calcul pour les applications aérothermodynamiques. Le travail de recherche sur les modèles multi-échelles met l’accent sur l’excitation énergétique et la dissociation. L’objectif était double : mieux comprendre la dynamique des processus d'excitation énergétique et dissociation et développer des modèles réduits en diminuant la résolution d’un modèle détaillé de collisions rovibrationnelles. Les résultats obtenus ont montré que les modèles réduits permettent de reproduire avec précision la dynamique d’écoulement prédites par le modèle détaillé de collisions rovibrationnelles. Le travail de recherche sur les méthodes de calcul a porté sur les écoulements raréfiés. L’objectif était de formuler une méthode numérique de type déterministe pour résoudre l’équation de Boltzmann dans le cas de gaz à plusieurs composants y compris l’énergie interne. La méthode numérique est basée sur la structure de convolution pondérée de la transformée de Fourier de l’équation de Boltzmann. La précision de la méthode numérique proposée a été évaluée en comparant les moments extraits de la fonction de distribution de vitesse avec les prédictions de la méthode de simulation directe Monte Carlo (DSMC). Dans toutes les applications étudiées, un excellent accord a été trouvé. / This thesis aimed at developing multi-scale models and computational methods for aerother-modynamics applications. The research on multi-scale models has focused on internal energy excitation and dissociation of molecular gases in atmospheric entry flows. The scope was two-fold: to gain insight into the dynamics of internal energy excitation and dissociation in the hydrodynamic regime and to develop reduced models for Computational Fluid Dynamics applications. The reduced models have been constructed by coarsening the resolution of a detailed rovibrational collisional model developed based on ab-initio data for the N2 (1Σ+g)-N (4Su) system provided by the Computational Quantum Chemistry Group at NASA Ames Research Center. Different mechanism reduction techniques have been proposed. Their appli-cation led to the formulation of conventional macroscopic multi-temperature models and vi-brational collisional models, and innovative energy bin models. The accuracy of the reduced models has been assessed by means of a systematic comparison with the predictions of the detailed rovibrational collisional model. Applications considered are inviscid flows behind normal shock waves, within converging-diverging nozzles and around axisymmetric bodies, and viscous flows along the stagnation-line of blunt bodies. The detailed rovibrational colli-sional model and the reduced models have been coupled to two flow solvers developed from scratch in FORTRAN 90 programming language (SHOCKING_F90 and SOLV-ER_FVMCC_F90). The results obtained have shown that the innovative energy bin models are able to reproduce the flow dynamics predicted by the detailed rovibrational collisional model with a noticeable benefit in terms of computing time. The energy bin models are also more accurate than the conventional multi-temperature and vibrational collisional models. The research on computational methods has focused on rarefied flows. The scope was to formu-late a deterministic numerical method for solving the Boltzmann equation in the case of multi-component gases with internal energy by accounting for both elastic and inelastic collisions. The numerical method, based on the weighted convolution structure of the Fourier trans-formed Boltzmann equation, is an extension of an existing spectral-Lagrangian method, valid for a mono-component gas without internal energy. During the development of the method, particular attention has been devoted to ensure the conservation of mass, momentum and en-ergy while evaluating the collision operators. Conservation is enforced through the solution of constrained optimization problems, formulated in a consistent manner with the collisional in-variants. The extended spectral-Lagrangian method has been implemented in a parallel com-putational tool (best; Boltzmann Equation Spectral Solver) written in C programming lan-guage. Applications considered are the time-evolution of an isochoric gaseous system initially set in a non-equilibrium state and the steady flow across a normal shock wave. The accuracy of the proposed numerical method has been assessed by comparing the moments extracted from the velocity distribution function with Direct Simulation Monte Carlo (DSMC) method predictions. In all the cases, an excellent agreement has been found. The computational results obtained for both space homogeneous and space inhomogeneous problems have also shown that the enforcement of conservation is mandatory for obtaining accurate numerical solutions.

Desenvolvemos urna teoria cinética para urna mistura de gases ionizados em presença de campos elétricos e magnéticos. As leis de Ohm, Fourier e Navier-Stokes são obtidas por dois métodos distintos que se baseiam na equação de Boltzmann. Verificamos que o emprego de teoremas de representação torna o método de Chapman-Enskog mais direto. Entretanto o método combinado mostrou-se extremamente simples, onde os coeficientes de transporte são determinados através da inversão de tensores de segunda e quarta ordens. Calculamos também a integral de colisão para as possíveis interações em gases ionizados tais como, entre partículas carregadas, partícula carregada e partícula neutra e entre partículas neutras. Como uma aplicação do método combinado, determinamos os coeficientes de condutividade elétrica, condutividade térmica, coeficiente termo-elétrico e o coeficiente de viscosidade cisalhante para um gás totalmente ionizado. Obtemos seus respectivos gráficos, considerando então um gás ionizado formado a partir do gás de hélio. / We develop a kinetic theory for ionized gases mixtures under the presence of electric and magnetic fields. The laws of Ohm, Fourier and Navier-Stokes are obtained by two different methods based on the Boltzmann equation. We verify that the use of representation theorems makes the Chapman-Enskog method more direct. However the combined method shows up as extremely simple where the transport coefficients are determined through inversion of second-order and fourth order tensors. We calculate also the collision integrals for possible interactions in ionized gases like: between charged particles, between charged particles and neutral particles and between neutral particles. As an application of the combined method, we determine the electrical and thermal conductivity coefficients, thermo-electric and shear viscosity coefficients for a completely ionized gas. We obtain their respective graphics considering an ionized gas of helium.

Apesar dos avanços na última metade de século na teoria de transporte em Física de Plasmas, muitos de seus aspectos ainda são pouco compreendidos. Grande parte dessa limitação se deve à carência de modelos de primeiros princípios minimamente capazes de reproduzir os resultados experimentais. De fato, sem o embasamento em hipóteses fundamentais, os modelos devem se restringir à descrição do comportamento observado nos diferentes regimes de transporte no plasma, sem necessariamente especificar por que ou quais são os mecanismos envolvidos; até mesmo a identificação dos elementos envolvidos no transporte, por exemplo, se partículas ou células convectivas, é prejudicada. Uma abordagem que vem ganhando destaque na comunidade de Física de Plasmas ao longo dos anos é a estatística não-extensiva. Em particular, o interesse na teoria de Tsallis está na sua capacidade de descrever sistemas distantes do equilíbrio termodinâmico, uma característica comum à maioria dos plasmas de laboratório e astrofísicos. De fato, nessas circunstâncias, é sabido que as funções de distribuição das partículas são distantes das distribuições Maxwellianas, com longas-caudas, especialmente para os elétrons. A capacidade da teoria de Tsallis em descrever fenômenos da Física de Plasmas é retratada nas suas diversas aplicações encontradas na literatura, por exemplo, o transporte anômalo, oscilações eletrostáticas, ventos solares, plasmas empoeirados, onde é sabido que as previsões dadas pela estatística de Maxwell-Boltzmann não são capazes de descrever corretamente os resultados experimentais. A proposta desta tese de doutoramento é utilizar a estatística não-extensiva para determinar o transporte colisional em plasmas intensamente magnetizados. O desenvolvimento completo do modelo de transporte no contexto não-extensivo é estabelecido rigorosamente: partindo da definição da entropia de Tsallis e da hipótese das interações fracas (a condição do transporte colisional), somos capazes de deduzir as equações de fluidos utilizando apenas métodos estatísticos genéricos, e sem hipóteses adicionais. Nesse percurso, apresentamos, sempre de maneira consistente com a estatística não-extensiva, a definição da temperatura; a dedução da equação cinética com o operador colisional para plasmas; a generalização do método utilizado por Braginskii para determinar as soluções aproximadas da equação cinética; e o cálculo dos coeficientes de transporte. Porém, também apresentamos a aplicação de nosso modelo no transporte de calor em ventos solares e no pulso frio em plasmas de laboratório. / Despite the advances in the last half century in the plasma transport theory, many aspects of such phenomena remain poorly understood. Most of this limitation is due to the lack o first principles models capable of reproducing experimental observations. In fact, without a fundamental hypothesis, the models are restricted to describing the behavior of the observed plasma transport in diferent regimes, without specifying why or which mechanisms take part in the process; even the determination of the elements involved in the transport, for instance, whether particles or convective cells, is impaired. One approach that has been attracting attention in Plasma Physics community over the years is the non-extensive statistics. In particular, the interest in the Tsallis\'s theory lies in its ability to describe systems far from thermodynamic equilibrium, a common feature in most laboratory and astrophysical plasmas. The capability of the non-extensive statistics in describing phenomena of Plasma Physics is portrayed in various applications, for example, the anomalous transport, electrostatic oscillations, solar winds, dusty plasmas, where it is know that the predictions given by Maxwell-Boltzmann statistics cannot describe the experimental results. Indeed, under such cases, it is well known that the particle distribution functions are quite distant from Maxwellian distributions, with long tails, especially for electrons. The purpose of this doctoral thesis is to use the non-extensive statistics in order to obtain a model for the collisional transport in strongly magnetized plasmas. The complete development of the model in the non-extensive context is strictly established; starting with the definition of the Tsallis entropy and the weak interactions hypothesis (the collisional transport condition), we are able to derive the fluid equations using only generic statistical methods, without additional hypotheses. For such task, we present, consistently with non-extensive statistics, the definition of temperature; the deduction of the kinetic equation with the collision operator for plasmas, which are also appropriated for the determination of the fluid equations; the generalization of the method used by Braginskii to approximate the solution of the kinetic equation for electrons; and the calculation of electron transport coeficients. Lastly, we present the application of our model in the heat transport in the solar winds and in the phenomena of the cold pulse in laboratory plasmas.

Cette thèse est dédiée à l'étude mathématique et numérique d'une classe d'équations cinétiques collisionnelles, de type équation de Boltzmann. Nous avons porté un intérêt tout particulier à l'équation des milieux (ou gaz) granulaires, initialement introduite dans la littérature physique pour décrire le comportement hors équilibre de matériaux composés d'un grand nombre de grains, ou particules, non nécessairement microscopiques, et interagissant par des collisions dissipant l'énergie cinétique. Ces modèles se sont révélés avoir une structure mathématique très riche. Cette thèse se structure en trois partie pouvant être lues de manière indépendante, mais néanmoins en rapport avec des équations cinétiques collisionnelles en général, et l'équation des milieux granulaires en particulier. La première partie est dédiée à l'étude mathématique du comportement asymptotique de certaines équations cinétiques collisionnelles dans un cadre homogène en espace. Nous y montrons des résultats de type explosion et convergence vers la solution autosimilaire avec calcul explicite des taux, pour des opérateurs de type Boltzmann, grâce à l'utilisation (entre autre) d'une nouvelle méthode de changement de variables dépendant directement de la solution de l'équation considérée. En particulier, nous démontrons que pour un modèle de gaz granulaire - dit anormal - il est possible d'observer une explosion en temps fini. Dans la deuxième partie, orientée analyse numérique et calcul scientifique, nous nous intéressons développement et à l'étude de méthodes spectrales pour la résolution de problèmes multi-échelles, issus de la théorie des équations cinétiques collisionnelles. Les méthodes de changement de variables tiennent aussi une place importante dans cette partie, et permettent d'observer numériquement des phénomènes non triviaux qui apparaissent lors de l'étude de gaz granulaires, comme la création d'amas de matière ou la caractérisation précise du retour vers l'équilibre. La troisième et dernière partie est dédiée à l'étude spectrale de l'opérateur des milieux granulaires avec bain thermique, linéarisé au voisinage d'un équilibre homogène en espace, afin d'établir des résultats de type stabilité et convergence vers une limite hydrodynamique. Ce travail est en fait la généralisation d'un résultat célèbre dans la théorie de l'équation de Boltzmann, dû à R. Ellis et M. Pinsky, et établissant rigoureusement la première limite hydrodynamique vers les équations d'Euler compressibles linéaires puis Navier-Stokes de cette équation / This dissertation is dedicated to the mathematical and numerical study of a class of collisional kinetic equations, such as the Boltzmann equation of perfect gases. We took a particular interest in the granular media (or gases) equation, which has been first introduced in the physical literature to describe the nonnequilibrium behavior of materials composed of a large number of grains (the particles) of macroscopic size, interacting through energy dissipative collisions. These models have a very rich mathematical structure. This dissertation is divided in three independent part, all related to the theory of collisional kinetic equation, with a strong emphasis on granular media. The first part concerns the mathematical study of the asymptotic behavior of space homogeneous Boltzmann-like kinetic equations. We prove some blow up results, as well as convergence towards self-similarity, with explicit rates for two different models. One of the key tools of our proofs is the use of a new scaling method, where the scaling function depends on the solution itself. We especially prove that for a particular model of granular gases (also know as anomalous), finite time blow up occurs. The second part is dedicated to the development and study of spectral methods for the resolution of multi-scale problems, coming from the theory of collisional kinetic equations. Some rescaling methods take a very important place in this part, allowing to observe numerically some nontrivial phenomena such as the clustering in space which occurs in the time evolution of a space inhomogeneous granular gas, or to investigate numerically the trend to equilibrium for this equation. The whole third (and last) part is dedicated to the spectral study of the granular gases operator with a thermal bath, linearized near a space homogeneous self-similar profile. The goal of this work is to prove some stability results for the complete space inhomogeneous equation, and to investigate the hydrodynamic limit of the model. This work is based and extend the famous result of R. Ellis and M. Pinsky on the spectrum of the linearized Boltzmann equation, intended to establish rigorously the hydrodynamic limit of this equation towards the linearized Euler and Navier-Stokes equations

Apesar dos avanços na última metade de século na teoria de transporte em Física de Plasmas, muitos de seus aspectos ainda são pouco compreendidos. Grande parte dessa limitação se deve à carência de modelos de primeiros princípios minimamente capazes de reproduzir os resultados experimentais. De fato, sem o embasamento em hipóteses fundamentais, os modelos devem se restringir à descrição do comportamento observado nos diferentes regimes de transporte no plasma, sem necessariamente especificar por que ou quais são os mecanismos envolvidos; até mesmo a identificação dos elementos envolvidos no transporte, por exemplo, se partículas ou células convectivas, é prejudicada. Uma abordagem que vem ganhando destaque na comunidade de Física de Plasmas ao longo dos anos é a estatística não-extensiva. Em particular, o interesse na teoria de Tsallis está na sua capacidade de descrever sistemas distantes do equilíbrio termodinâmico, uma característica comum à maioria dos plasmas de laboratório e astrofísicos. De fato, nessas circunstâncias, é sabido que as funções de distribuição das partículas são distantes das distribuições Maxwellianas, com longas-caudas, especialmente para os elétrons. A capacidade da teoria de Tsallis em descrever fenômenos da Física de Plasmas é retratada nas suas diversas aplicações encontradas na literatura, por exemplo, o transporte anômalo, oscilações eletrostáticas, ventos solares, plasmas empoeirados, onde é sabido que as previsões dadas pela estatística de Maxwell-Boltzmann não são capazes de descrever corretamente os resultados experimentais. A proposta desta tese de doutoramento é utilizar a estatística não-extensiva para determinar o transporte colisional em plasmas intensamente magnetizados. O desenvolvimento completo do modelo de transporte no contexto não-extensivo é estabelecido rigorosamente: partindo da definição da entropia de Tsallis e da hipótese das interações fracas (a condição do transporte colisional), somos capazes de deduzir as equações de fluidos utilizando apenas métodos estatísticos genéricos, e sem hipóteses adicionais. Nesse percurso, apresentamos, sempre de maneira consistente com a estatística não-extensiva, a definição da temperatura; a dedução da equação cinética com o operador colisional para plasmas; a generalização do método utilizado por Braginskii para determinar as soluções aproximadas da equação cinética; e o cálculo dos coeficientes de transporte. Porém, também apresentamos a aplicação de nosso modelo no transporte de calor em ventos solares e no pulso frio em plasmas de laboratório. / Despite the advances in the last half century in the plasma transport theory, many aspects of such phenomena remain poorly understood. Most of this limitation is due to the lack o first principles models capable of reproducing experimental observations. In fact, without a fundamental hypothesis, the models are restricted to describing the behavior of the observed plasma transport in diferent regimes, without specifying why or which mechanisms take part in the process; even the determination of the elements involved in the transport, for instance, whether particles or convective cells, is impaired. One approach that has been attracting attention in Plasma Physics community over the years is the non-extensive statistics. In particular, the interest in the Tsallis\'s theory lies in its ability to describe systems far from thermodynamic equilibrium, a common feature in most laboratory and astrophysical plasmas. The capability of the non-extensive statistics in describing phenomena of Plasma Physics is portrayed in various applications, for example, the anomalous transport, electrostatic oscillations, solar winds, dusty plasmas, where it is know that the predictions given by Maxwell-Boltzmann statistics cannot describe the experimental results. Indeed, under such cases, it is well known that the particle distribution functions are quite distant from Maxwellian distributions, with long tails, especially for electrons. The purpose of this doctoral thesis is to use the non-extensive statistics in order to obtain a model for the collisional transport in strongly magnetized plasmas. The complete development of the model in the non-extensive context is strictly established; starting with the definition of the Tsallis entropy and the weak interactions hypothesis (the collisional transport condition), we are able to derive the fluid equations using only generic statistical methods, without additional hypotheses. For such task, we present, consistently with non-extensive statistics, the definition of temperature; the deduction of the kinetic equation with the collision operator for plasmas, which are also appropriated for the determination of the fluid equations; the generalization of the method used by Braginskii to approximate the solution of the kinetic equation for electrons; and the calculation of electron transport coeficients. Lastly, we present the application of our model in the heat transport in the solar winds and in the phenomena of the cold pulse in laboratory plasmas.

In this thesis, we investigate the emergence of kinetic processes in finite quantum systems. We first generalize the Redfield theory to describe the dynamics of a small quantum system weakly interacting with an environment of finite heat capacity. We then study in detail the spin-GORM model, a model made of a two-level system interacting with a random matrix environment. By doing this, we verify our new theory and find a critical size of the environment over which kinetic processes occur. We finally study the emergence of a diffusive transport process, on a finite tight-binding subsystem interacting with a fast environment, when the size of subsystem exceeds a critical value. / Doctorat en sciences, Spécialisation chimie / info:eu-repo/semantics/nonPublished

Convection in a layer of fluid with a free surface due to a combination of thermocapillary stresses and buoyancy is a classic problem of fluid mechanics. It has attracted increasing attentions recently due to its relevance for two-phase cooling. Many of the modern thermal management technologies exploit the large latent heats associated with phase change at the interface of volatile liquids, allowing compact devices to handle very high heat fluxes. To enhance phase change, such cooling devices usually employ a sealed cavity from which almost all noncondensable gases, such as air, have been evacuated. Heating one end of the cavity, and cooling the other, establishes a horizontal temperature gradient that drives the flow of the coolant. Although such flows have been studied extensively at atmospheric conditions, our fundamental understanding of the heat and mass transport for volatile fluids at reduced pressures remains limited. A comprehensive and quantitative numerical model of two-phase buoyancy-thermocapillary convection of confined volatile fluids subject to a horizontal temperature gradient has been developed, implemented, and validated against experiments as a part of this thesis research. Unlike previous simplified models used in the field, this new model incorporates a complete description of the momentum, mass, and heat transport in both the liquid and the gas phase, as well as phase change across the entire liquid-gas interface. Numerical simulations were used to improve our fundamental understanding of the importance of various physical effects (buoyancy, thermocapillary stresses, wetting properties of the liquid, etc.) on confined two-phase flows. In particular, the effect of noncondensables (air) was investigated by varying their average concentration from that corresponding to ambient conditions to zero, in which case the gas phase becomes a pure vapor. It was found that the composition of the gas phase has a crucial impact on heat and mass transport as well as on the flow stability. A simplified theoretical description of the flow and its stability was developed and used to explain many features of the numerical solutions and experimental observations that were not well understood previously. In particular, an analytical solution for the base return flow in the liquid layer was extended to the gas phase, justifying the previous ad-hoc assumption of the linear interfacial temperature profile. Linear stability analysis of this two-layer solution was also performed. It was found that as the concentration of noncondensables decreases, the instability responsible for the emergence of a convective pattern is delayed, which is mainly due to the enhancement of phase change. Finally, a simplified transport model was developed for heat pipes with wicks or microchannels that gives a closed-form analytical prediction for the heat transfer coefficient and the optimal size of the pores of the wick (or the width of the microchannels).

<p align="justify">Ce travail consiste à étudier des systèmes constitués par un grand nombre de grains, auxquels de l’énergie cinétique est fournie, et à étudier leurs similarités et leurs différences avec des fluides traditionnels. Je me concentre principalement sur la nature de non-équilibre de ces fluides granulaires, en montrant que, même si les méthodes de méchanique statistique y sont applicables, leurs propriétés sont très différentes de celles de systèmes à l’équilibre ou proches de l’équilibre :</p><p><p><ul><li>Les fluides granulaires présentent des phénomènes de transport qui n’ont pas d’équivalent dans des fluides moléculaires, tels qu’un couplage spécifique entre flux de chaleur et gradient de densité.<p><li>Leur distribution de vitesse est en général différente de la distribution de Maxwell-Boltzmann, et présente une surpopulation pour les grandes vitesses. <p><li>Dans le cas de mélanges, différentes espèces de grains sont en général caractérisées par des énergies cinétiques différentes, i.e. ces systèmes sont sujet à une non-equipartition de leur énergie.<p><li>Ces fluides ont tendance à former des inhomogénéités spatiales spontanément. Cette propriété est illustrée en étudiant l’expérience du Demon de Maxwell appliquée aux systèmes granulaires.</ul><p><p align="justify">Chacune de ces particularités est discutée en détail dans des chapitres distincts, où l’on applique différentes méthodes de méchanique statistique (équation de Boltzmann, transition de phase, mean field models…) et où l’on vérifie les prédictions théoriques par simulations numériques (MD, Monte Carlo…).</p> / Doctorat en sciences, Spécialisation physique / info:eu-repo/semantics/nonPublished