[pt] Consideramos a equação de Boltzmann espacialmente não homogênea para esferas duras inelásticas, com coeficiente de restituição constante alfa pertence (0, 1), sob a termalização induzida por um meio hospedeiro com uma distribuição Maxwelliana fixa e fixando e pertence (0, 1) qualquer. Quando o coeficiente de restituição alfa é próximo de 1, comprovamos a existência de soluções globais considerando o regime próximo ao equilíbrio. Também estudamos o comportamento de longo prazo dessas soluções e comprovamos uma convergência para o equilíbrio com uma taxa exponencial. / [en] We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient alpha element of (0, 1), under the thermalization induced by a host medium with a fixed Maxwellian distribution and any fixed e element of (0, 1). When the restitution coefficient alpha is close to 1 we prove existence of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.

O objetivo deste trabalho e a determinação dos coeficientes de viscosidade de cisalhamento e condutividade térmica de gases ideais quânticos. No cálculo dos coeficientes de transporte foram considerados dois aspectos: uma estatística quântica com seções transversais quânticas e uma estatística quântica com seções transversais quânticas. No primeiro caso, e utilizado um método alternativo para a determinação das aproximações sucessivas (ate a quinta ordem) para os coeficientes de transporte dos gases Helio 4, Helio 3, para-hidrogênio e orto-hidrogênio. No caso de estatística quântica desenvolvida uma teoria com base no método dos momentos de Grad e na equação de Uehling- Uhlenbeck, com a finalidade de determinar as express6es algébricas para os coeficientes de transporte. Esta teoria e baseada em 13 momentos, de densidade, velocidade, tensor pressão e fluxo de calor. Do conhecimento da função de distribuição em termos dos momentos, as equações constitutivas são determinadas e os coeficientes de transporte seguem de um método iterativo semelhante ao procedimento Maxwelliano. / The aim of this work is the determination of the coefficients of shear viscosity and thermal conductivity of quantum ideal gases. In the calculation of the transport coefficients two aspects have been taken into account: a classical statistical with quantum cross-sections and a quantum statistical with quantum cross-sections. In the first case, an alternative method is used for the determination of the successive approximations (up to the fifth order) to the transport coefficients of the gases helium 4, helium 3, para-hydrogen, ortho-hidrogen. In the case of quantum statistical a theory based on method of moments of Grad and on the Uehling-Uhlenbeck equation is developed in order to determine the algebraic expressions to the transport coefficients. This theory is based on 13 moments of density, velocity, pressure tensor and heat flux. From the knowledge of the distribution function the constitutive equations are determinate and the transport coefficients follow from an iterative method akin to the 11axwellian procedure.

No presente trabalho desenvolve-se um estudo de modelagem matemática e simulação numérica do escoamento bifásico gás-sólido na coluna ascendente de um leito fluidizado circulante. Utiliza-se o modelo euleriano de duas fases separadas considerando dois procedimentos diferentes para a modelagem do tensor das tensões da fase sólida: modelo tradicional e a teoria cinética dos escoamentos granulares (TCEG). As simulações numéricas são conduzidas com a utilização do código MFIX que é um software livre e disponível na rede (Internet). Os resultados da simulação numérica são avaliados por meio da análise da influência dos seguintes parâmetros: malha computacional, correlações para o computo do tensor das tensões da fase sólida e esquemas de discretização dos termos advectivos. Também se desenvolve estudo de caracterização de estruturas coerentes - \"clusters\". De forma complementar foram realizadas duas análises teóricas compreendendo: uma análise da influência das diversas correlações utilizadas na TCEG para o computo da viscosidade dinâmica do sólido; e uma análise enfocando o emprego de diversos esquemas de discretização para os termos advectivos presentes nas equações de conservação (Foup, Muscl, Van Leer, Minmod e Superbee). De todos os estudos e resultados apresentados no trabalho conclui-se que os escoamentos gás-sólido em leitos fluidizados circulantes são muito complexos, sendo necessário a realização de futuras pesquisas para uma melhor compreensão dos fenômenos físicos inerentes a esses escoamentos. / In the present work is described a mathematical model and numerical study simulation of the gas-solid flow in the riser of a circulating fluidized bed. It is used the two fluids eulerian model considering two different procedures for the solid phase stress tensor modeling: the traditional model and the kinetic theory of granular flows (KTGF). The numerical simulation results are evaluated through the influence analysis of the following parameters: computational mesh, correlations for computing the solid phase stress tensor and the discretization of the advective terms. It is also presented a study concerning the characterization coherent structures - \"clusters\". Complementing the above studies were accomplished two theoretical analyses comprehending: an influence analysis of several correlations used in the KTGF for computing the dynamic viscosity of the solid phase; and an analysis concerning several discretization schemes for the advective terms present in the conservative equations. Considering the developed studies and the obtained results it is concluded that the gas-solid flows in circulating fluidized beds are very complex, being necessary future research works for a better comprehension of the inherent physical phenomena to these flows.

No presente trabalho desenvolve-se um estudo de modelagem matemática e simulação numérica do escoamento bifásico gás-sólido na coluna ascendente de um leito fluidizado circulante. Utiliza-se o modelo euleriano de duas fases separadas considerando dois procedimentos diferentes para a modelagem do tensor das tensões da fase sólida: modelo tradicional e a teoria cinética dos escoamentos granulares (TCEG). As simulações numéricas são conduzidas com a utilização do código MFIX que é um software livre e disponível na rede (Internet). Os resultados da simulação numérica são avaliados por meio da análise da influência dos seguintes parâmetros: malha computacional, correlações para o computo do tensor das tensões da fase sólida e esquemas de discretização dos termos advectivos. Também se desenvolve estudo de caracterização de estruturas coerentes - \"clusters\". De forma complementar foram realizadas duas análises teóricas compreendendo: uma análise da influência das diversas correlações utilizadas na TCEG para o computo da viscosidade dinâmica do sólido; e uma análise enfocando o emprego de diversos esquemas de discretização para os termos advectivos presentes nas equações de conservação (Foup, Muscl, Van Leer, Minmod e Superbee). De todos os estudos e resultados apresentados no trabalho conclui-se que os escoamentos gás-sólido em leitos fluidizados circulantes são muito complexos, sendo necessário a realização de futuras pesquisas para uma melhor compreensão dos fenômenos físicos inerentes a esses escoamentos. / In the present work is described a mathematical model and numerical study simulation of the gas-solid flow in the riser of a circulating fluidized bed. It is used the two fluids eulerian model considering two different procedures for the solid phase stress tensor modeling: the traditional model and the kinetic theory of granular flows (KTGF). The numerical simulation results are evaluated through the influence analysis of the following parameters: computational mesh, correlations for computing the solid phase stress tensor and the discretization of the advective terms. It is also presented a study concerning the characterization coherent structures - \"clusters\". Complementing the above studies were accomplished two theoretical analyses comprehending: an influence analysis of several correlations used in the KTGF for computing the dynamic viscosity of the solid phase; and an analysis concerning several discretization schemes for the advective terms present in the conservative equations. Considering the developed studies and the obtained results it is concluded that the gas-solid flows in circulating fluidized beds are very complex, being necessary future research works for a better comprehension of the inherent physical phenomena to these flows.

O objetivo deste trabalho e a determinação dos coeficientes de viscosidade de cisalhamento e condutividade térmica de gases ideais quânticos. No cálculo dos coeficientes de transporte foram considerados dois aspectos: uma estatística quântica com seções transversais quânticas e uma estatística quântica com seções transversais quânticas. No primeiro caso, e utilizado um método alternativo para a determinação das aproximações sucessivas (ate a quinta ordem) para os coeficientes de transporte dos gases Helio 4, Helio 3, para-hidrogênio e orto-hidrogênio. No caso de estatística quântica desenvolvida uma teoria com base no método dos momentos de Grad e na equação de Uehling- Uhlenbeck, com a finalidade de determinar as express6es algébricas para os coeficientes de transporte. Esta teoria e baseada em 13 momentos, de densidade, velocidade, tensor pressão e fluxo de calor. Do conhecimento da função de distribuição em termos dos momentos, as equações constitutivas são determinadas e os coeficientes de transporte seguem de um método iterativo semelhante ao procedimento Maxwelliano. / The aim of this work is the determination of the coefficients of shear viscosity and thermal conductivity of quantum ideal gases. In the calculation of the transport coefficients two aspects have been taken into account: a classical statistical with quantum cross-sections and a quantum statistical with quantum cross-sections. In the first case, an alternative method is used for the determination of the successive approximations (up to the fifth order) to the transport coefficients of the gases helium 4, helium 3, para-hydrogen, ortho-hidrogen. In the case of quantum statistical a theory based on method of moments of Grad and on the Uehling-Uhlenbeck equation is developed in order to determine the algebraic expressions to the transport coefficients. This theory is based on 13 moments of density, velocity, pressure tensor and heat flux. From the knowledge of the distribution function the constitutive equations are determinate and the transport coefficients follow from an iterative method akin to the 11axwellian procedure.

A numerical model for inductive plasma wind tunnels is developed. This model provides the flow conditions at the edge of a boundary layer in front of a thermal protection material placed in the plasma jet stream at the outlet of an inductive torch. The governing equations for the hydrodynamic field are derided from the kinetic theory. The electromagnetic field is deduced from the Maxwell equations. The transport properties of partially ionized and unmagnetized plasma in weak thermal nonequilibrium are derived from the Boltzmann equation. A kinetic data base of transport collision integrals is given for the Martian atmosphere. Multicomponent transport algorithms based upon Krylov subspaces are compared to mixture rules in terms of accuracy and computational cost. The composition and thermodynamic properties in local thermodynamic<p>equilibrium are computed from the semi-classical statistical mechanics.<p>The electromagnetic and hydrodynamic fields of an inductive wind tunnel is presented. A total pressure measurement technique is thoroughly investigated by means of numerical simulations.<p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

Les propriétés de transport à haute température dans les gaz et/ou dans les plasmas ont une importance capitale dans différents domaines, à savoir dans le domaine de technologie de coupure à arc, plasmas de coupure, de soudure ou de gravure. La connaissance des coefficients de transport est nécessaire pour toute modélisation faisant intervenir les équations hydrodynamiques. Dans le cadre de la théorie cinétique des gaz dilués, une solution approchée de l’équation intégro-différentielle de Boltzmann régissant les fonctions de distribution a été proposée par Chapman-Enskog. Les coefficients de transport sont calculés classiquement par la méthode de Chapman-Enskog via les intégrales de collision. Dans le cadre de notre étude nous avons développé, dans un premier temps, un code numérique permettant l’obtention de ces intégrales de collision en tenant compte des singularités qui peuvent apparaître dans le calcul des sections efficaces relatives aux interactions entre les particules constituant les gaz et/ou les plasmas. Dans un second temps nous avons étudié l’influence du choix des paramètres des potentiels d’interaction sur les coefficients de transport. Par la suite, nous avons utilisé le code numérique ainsi développé pour évaluer les coefficients de transport du plasma d’hélium en étudiant l’influence du choix de la méthode de calcul de composition chimique sur ces coefficients. Enfin, un modèle simplifié d’une interaction entre une onde électromagnétique et un plasma d’hélium a été proposé comme une application directe des coefficients de transport. / Transport properties at high temperature in gases and/or in plasmas are of very importance in various fields, namely in the field of breaking technology in arc, cutting plasma, welding or burning. Knowledge of transport coefficients is necessary for any modeling involving hydrodynamic equations. As part of the kinetic theory of diluted gas, an approximate solution of the integro-differential Boltzmann equation governing distribution functions was proposed by Chapman-Enskog. Transport coefficients are classically computed using the method of Chapman-Enskog through the collision integrals. In our study we have developed, initially, a numerical code to obtain these collision integral taking into account the singularities that may occur in the calculation of the cross sections relating to interactions between particles forming the gas and/or plasmas. Secondly, we have studied the influence of the choice of parameters of interaction potentials on transport coefficients. Subsequently, we have used the numerical code developed for evaluating and helium plasma transport coefficients by studying the influence of the choice of method for calculating chemical composition on these coefficients. Finally, a simplified model of an interaction between an electromagnetic wave and a helium plasma has been proposed as a direct application of the transport coefficients.

Les travaux de cette thèse s'inscrivent dans le domaine des équations aux dérivées partielles et sont liés à la problématique des changements d'échelle dans le contexte de la cinétique des gaz. En effet, sachant qu'il existe plusieurs niveaux de description pour un gaz, on cherche à relier les différentes échelles associées dans un cadre où une part d'aléa intervient. Dans une première partie, on établit la dérivation rigoureuse de l'équation de Boltzmann linéaire sans cut-off en partant d'un système de particules interagissant via un potentiel à portée infinie en partant d'un équilibre perturbé.La deuxième partie traite du passage d'un modèle BGK stochastique avec champ fort à une loi de conservation scalaire avec forçage stochastique. D'abord, on établit l'existence d'une solution au modèle BGK considéré. Sous une hypothèse additionnelle, on prouve alors la convergence vers une formulation cinétique associée à la loi de conservation avec forçage stochastique.Au cours de la troisième partie, on quantifie dans le cas à vitesses discrètes le défaut de régularité dans les lemmes de moyenne et on établit un lemme de moyenne stochastique dans ce même cas. On applique alors le résultat au cadre de l'approximation de Rosseland pour établir la limite diffusive associée à ce modèle.Enfin, on s'intéresse à l'étude numérique du modèle de Uchiyama de particules carrées à quatre vitesses en dimension deux. Après avoir adapté les méthodes de simulation développées dans le cas des sphères dures, on effectue une étude statistique des limites à différentes échelles de ce modèle. On rejette alors l'hypothèse d'un mouvement Brownien fractionnaire comme limite diffusive / The work of this thesis belongs to the field of partial differential equations and is linked to the problematic of scale changes in the context of kinetic of gas. Indeed, knowing that there exists different scales of description for a gas, we want to link these different associated scales in a context where some randomness acts, in initial data and/or distributed on all the time interval. In a first part, we establish the rigorous derivation of the linear Boltzmann equation without cut-off starting from a particle system interacting via a potential of infinite range starting from a perturbed equilibrium. The second part deals with the passage from a stochastic BGK model with high-field scaling to a scalar conservation law with stochastic forcing. First, we establish the existence of a solution to the considered BGK model. Under an additional assumption, we prove then the convergence to a kinetic formulation associated to the conservation law with stochastic forcing. In the third part, first we quantify in the case of discrete velocities the defect of regularity in the averaging lemmas. Then, we establish a stochastic averaging lemma in that same case. We apply then the result to the context of Rosseland approximation to establish the diffusive limit associated to this model.Finally, we are interested into the numerical study of Uchiyama's model of square particles with four velocities in dimension two. After adapting the methods of simulation which were developed in the case of hard spheres, we carry out a statistical study of the limits at different scales of this model. We reject the hypothesis of a fractional Brownian motion as diffusive limit

Nous nous intéressons dans cette thèse à la théorie cinétique et aux systèmes de particules dans le cadre des équations de Boltzmann et Landau. Premièrement, nous étudions la dérivation des équations cinétiques comme des limites de champ moyen des systèmes de particules, en utilisant le concept de propagation du chaos. Plus précisément, nous étudions les probabilités chaotiques sur l'espace de phase de ces systèmes de particules : la sphère de Boltzmann, qui correspond à l'espace de phase d'un système de particules qui évolue conservant le moment et l'énergie ; et la sphère de Kac, correspondant à un système de particules qui conserve seulement l'énergie. Ensuite, nous nous intéressons à la propagation du chaos, avec des estimations quantitatives et uniforme en temps, pour les équations de Boltzmann et Landau. Deuxièmement, nous étudions le comportement asymptotique en temps grand des solutions de l'équation de Landau. / This thesis is concerned with kinetic theory and many-particle systems in the setting of Boltzmann and Landau equations. Firstly, we study the derivation of kinetic equation as mean field limits of many-particle systems, using the concept of propagation of chaos. More precisely, we study chaotic probabilities on the phase space of such particle systems : the Boltzmann's sphere, which corresponds to the phase space of a many-particle system undergoing a dynamics that conserves momentum and energy ; and the Kac's sphere, which corresponds to the energy conservation only. Then we are concerned with the propagation of chaos, with quantitative and uniform in time estimates, for Boltzmann and Landau equations. Secondly, we study the long-time behaviour of solutions to the Landau equation.

The fast-paced growth in microelectromechanical systems (MEMS), microfluidic fabrication, porous media applications, biomedical assemblies, space propulsion, and vacuum technology demands accurate and practical transport equations for rarefied gas flows. It is well-known that in rarefied situations, due to strong deviations from the continuum regime, traditional fluid models such as Navier-Stokes-Fourier (NSF) fail. The shortcoming of continuum models is rooted in nonequilibrium behavior of gas particles in miniaturized and/or low-pressure devices, where the Knudsen number (Kn) is sufficiently large. Since kinetic solutions are computationally very expensive, there has been a great desire to develop macroscopic transport equations for dilute gas flows, and as a result, several sets of extended equations are proposed for gas flow in nonequilibrium states. However, applications of many of these extended equations are limited due to their instabilities and/or the absence of suitable boundary conditions. In this work, we concentrate on regularized 13-moment (R13) equations, which are a set of macroscopic transport equations for flows in the transition regime, i.e., Kn≤1. The R13 system provides a stable set of equations in Super-Burnett order, with a great potential to be a powerful CFD tool for rarefied flow simulations at moderate Knudsen numbers. The goal of this research is to implement the R13 equations for problems of practical interest in arbitrary geometries. This is done by transformation of the R13 equations and boundary conditions into general curvilinear coordinate systems. Next steps include adaptation of the transformed equations in order to solve some of the popular test cases, i.e., shear-driven, force-driven, and temperature-driven flows in both planar and curved flow passages. It is shown that inexpensive analytical solutions of the R13 equations for the considered problems are comparable to expensive numerical solutions of the Boltzmann equation. The new results present a wide range of linear and nonlinear rarefaction effects which alter the classical flow patterns both in the bulk and near boundary regions. Among these, multiple Knudsen boundary layers (mechanocaloric heat flows) and their influence on mass and energy transfer must be highlighted. Furthermore, the phenomenon of temperature dip and Knudsen paradox in Poiseuille flow; Onsager's reciprocity relation, two-way flow pattern, and thermomolecular pressure difference in simultaneous Poiseuille and transpiration flows are described theoretically. Through comparisons it is shown that for Knudsen numbers up to 0.5 the compact R13 solutions exhibit a good agreement with expensive solutions of the Boltzmann equation.