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Tese de Doutoramento em Matemática Aplicada, O principal objetivo desta tese é o estudo de misturas gasosas multicomponentes – sistemas gasosos com três ou mais espécies misturadas ao nível molecular – e a dedução de modelos macroscópicos da mecânica de fluidos a partir de uma descrição cinética das partículas que constituem a mistura. Mais especificamente, consideram-se equações cinéticas de tipo Boltzmann que descrevem a evolução de misturas quimicamente reativas, com estrutura monatómica ou poliatómica, que participam numa reação química reversível de tipo bi-molecular e deduzem-se, no limite hidrodinâmico, modelos macroscópicos que descrevem como as espécies se propagam no espaço devido ao processo de difusão e como se transformam quimicamente devido ao processo de reação. Para obter estes modelos macroscópicos, diferentes regimes de adimensionamento das equações cinéticas são considerados, correspondendo a diferentes regimes químicos. O primeiro regime descreve um estado físico em que as colisões mecânicas são dominantes relativamente à reação química, enquanto o segundo regime corresponde a um estado em que as colisões mecânicas e reativas têm a mesma relevância na evolução. No primeiro regime, obtémse, como limite hidrodinâmico das equações cinéticas, um sistema de reação-difusão em que o processo de difusão é modelado pelas equações de Maxwell-Stefan. No segundo regime, obtêm-se as equações de Maxwell-Stefan reativas, com termos adicionais que descrevem a difusão das espécies reativas. As propriedades matemáticas dos modelos cinéticos considerados, relacionadas com as leis de conservação, o equilíbrio termodinâmico, o teorema H e as taxas de produção química, que são fundamentais para a análise desenvolvida nesta tese, são estabelecidas e estudadas rigorosamente., The purpose of this thesis is to study multicomponent gaseous mixtures – systems composed of three or more species mixed at a molecular level – and derive macroscopic models of fluid mechanics from a kinetic description of particles that constitute the mixture. Specifically, this thesis considers Boltzmann-type kinetic equations that model the evolution of reactive monatomic and polyatomic gaseous mixtures participating in a bi-molecular reversible chemical reaction and derive, in the hydrodynamic limit, macroscopic models that describe how the species spread in space due to diffusion and how the species are transformed into each other as a result of the chemical reaction. To obtain these macroscopic models, different scalings of the Boltzmann-type kinetic equations, which correspond to different chemical regimes, are considered. The first regime describes a physical situation in which mechanical collisions are dominant with respect to chemical reaction, whereas the other regime corresponds to a situation in which mechanical and reactive collisions are of the same relevance in the evolution. In the former regime, reaction-diffusion systems where the diffusion process is modelled by the Maxwell-Stefan equations are obtained as hydrodynamic limits of the scaled kinetic equations, while in the latter regime, the reactive Maxwell-Stefan equations, with additional terms describing the diffusion of the reactive species, are obtained. Moreover, the mathematical properties of the considered kinetic models related to the conservation laws, thermodynamic equilibrium, H-theorem, and chemical production rates, which are fundamental for the analysis developed in this thesis, are stated and studied in detail., This thesis was supported by Centro de Matemática, Universidade do Minho, Fundação para a Ciência e a Tecnologia (FCT) through the PhD grant PD/BD/128188/2016, and Program Pessoa of Cooperation between Portugal and France reference 406/4/4/2017/S.

International audience; We discuss transport coefficients in weakly ionized mixtures. We investigate the situations of weak and strong magnetic fields as well as electron temperature nonequilibrium. We present in each regime the Boltzmann equations, examples of transport fluxes, the structure of transport linear systems and discuss their solution by efficient iterative techniques. Numerical simulations are presented for partially ionized high-temperature air.

We address the problem of recollisions in cooling granular gases. To this aim, we dynamically construct the interaction network in a granular gas, using the sequence of collisions collected in an event driven simulation of inelastic hard disks from time 0 until time t . The network is decomposed into its k -core structure: particles in a core of index k have collided at least k times with other particles in the same core. The difference between cores k + 1 and k is the so-called k -shell, and the set of all shells is a complete and nonoverlapping decomposition of the system. Because of energy dissipation, the gas cools down: its initial spatially homogeneous dynamics, characterized by the Haff law, i.e., a t − 2 energy decay, is unstable toward a strongly inhomogeneous phase with clusters and vortices, where energy decays as t − 1 . The clear transition between those two phases appears in the evolution of the k -shells structure in the collision network. In the homogeneous state the k -shell structure evolves as in a growing network with a fixed number of vertices and randomly added links: the shell distribution is strongly peaked around the most populated shell, which has an index k max ∼ 0.9 ⟨ d ⟩ with ⟨ d ⟩ the average number of collisions experienced by a particle. During the final nonhomogeneous state a growing fraction of collisions is concentrated in small, almost closed, communities of particles: k max is no more linear in ⟨ d ⟩ and the distribution of shells becomes extremely large developing a power-law tail ∼ k − 3 for high shell indexes. We conclude proposing a simple algorithm to build a correlated random network that reproduces, with few essential ingredients, the whole observed phenomenology, including the t − 1 energy decay. It consists of two kinds of collisions (links): single random collisions with any other particle and long chains of recollisions with only previously encountered particles. The algorithm disregards the exact spatial arrangement of clusters, suggesting that the observed stringlike structures are not essential to determine the statistics of recollisions and the energy decay. Fil: Alvarez Hamelin, José Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina Fil: Puglisi, Andrea. Università La Sapienza. Dipartimento di Fisica; Italia

Test particles interact with a medium by means of a bimolecular reversible chemical reaction. Two species are assumed to be much more numerous so that they are distributed according fixed distributions: Maxwellians and Dirac's deltas. Equilibrium and its stability are investigated in the first case. For the second case, a system is constructed, in view of an approximate solution., 5 pages, 1 figure, RevTeX4; contribution to the international conference "Next Sigma Phi" on News, EXpectations, and Trends in statistical physics, Crete 2005