Made available in DSpace on 2016-12-23T14:08:13Z (GMT). No. of bitstreams: 1 Dissertacao de Felipe Patricio das Neves.pdf: 512132 bytes, checksum: d42e56e29214989211674c822342bbea (MD5) Previous issue date: 2009-12-04 / In this work is implemented a numerical model to simulate computationally the distribution of pressures, velocities, temperatures and heat flows in two-dimensional stationary control volumes. The relation between temperatures and velocities is established by the advective-diffusive Equation, using the Dual Reciprocity Boundary Element Method formulation... / Neste trabalho é desenvolvido um modelo numérico para simular computacionalmente a distribuição de pressões, velocidades, temperaturas e fluxos de calor estacionários em volumes de controle bidimensionais. A relação do campo de temperaturas e velocidades é governada pela equação da Difusão-Advecção, resolvida através da formulação com Dupla Reciprocidade do Método dos Elementos de Contorno. Admite-se a lei de Darcy para associar pressão e velocidade, resultando num modelo matemático dado pela Equação de Laplace, no caso linear. Na análise não-linear insere-se a dependência entre do campo de velocidades e as temperaturas, resultando num campo matematicamente representado pela Equação de Poisson. Os resultados da solução desse problema são então implementados no modelo difusivo-advectivo, gerando temperaturas e fluxos de calor

Cette thèse est consacrée à l’étude des modèles microscopiques pour la dérivation de la conduction de la chaleur. Démontrer rigoureusement une équation diffusive macroscopique à partir d’une description microscopique du système est à aujourd’hui encore un problème ouvert. On étudie un système décrit par l’équation de Schrödinger linéaire discrète (DLS) en dim 1, perturbé par une dynamique stochastique conservative. On peut montrer que le système a une limite hydrodynamique donnée par la solution de l’équation de la chaleur. Quand le système est rattaché aux bords à deux réservoirs de Langevin à deux différents potentiels chimiques, on peut montrer que l’état stationnaire, dans la limite vers l'infinie, satisfait la loi de Fourier. On étudie une chaine des oscillateurs anharmonique immergée en un réservoir de chaleur avec un gradient de température. On exerce une tension, variable dans le temps, à une des deux extrémités de la chaine, et l’autre reste fixe. On montre que sous un changement d’échelle diffusive dans l’espace et dans le temps, la distribution d’étirement de la chaine évolue selon un équation diffusive non-linéaire. On développe des estimations qui reposent sur l’hypocoercitivité entropique. La limite macroscopique peut être utilisée pour modéliser les transformations thermodynamique isothermiques entre états stationnaire de non-équilibre. / The object of research of this thesis is the derivation of heat equation from the underlying microscopic dynamics of the system. Two main models have been studied: a microscopic system described by the discrete Schrödinger equation and an anharmonic chain of oscillators in presence of a gradient of temperature. The first model considered is the one-dimensional discrete linear Schrödinger (DLS) equation perturbed by a conservative stochastic dynamics, that changes the phase of each particles, conserving the total norm (or number of particles). The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. It has been shown that the system has a hydrodynamical limit given by the solution of the heat equation. When it is coupled at the boundaries to two Langevin thermostats at two different chemical potentials, it has been proven that the stationary state, in the limit to infinity, satisfies the Fourier’s law. The second model considered is a chain of anharmonic oscillators immersed in a heat bath with a temperature gradient and a time varying tension applied to one end of the chain while the other side is fixed to a point. We prove that under diffusive space-time rescaling the volume strain distribution of the chain evolves following a non-linear diffusive equation. The stationary states of the dynamics are of non-equilibrium and have a positive entropy production, so the classical relative entropy methods cannot be used. We develop new estimates based on entropic hypocoercivity, that allows to control the distribution of the positions configurations of the chain. The macroscopic limit can be used to model isothermal thermodynamic transformations between non-equilibrium stationary states. CEMRACS project on simulating Rayleigh- Taylor and Richtmyer-Meshkov turbulent mixing zones with a probability density function method at last.