Neste trabalho é apresentada uma nova modelagem matemática para a descrição do escoamento de um líquido incompressível através de um meio poroso rígido homogêneo e isotrópico, a partir do ponto de vista da Teoria Contínua de Misturas. O fenômeno é tratado como o movimento de uma mistura composta por três constituintes contínuos: o primeiro representando a matriz porosa, o segundo representando o líquido e o terceiro representando um gás de baixíssima densidade. O modelo proposto possibilita uma descrição matemática realista do fenômeno de transição insaturado/saturado a partir de uma combinação entre um sistema de equações diferenciais parciais e uma desigualdade. A desigualdade representa uma limitação geométrica oriunda da incompressibilidade do líquido e da rigidez do meio poroso. Alguns casos particulares são simulados e os resultados comparados com resultados clássicos, mostrando as consequências de não levar em conta as restrições inerentes ao problema. / This work is concerned with a new mathematical modelling for describing the flow of an incompressible fluid (a liquid) through a rigid, homogeneous and isotropic porous medium, from a Continuum Mixtures point of view. The phenomenon is regarded as the motion of a mixture composed by three overlaping continuous constituents: the first one representing the porous matrix, the second one representing the liquid and the third one representing a (very) low density gas. The proposed mathematical modelling allows a realistic mathematical description for the unsaturated/saturated transition process by means of a combination between a system of partial differential equations and an inequality. This inequality represents a geometrical constraint arising from the liquid incompressibility merged with the porous matrix rigidity. The simulation of some interesting particular cases is carried out presenting a comparison between the obtained results and the classical ones, showing the consequences of disregarding the constraints associated to the phenomenon.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-11T13:32:15Z No. of bitstreams: 1 ERIVALDO DINIZ DE LIMA - DISSERTAÇÃO PPGMAT 2015..pdf: 2368610 bytes, checksum: 3db3b8e45efcb83c955dae60371f8f4b (MD5) / Made available in DSpace on 2018-08-11T13:32:15Z (GMT). No. of bitstreams: 1 ERIVALDO DINIZ DE LIMA - DISSERTAÇÃO PPGMAT 2015..pdf: 2368610 bytes, checksum: 3db3b8e45efcb83c955dae60371f8f4b (MD5) Previous issue date: 2015-08 / Neste trabalho consideramos um sistema de leis de conservação proveniente da modelagem matemática de um escoamento bifásico unidimensional num meio poroso, preenchido de óleo e água com polímero dissolvido nela e levando em conta a adsorção de parte do polímero pela rocha. Usando a técnica das curvas de onda apresentamos a construção detalhada da solução do problema de Riemann para dados iniciais arbitrários no espaço de estados. Usamos a condição de entropia do per l viscoso para as ondas de choque com salto na concentração do polímero e a condição de Oleinik-Liu para os choques com concentração constante do polímero e salto na saturação da água / In this work we consider a system of conservation laws from the mathematical modeling of a one-dimensional two-phase flow in porous media, filled with oil and water with dissolved polymer in it and taking into account the adsorption of part of the polymer by the rock. Using the wave curves technique, we present a detailed construction of the Riemann problem solution for arbitrary initial data on the state space. We use the entropy condition of the viscous pro le for the shock waves with jumps in the polymer concentration and Oleynik-Liu condition for the shocks with constant concentration of polymer and jumps on the water saturation.

Neste trabalho é apresentada uma nova modelagem matemática para a descrição do escoamento de um líquido incompressível através de um meio poroso rígido homogêneo e isotrópico, a partir do ponto de vista da Teoria Contínua de Misturas. O fenômeno é tratado como o movimento de uma mistura composta por três constituintes contínuos: o primeiro representando a matriz porosa, o segundo representando o líquido e o terceiro representando um gás de baixíssima densidade. O modelo proposto possibilita uma descrição matemática realista do fenômeno de transição insaturado/saturado a partir de uma combinação entre um sistema de equações diferenciais parciais e uma desigualdade. A desigualdade representa uma limitação geométrica oriunda da incompressibilidade do líquido e da rigidez do meio poroso. Alguns casos particulares são simulados e os resultados comparados com resultados clássicos, mostrando as consequências de não levar em conta as restrições inerentes ao problema. / This work is concerned with a new mathematical modelling for describing the flow of an incompressible fluid (a liquid) through a rigid, homogeneous and isotropic porous medium, from a Continuum Mixtures point of view. The phenomenon is regarded as the motion of a mixture composed by three overlaping continuous constituents: the first one representing the porous matrix, the second one representing the liquid and the third one representing a (very) low density gas. The proposed mathematical modelling allows a realistic mathematical description for the unsaturated/saturated transition process by means of a combination between a system of partial differential equations and an inequality. This inequality represents a geometrical constraint arising from the liquid incompressibility merged with the porous matrix rigidity. The simulation of some interesting particular cases is carried out presenting a comparison between the obtained results and the classical ones, showing the consequences of disregarding the constraints associated to the phenomenon.

CAPES / Este trabalho traz um relato a respeito da Hipótese de Riemann, com o objetivo de tornar os conceitos referentes a esse problema acessíveis ao professor da educação básica, que pretenda abordá-los em sala de aula quando tratar de conteúdos a ele relacionados. A pesquisa foi inteira bibliográfica, apoiada em sua grande parte em textos de História da Matemática, tornando este trabalho divulgador dos problemas que ocupam parte das pesquisas matemáticas deste século, em especial da Hipótese de Riemann. / This study presents a report about the Riemann Hypothesis, leaving the underlying concepts behind this problem more accessible to a high school teacher. The literature review was based mainly on History of Mathematics texts. This research aims to study significant topics of mathematical research throughout this century, particularly to popularize the Riemann Hypothesis.

Les modèles aux moments angulaires constituent des descriptions intermédiaires entre les modèles cinétiques et les modèles fluides. Dans ce manuscrit, les modèles aux moments angulaires basés sur un principe de minimisation d'entropie sont étudiés pour des applications en physique des plasmas. Ce mémoire se découpe en trois parties. La première est une contribution à la modélisation en physique des plasmas à travers le formalisme des modèles aux moments angulaires. Dans celle-ci, le domaine de validité de ces modèles est étudié en régimes non-collisionels. Il est également montré que les opérateurs de collisions proposés pour le modèle M1 permettent de retrouver des coefficients de transport plasma précis. La deuxième partie de ce document concerne la dérivation de méthodes numériques pour l'étude du transport de particules en temps long. Dans ce cadre, des schémas numériques appropriés pour le modèle M1, préservant l'asymptotique, sont construits et validés numériquement. La troisième partie représente un premier pas significatif vers la modélisation multi-espèces. Ici, le modèle aux moments angulaire M1, construit dans un référentiel mobile, est appliqué à la dynamique des gaz raréfiés. Les propriétés de ce modèle sont détaillées, un schéma numérique est proposé et une validation numérique est menée. / Angular moments models represent alternative descriptions situated in between the kinetic and the fluid models. In this work, angular moments models based on an entropy minimisation principle are considered for plasma physics applications. This manuscript is organised in three parts. The first one is a contribution to plasma physics modelling within the formalism of angular moments models. The validity domain of angular moments models in collisionless regimes is studied. It is also shown that the collisional operators proposed for the M1 angular moments model enable to recover accurate plasma transport coefficients. The second part of this document deals with the derivation of numerical methods for the long timescales particle transport. Appropriate asymptotic-preserving numerical schemes are designed for the M1 angular moments model and numerical validations are performed. The third part represents a first important step toward multi-species modelling. The M1 angular moments model in a moving frame is introduced and applied to rarefied gas dynamics. The model properties are highlighted, a numerical scheme is proposed and a numerical validation is carried out.

Ce travail porte sur la simulation numérique des écoulements multiphasiques compressibles en déséquilibre de vitesses. Un solveur de Riemann diphasique de type HLLC, à la fois robuste, simple et précis est développé et validé à partir de solutions exactes et de données expérimentales. Cette méthode numérique est étendue au cas 3D non-structuré. Par ailleurs, la construction d’une technique numérique pour la répartition de l’énergie d’une onde de choc dans les différentes phases constituant le milieu est établie et permet le respect des conditions de choc multiphasiques. L’extension multiphasique du solveur de Riemann de type HLLC est réalisée, permettant ainsi la simulation d’une plus large gamme d’applications. Enfin, un modèle de transfert de chaleur et de masse dans un brouillard de gouttes ou nuage de bulles, en présence d’effets couplés de diffusion thermique et massiques, est proposé et dévoile des résultats intéressants. / This work deals with the numerical simulation of compressible multiphase flows in velocity disequilibrium. A HLLC-type two-phase Riemann solver is developed and validated against exact solutions and experimental data. This solver is robust, simple, accurate and entropy preserving. The numerical method is then implemented in 3D unstructured meshes. Furthermore, a numerical technique consisting in enforcing the correct energy partition at a discrete level in agreement with the multiphase shock relations is built. The multiphase extension of the HLLC-type Riemann solver is realized and allows the simulation of a wide range of applications. Finally, a droplet heat and mass transfer model with large range of validity is derived. It is valid in any situation: evaporation, flashing and condensation. It accounts for coupled heat and mass diffusion in the gas phase, thermodynamics of the multi-component gas mixture and heat diffusion inside the liquid droplet, enabling in this way consideration of both droplets heating and cooling phenomena.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-11T14:36:03Z No. of bitstreams: 1 edwinmarcosmaravipercca.pdf: 1012447 bytes, checksum: f4600cdbca54aedbb08335b949e92788 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-04-17T20:09:05Z (GMT) No. of bitstreams: 1 edwinmarcosmaravipercca.pdf: 1012447 bytes, checksum: f4600cdbca54aedbb08335b949e92788 (MD5) / Made available in DSpace on 2017-04-17T20:09:05Z (GMT). No. of bitstreams: 1 edwinmarcosmaravipercca.pdf: 1012447 bytes, checksum: f4600cdbca54aedbb08335b949e92788 (MD5) Previous issue date: 2017-02-20 / As leis de balanço expressam de uma maneira mais geral as leis de conservação e, portanto, é natural que coincidam em algumas definições ou resultados que vamos mostrar aqui. Um sistema de leis de conservação estritamente hiperbólico numa dimensão espacial sob certas condições é um sistema simetrizável, portanto, possui uma entropia convexa. Isto induz a definiroparentropia-fluxodeentropiaeaproduçãodeentropia,ingredientesmínimospara usar o critério de admissibilidade da taxa de entropia e conferir se a solução do problema de Riemann respectivo é ótimo. A taxa de entropia definida aqui em termos da entropia é um funcional que pode ser minimizada nos leques de ondas com estados constantes do problema de Riemann, usando as equações de Euler-Lagrange. Primeiramente, mostramos que as soluções do problema de Riemann são funções de variação limitada, resultando num método variacional para resolver o problema. Neste trabalho será mostrado que a solução obtida pelo método variacional, coincide com a solução obtida pelo método das curvas caraterísticas. / The balance laws express in a more general way the conservation laws and therefore it is naturalthattheycoincideinsomedefinitionsorresultsthatwewillshowhere. Thestrictly hyperbolic systems of conservation laws in a spatial dimension under certain conditions is a symmetrizable system, therefore it has a convex entropy. This induces to define the entropy-entropy flux pair and the entropy production, minimum ingredients to use the Entropy rate admissibility criterion and check whether the solution of the respective Riemann problem is optimal. The entropy rate defined here in terms of entropy is a functional that can be minimized in the wave fans with constant states of the Riemann problem using the Euler-Lagrange equations, we show that the solutions of the Riemann problem are functions of bounded variation, resulting in a variational method to solve the respective problem. In this work it will be shown that the solution obtained by the variational method, coincides with the solution obtained by the method of characteristics.

L'équation de Weinstein est une équation régissant les Potentiels à Symétrie Axiale (PSA) qui est $L_m[u]=Delta u+(m/x)partial_x u=0$, $minmathbb{C}$. On généralise des résultats connus pour $min mathbb{R}$ au cas $minmathbb{C}$. On donne des expressions de solutions fondamentales des opérateurs $L_m[u]$ et leurs estimations, on démontre une formule de Green pour les PSA dans le demi-plan droit $mathbb{H}^+$ pour Re $m< 1$. On prouve un nouveau théorème de décomposition des PSA dans des anneaux quelconques pour $minmathbb{C}$ et dans une géométrie annulaire particulière utilisant les coordonnées bipolaires, on prouve qu'une famille de solutions des PSA en termes de fonctions de Legendre Associées de 1re et 2de espèce est complète, on montre lorsque $min mathbb{R}$ que celle-ci est une base de Riesz.Dans la 2e partie, par une méthode qui est due à A. S. Fokas, on donne des formules des PSA dans un disque de $mathbb{H}^+$, avec $minmathbb{Z}$. Ces représentations sont obtenues par la résolution d'un problème de Riemann-Hilbert sur $mathbb{C}$ ou sur une surface de Riemann à deux feuillets.Dans la 3e partie, on étudie les fonctions pseudo-holomorphes, {it i. e.} les solutions de l'équation $overline{partial} w=alphaoverline{w}$, $alphain L^r$, $2leq r<infty$. Une nouvelle extension de la régularité du principe de similarité et une réciproque de celui-ci qui conduit à un paramétrage analytique de ces fonctions dans le cas critique $r=2$ ont été obtenues. On résoud un problème de Dirichlet à données $L^p$ pondérées sur des domaines lisses pour des équations du type conductivité à coefficient dont le log appartient à l'espace de Sobolev $W^{1,2}$. / The Weinstein equation with complex coefficients is the equation governing axisymmetric potentials (PSA) which can be written as $L_m[u]=Delta u+left(m/xright)partial_x u =0$, where $minmathbb{C}$. We generalize results known for $minmathbb{R}$ to $minmathbb{C}$. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for PSA in the right half-plane $mathbb{H}^+$ for Re $m<1$. We establish a new decomposition theorem for the PSA in any annular domains for $minmathbb{C}$. In particular, using bipolar coordinates, we prove for annuli that a family of solutions for PSA equation in terms of associated Legendre functions of first and second kind is complete. For $minmathbb{R}$, we show that this family is even a Riesz basis in some non-concentric circular annulus. In the second part, basing on a method due to A. S. Fokas, we give formulas for PSA in a circular domain of $mathbb{H}^+$ when $m$ is an integer. These representations are obtained by solving a Riemann-Hilbert problem on the complex plane or on a Riemann surface with two sheets according to the parity of $m$.In the last part, we study the pseudo-holomorphic functions, i.e. solutions of the complex equation $overline{partial} w=alpha overline{w}$, with $alphain L^r$, $2leq r<infty$. We extend the Bers similarity principle and a converse of this principle to the critical regularity case $r=2$. We establish well-posedness of Dirichlet problem in smooth domains with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in the Sobolev space $W^{1,2}$.

In this paper I present a theoretical analysis (genetic decomposition) of the cognitive constructions for the concept of infinite Riemann sums following Piaget\''s model of epistemology. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my continual observations of students in the process of learning. Based on this analysis I plan to suggest instructional procedures that motivate the mental activities described in the proposed genetic decomposition. In a later study, I plan to present empirical data in the form of informal interviews with students at different stages of learning. The analysis of those interviews may suggest a review of my initial genetic decomposition.

The discourse presented here is aimed at examining the justification of applications of current analysis to real world problems.