Submitted by Jeane Batista de Carvalho null (carvalhojeanetiagina@yahoo.com.br) on 2017-05-23T15:08:27Z No. of bitstreams: 1 Dissertação_CarvalhoJeane.pdf: 8189158 bytes, checksum: ebb9bc23a7d48fbc4154db51680f263e (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-05-23T18:27:50Z (GMT) No. of bitstreams: 1 carvalho_jb_me_ilha.pdf: 8189158 bytes, checksum: ebb9bc23a7d48fbc4154db51680f263e (MD5) / Made available in DSpace on 2017-05-23T18:27:50Z (GMT). No. of bitstreams: 1 carvalho_jb_me_ilha.pdf: 8189158 bytes, checksum: ebb9bc23a7d48fbc4154db51680f263e (MD5) Previous issue date: 2017-03-24 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Nas últimas décadas, há uma crescente preocupação em mensurar os parâmetros dinâmicos do sangue, dadas as imensas perdas de vidas por doenças cardiovasculares sendo, muitas delas, por aneurismas. A formação e desenvolvimento de um aneurisma é, predominantemente, degenerativo e provém de uma complexa interação entre os efeitos biológicos da parede arterial, os estímulos de escoamento e tensões provenientes da hemodinâmica. A tensão cisalhante na parede e o cíclico campo de pressão são um dos principais fatores responsáveis pela formação e crescimento de um aneurisma. Logo, há a necessidade de se conhecer os campos de velocidades e pressão além das tensões cisalhantes e efetivas na parede. Uma análise numérica é mais promissora que uma experimental. Uma análise experimental in-vivo possui impasses éticos e morais, sem contar a necessidade de um grande investimento. Outra vantagem de um estudo numérico é a disponibilidade de softwares livres de extração de tomografias que permite a extração da geometria sem a necessidade de um método invasivo que ocorreria em estudo experimental in vivo. Portanto com o auxílio de simulações numéricas (Ansys®), considerando o efeito multi-física de interação fluido estrutura (FSI) pela metodologia de elementos e volumes finitos foi possível verificar o efeito dos fatores que levam a formação e crescimento de aneurisma na aorta abdominal. Os aneurismas estudados foram modelos geométricos e reais sendo um dos reais obtidos através de imagens DICOM. / In recent decades, there is growing concern in measuring the dynamic parameters of the blood, given the immense loss of life from cardiovascular disease in human history, including, aneurysms. The formation and development of an aneurysm is predominantly degenerative and results from a complex interaction between the biological effects of the arterial wall and the flow and stress effects from hemodynamics. The stress in the wall and in the cyclic field of pressure is one of the main factors for the formation and growth of an aneurysm that degenerates until its rupture. Therefore, it is necessary to know the velocity and pressure fields as well as the shear stress and effective stress on the wall. A numerical analysis is more promising than an experimental one since an in-vivo experimental analysis has ethical and moral impasses, not counting the need for a large investment. Another advantage of a numerical study is the availability of free softwares for tomography analysis that allows the extraction of the geometry without the need for an invasive method that would occur in an in vivo experimental study. Therefore, with the aid of numerical simulations (Ansys®), considering the multi-physical effect of fluid structure interaction (FSI) by the methodology of finite elements and finite volumes it was possible to verify the effect of factors that lead to the formation and growth of abdominal aortic aneurysm. The studied aneurysms came from geometric models and from real examples, being one of the real ones obtained through DICOM images. / CNPq: 131153/2015-3

Cette thèse présente quelques contributions à l'étude qualitative de solutions d'équations aux dérivées partielles non linéaires dans des modèles issus de l'optique et de la mécanique des fluides. Nous nous intéressons plus précisément à l'existence de solutions et à leur stabilité temporelle. Le Chapitre 1 est consacré à l'équation de Lugiato-Lefever, qui est une variante de l'équation de Schrödinger non linéaire et qui a été dérivée dans plusieurs contextes en optique. En utilisant des outils de la théorie des bifurcations et des formes normales, nous procédons à une étude systématique des solutions stationnaires de cette équation, et prouvons l'existence de solutions périodiques et localisées. Dans le Chapitre 2, nous présentons un critère simple d'instabilité linéaire pour des ondes non linéaires. Nous appliquons ce résultat aux équations de Lugiato-Lefever, de Kadomtsev-Petviashvili-I et de Davey-Stewartson. Ces deux dernières équations sont des équations modèles dérivées en mécanique des fluides. Dans le Chapitre 3, nous montrons un critère d'instabilité linéaire pour des solutions périodiques de petite amplitude, par rapport à certaines perturbations quasipériodiques. Ce résultat est ensuite appliqué à l'équation de Lugiato-Lefever. / In this thesis we present several contributions to qualitative study of solutions of nonlinear partial differential equations in optics and fluid mechanics models. More precisely, we focus on the existence of solutions and their stability properties. In Chapter 1, we study the Lugiato-lefever equation, which is a variant of the nonlinear Schrödinger equation arising in sereval contexts in nonlinear optics. Using tools from bifurcation and normal forms theory, we perfom a systematic analysis of stationary solutions of this equation and prove the existence of periodic and localized solutions. In Chapter 2, we present a simple criterion for linear instability of nonlinear waves. We then apply this result to the Lugiato-Lefever equation, to the Kadomtsev-Petviashvili-I equation and the Davey-Stewartson equations. These last two equations are model equations arising in fluid mechanics. In Chapter 3, we prove a criterion for linear instability of periodic solutions with small amplitude, with respect to certain quasiperiodic perturbations. This result is then applied to the Lugiato-Lefever equation.

Introdução – Pouco se publicou na literatura médica a respeito de projeções angiográficas dedicadas às anatomias especiais. Neste cenário, propomos uma projeção angiográfica para a exposição da bifurcação do tronco da artéria coronária esquerda (TCE) quando o conjunto de projeções rotineiras previamente utilizadas foi ineficaz para exibir com clareza essa região. Pacientes e Métodos – 84 pacientes foram submetidos à projeção Lateral Direita (90-120°) associada a Cranial (30-40°). Estudo de reprodutibilidade, realizado com a participação de dois observadores independentes, julgou a eficácia da projeção proposta. O índice kappa, ajustado para a prevalência e viés (PABAK) com IC de 95%, foi utilizado para demonstrar a intensidade da concordância inter e intraobservador: ≤ 0,20 = ruim, 0,21 – 0,40 = fraca, 0,41 – 0,60 = moderada, 0,61 – 0,80 = substancial e 0,81 – 1,0 = quase perfeita. Resultados – A projeção proposta foi eficaz em 79% dos angiografias com concordância de 0,76 (0,6 – 0,9; P ≤ 0,001). A origem e o segmento proximal da artéria coronária descendente anterior foram expostos em 89% com concordância de 0,86 (0,7 – 1,0; P ≤ 0,001), a origem e o segmento proximal da artéria circunflexa foram expostos em 83% com concordância de 0,72 (0,5 – 1,0; p ≤ 0,001) e a origem e o segmento proximal do ramo intermediário, quando presente, foram expostos em 89% com concordância de 0,79 (0,6 – 1,0; p ≤ 0,001). Conclusão – A projeção proposta é eficaz, segura e reproduzível. Em situações especiais, onde projeções rotineiras falham, essa poderá exibir detalhes importantes da anatomia da bifurcação do TCE à cinecoronariografia ou ser a projeção de trabalho durante a intervenção coronariana percutânea. / Objectives – We propose a 90 – 120° right lateral with 30-40° cranial angiographic view to expose the bifurcation of the left main coronary artery (LMCA) when previously used routine projections were inefficient at clearly showing this region. Background – Little has been published in the medical literature regarding angiographic projections dedicated to special anatomies. Methods – A total of 84 patients were subjected to the proposed projections. A reproducibility study, conducted with the participation of two independent observers, judged the effectiveness of the proposed projection. The Prevalence and Bias Adjusted Kappa (PABAK) index, with a 95% confidence interval (CI), was used to demonstrate the intensity of intra- and inter-observer agreement: ≤ 0.20 = poor, 0.21 – 0.40 = slight, 0.41 – 0.60 = moderate, 0.61 – 0.80 = substantial and 0.81 – 1.0 = almost perfect. Results – The proposed projection was efficient in 79% of the angiograms, with agreement of 0.76 (0.6 – 0.9; P ≤ 0.001). The origin and the proximal segment of the anterior descending coronary artery were exposed in 89% of the angiograms, with agreement of 0.86 (0.7 – 1.0; P ≤ 0.001); the origin and the proximal segment of the circumflex artery were exposed in 83% of the angiograms, with agreement of 0.72 (0.5 – 1.0; P ≤ 0.001); and the origin and the proximal segment of the intermediate branch, when present, were exposed in 89% of the angiograms, with agreement of 0.79 (0.6 – 1.0; P ≤ 0.001). Conclusion – The proposed projection is effective, safe and reproducible. In special situations where routine projections fail, this proposed projection can reveal important details of the anatomy of the bifurcation of the LMCA during conventional / Tese (Doutorado)

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-06-14T11:12:47Z No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-06-15T10:25:16Z (GMT) No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-06-15T10:25:16Z (GMT). No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-05-18 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work we exhibit canonical forms for 2D codimension one piecewise smooth vector Fields (PSVF). All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. Also we present topological distinct canonical forms for a larger class for symmetric PSVF where the set of fixed points is contained in the variety os discontinuity. Finally we analyze the simultaneous occurrence of sliding and crossing limit cycle in the case where the piecewise linear vector fields presents a continuum of periodic orbits. / Neste trabalho exibiremos inicialmente as formas canônicas para campos vetoriais suaves por partes (PSVF) no plano. Todas os possíveis cenários de codimensão um são abordados. Também apresentamos formas canônicas topologicamente distintas para uma classe de PSVF com simetria onde o conjunto de pontos fixos está contido na variedade de descontinuidade. Finalmente, analisaremos a ocorrência simultânea de ciclos limite costurantes e deslizantes no caso linear por partes que apresentam um contínuo de órbitas periódicas.

Este trabalho apresenta um estudo sobre o método da continuação aplicado ao problema de fluxo de potência. Definições e conceitos de estabilidade de tensão são descritos de forma a explicitar as diferenças e semelhanças existentes com relação ao estudo de máximo carregamento em sistemas de energia elétrica. Uma síntese da teoria da bifurcação aborda sua importância em estudos de colapso de tensão. É proposta uma técnica de controle do tamanho do passo para o método da continuação com o objetivo de determinar o ponto de máximo carregamento (PMC) sem a necessidade de especificar, por meio do usuário, um valor para o tamanho inicial do passo. Os resultados dos estudos realizados em sistemas testes do IEEE (14, 30, 57 e 118 barras) mostram a aplicação do método da continuação convencional e de sua associação à técnica de controle do tamanho do passo. / This work presents a research about the continuation method applied to the power flow problem. Voltage stability definitions and concepts are described in a way to highlight and point out the differences and the similarities among several methods used to determine the maximum loading of electrical power systems. A short description of the bifurcation theory is also presented in order to show its importance to the voltage collapse studies. A technique based on automatically controlling the step size is proposed as an innovation of the continuation method. The objective of this technique is to determine the maximum loading point without the traditional need of asking the user for the initial step size. The results compare the performance between the conventional and the new method. These methods are analyzed using IEEE test systems (14, 30, 57 and 118-bus).

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-05-20T18:30:38Z No. of bitstreams: 2 Dissertação - Lucyjane de Almeida Silva - 2015.pdf: 1357945 bytes, checksum: c6e7c0e30627101c90a6eb4ae00a5c4d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-05-20T18:32:39Z (GMT) No. of bitstreams: 2 Dissertação - Lucyjane de Almeida Silva - 2015.pdf: 1357945 bytes, checksum: c6e7c0e30627101c90a6eb4ae00a5c4d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-20T18:32:39Z (GMT). No. of bitstreams: 2 Dissertação - Lucyjane de Almeida Silva - 2015.pdf: 1357945 bytes, checksum: c6e7c0e30627101c90a6eb4ae00a5c4d (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the global qualitative behavior of three predator-prey models. We analyze the existence of limit cycle and canard cycle and we investigate the kinds of bifurcation that can occur. In the first model, Gause predator-prey with a refuge, we analyze the effects of a prey refuge on the ecosystem qualitative behavior. Employing the carrying capacity of the prey population in the Gause Model with a refuge we obtain the second model, for which we analyze the effects of the carrying capacity and we compare the results. In the third model we consider the continuous threshold harvesting strategies ocurring when the predator density is above a certain threshold. We note that the model has a complex dynamics with multiple internal equilibria and different types of bifurcation. / Neste trabalho estudamos o comportamento qualitativo global de três modelos predadorpresa. Analisamos a existência de ciclos limite e ciclos de canard e investigamos os tipos de bifurcações que podem ocorrer. No primeiro modelo, modelo predador-presa de Gause com refúgio, analisamos os efeitos do refúgio para as presas no comportamento dinâmico do ecossistema. Empregando a capacidade de suporte para a população de presas no modelo de Gause com refúgio obtemos o segundo modelo, para o qual analisamos os efeitos da capacidade de suporte e comparamos os resultados obtidos. No terceiro modelo consideramos as estratégias de colheita com limiar contínuo que é aplicada quando a densidade de predadores está acima de um certo limite e investigamos o comportamento dinâmico global. Observamos que o modelo possui uma dinâmica complexa com múltiplos pontos de equilíbrio e diferentes tipos de bifurcações.

Let f : \'K POT. \' be a \'C POT. 2\' semi-algebraic mapping for K = R and a polynomial mapping for K = C. It is well-known that f is a locally trivial topological fibration over the complement of the bifurcation set B(f), also called atypical set. In this work, we consider the notion of t-regularity and \'ho E\'-regularity to study the bifurcation set of semi-algebraic mappings f : \'R POT. n\' \'ARROW\' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\'. We show that t-regularity is equivalent to regularity conditions at infinity which have been used by Rabier (1997), Gaffney (1999), Kurdyka, Orro and Simon (2000) and Jelonek (2003) in order to control the asymptotic behaviour of mappings. In addition, we prove that t-regularity implies \'ho E\'-regularity. The \'ho E\'-regularity enables one to define the set of asymptotic non \'ho E\'-regular values S(f) \'This contained\' \' K POT. p\', and the set \'A IND. \'ho E\'\' := f(Singf) U S(f). For \'C POT. 2\' semi-algebraic mappings f : \'R POT. n\' ARROW \' \'R POT. p\' and polynomial mappings f : \'C POT. n\' \'ARROW\' \'C POT. p\', based on a partial Thom stratification at infinity, we rove that S(f) and \'A IND. ho E\' are closed real semi-algebraic sets of dimension at most p - 1 (real dimension at most 2p - 2, for f : \'C POT. n\' \'ARROW\' \'C POT. p\'). Moreover, based on a new fibration theorem at infinity, i.e. holding in the complement of a sufficiently large ball, we obtain B(f) \'this contained\' \'A IND. ho E\'. We study two special classes of polynomial mappings f : \'R POT. n\' \"ARROW\' \'R POT. p\', the class of fair polynomial mappings and the class of Newton non-degenerate polynomial mappings. For fair polynomial mappings, we give an interpretation of t-regularity in terms of integral closure of modules, which is a real counterpart of Gaffney\'s result (1999). For non-degenerate polynomial mappings, we obtain an approximation for B(f) through a set which depends on the Newton polyhedron of f (results like this have been obtained by Némethi and Zaharia (1990) for polynomial functions f : \'C POT. n\' \'ARROW\' C and recently for mixed polynomial functions by Chen and Tibar (2012)). To finish, we discuss some simple consequences of our work: the equivalence t regularity Rabier (equivalently Gaffney, Kuo-KOS, Jelonek) condition for mappings f : X \'ARROW\' \'K POT. p\', where X \'this contained\' \'K POT. n\' is a smooth ane variety; the problem of bijectivity of semi-algebraic mappings; and a formula to compute the Euler characteristic of regular fibres of polynomial mappings f : \'R POT. n\' \'AROOW\' \'R POT. n-1\'. The above results are also extensions of some results obtained, for polynomial functions f : \'K POT. n\' \'ARROW K, by Némethi and Zaharia (1990), Siersma and Tibar (1995), Paunescu and Zaharia (1997), Parusinski (1995) and Tibar (1998). Title: Regularity at infinity and global fibrations of real algebraic maps / Considere f : \'K POT. n\' \"SETA\' \'K POT. p\' uma aplicação semi-algébrica de classe \'C POT. 2\' para K = R e uma aplicação polinomial para K = C. Por resultados clássicos, sabe-se que f é uma fibração topologicamente trivial sobre o complementar dos valores de bifurcação B(f), também chamado de valores atípicos. Neste trabalho, consideramos a t-regularidade e a \'ho E\'-regularidade no estudo dos valores de bifurcação de aplicações semi-algébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. n\' \'SETA\' \'C POT. p\'. Mostramos que t-regularidade é equivalente às condições de regularidade no infinito usadas por Rabier (1997), Gaffney (1999), Kurdyka, Orro e Simon (2000) e Jelonek (2003) no controle do comportamento assintótico de aplicações. Também mostramos que t-regularidade implica \'ho E\'-regularidade. Através da \'ho E\'-regularidade, definimos o conjunto dos valores assintóticos não \'ho E\'- regulares S(f) \'K POT. p\', e o conjunto \'A IND. ho E\' : = f(Singf) U S(f). Para aplicações semialgébricas f : \'R POT. n\' \'SETA\' \'R POT. p\' de classe \'C POT. 2\' e aplicações polinomiais f : \'C POT. \' \'SETA\' \'C POT. p\', baseados na existência de uma estraticação parcial de Thom no infinito, provamos que S(f) e \'A IND. ho E\' são conjuntos semi-algébricos reais de dimensão no máximo p - 1 (dimensão real no máximo 2p 2, para f : \'C POT. \' \'SETA\' \' C POT. p\'). Além disso, baseados em um novo teorema de fibração no infinito, ou seja na existência de fibração no complementar de uma bola de raio suficientemente grande, obtemos que o conjunto de bifurcação B(f) está contido no conjunto \'A IND. ho E\'. Estudamos também duas classes de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. p\', a classe de aplicações polinomiais fair e a classe de aplicações Newton não degeneradas. Para aplicações polinomiais fair, obtemos uma interpretação da t-regularidade em termos da teoria de fecho integral de módulos, estendendo para o caso real os resultados de Gaffney (1999). Para aplicações não degeneradas, obtemos uma aproximação de B(f) através de um conjunto que depende do poliedro de Newton de f (resultados deste tipo foram obtidos por Némethi e Zaharia (1990) para funções polinomiais f : \'C POT. \' \'SETA\' C e recentemente para funções polinomiais mistas por Chen e Tibar (2012)). No final, discutimos algumas consequências simples do nosso trabalho: a equivalência t-regularidade condição de Rabier (equivalentemente Gaffney, Kuo-KOS, Jelonek) para aplicações f : X \'SETA\' \'K POT. p\', onde X \'está contido\' \'K POT. n\' é uma variedade suave afim; o problema de bijetividade de aplicações semi-algébricas; e uma fórmula para o cálculo da característica de Euler de fibras regulares de aplicações polinomiais f : \'R POT. n\' \'SETA\' \'R POT. n-1\'. Os resultados acima também são extensões de alguns resultados obtidos para funções polinomiais f : \'K POT. n\' \'SETA\' K, por Némethi e Zaharia (1990), Siersma e Tibar (1995), Paunescu e Zaharia (1997), Parusinski (1995) e Tibar (1998). Título: Regularidade no infinito e fibrações globais de aplicações algébricas reais

Des questions fondamentales concernant les propriétés de diffusion des systèmes biologiques dans des conditions isothermes et non-isothermes restent en suspens en raison de l’absence de techniques expérimentales capables de visualiser et de mesurer les phénomènes de diffusion avec une très bonne précision. Il existe en conséquence un besoin de développer de nouvelles techniques expérimentales permettant d’approfondir notre compréhension des phénomènes de diffusion. La convection naturelle en cavité tridimensionnelle inclinée est elle-aussi très peu étudiée. Cette inclinaison de la cavité peut correspondre à un léger défaut expérimental ou être imposée volontairement. Dans cette thèse, nous étudions les phénomènes de transport de chaleur et de masse en cavité parallélépipédique, nous intéressant particulièrement à la thermodiffusion en situation sans convection et à la convection naturelle en fluide pur (sans thermodiffusion). La diffusion de masse est étudiée à l’aide d’une nouvelle technique optique, tandis que la convection naturelle est tout d’abord étudiée en détails avec une méthode numérique sophistiquée, puis visualisée expérimentalement à l’aide du même système optique que pour les mesures de diffusion. Nous présentons l’interféromètre optique de haute précision développé pour les mesures de diffusion. Cet interféromètre comprend un interféromètre polarisé de Mach–Zehnder, un polariseur tournant, une caméra CCD et un algorithme de traitement d’images original. Nous proposons aussi une méthode pour déterminer le coefficient de diffusion isotherme en fonction de la concentration. Cette méthode, basée sur une analyse inverse couplée à un calcul numérique, permet de déterminer les coefficients de diffusion à partir des profils de concentration transitoires obtenus par le système optique. Mentionnons de plus que c’est la première fois que la thermodiffusion est visualisée dans des solutions aqueuses de protéines. La méthode optique proposée présente trois avantages principaux par rapport aux autres méthodes similaires : (i) un volume d’échantillon réduit, (ii) un temps de mesure court, (iii) une stabilité hydrodynamique améliorée. Toutes ces méthodes ont été validées par des mesures sur des systèmes de référence. La technique optique est d’abord utilisée pour étudier la diffusion isotherme dans des solutions de protéines : (a) dans des solutions binaires diluées, (b) dans des solutions binaires sur un large domaine de concentration, (c) dans des solutions ternaires diluées. Les résultats montrent que (a) le coefficient de diffusion isotherme dans les systèmes dilués décroit avec la masse moléculaire, comme prédit grossièrement par l’équation de Stokes-Einstein ; (b) la protéine BSA a un comportement diffusif de type sphère dure et la protéine lysozyme de type sphère molle ; (c) l’effet de diffusion croisée est négligeable dans les systèmes ternaires dilués. La technique optique est aussi utilisée (d) dans des solutions binaires diluées non-isothermes, révélant que les molécules d’aprotinin (6.5 kDa) et de lysozyme (14.3 kDa) sont, respectivement, thermophiliques et thermo-phobiques, quand elles sont en solutions aqueuses à température ambiante. Enfin, la technique optique est utilisée pour l’étude de la convection de Rayleigh-Bénard en cavité cubique horizontale. Puisque la convection peut aussi être étudiée de façon réaliste en utilisant les équations de Navier-Stokes, une analyse numérique de bifurcation est proposée, permettant une étude approfondie de la convection naturelle dans des cavités tridimensionnelles parallélépipédiques. Pour cela, une méthode de continuation a été développée à partir d’un code aux éléments finis spectraux. La méthode numérique proposée est particulièrement bien adaptée aux études de convection correspondant à des diagrammes de bifurcation complexes. [...] / Fundamental questions concerning the mass diffusion properties of biological systems under isothermal and non-isothermal conditions still remain due to the lack of experimental techniques capable of visualizing and measuring mass diffusion phenomena with a high accuracy. As a consequence, there is a need to develop new experimental techniques that can deepen our understanding of mass diffusion. Moreover, steady natural convection in a tilted three-dimensional rectangular enclosure has not yet been studied. This tilt can be a slight defect of the experimental device or can be imposed on purpose. In this dissertation, heat and mass transfer phenomena in parallelepiped enclosures are studied focusing on convectionless thermodiffusion and on natural convection of pure fluids (without thermodiffusion). Mass diffusion is studied with a novel optical technique, while steady natural convection is first studied in detail with an improved numerical analysis and then with the same optical technique initially developed for diffusion measurements. A construction of a precise optical interferometer to visualize and measure mass diffusion is described. The interferometer comprises a polarizing Mach–Zehnder interferometer, a rotating polariser, a CCD camera, and an original image-processing algorithm. A method to determine the isothermal diffusion coefficient as a function of concentration is proposed. This method uses an inverse analysis coupled with a numerical calculation in order to determine the diffusion coefficients from the transient concentration profiles measured with the optical system. Furthermore, thermodiffusion of protein molecules is visualized for the first time. The proposed method has three main advantages in comparison to similar methods: (i) reduced volume sample, (ii) short measurement time, and (iii) increased hydrodynamic stability of the system. These methods are validated by determining the thermophysical properties of benchmark solutions. The optical technique is first applied to study isothermal diffusion of protein solutions in: (a) dilute binary solutions, (b) binary solutions with a wide concentration range, and (c) dilute ternary solutions. The results show that (a) the isothermal diffusion coefficient in dilute systems decreases with molecular mass, as roughly predicted by the Stokes-Einstein equation; (b) BSA protein has a hard-sphere-like diffusion behaviour and lysozyme protein a soft sphere characteristic; and (c) the cross-term effect between the diffusion species in a dilute ternary system is negligible. The optical technique is then applied to (d) non-isothermal dilute binary solutions, revealing that that the aprotinin (6.5 kDa) and lysozyme (14.3 kDa) molecules are thermophilic and thermophobic, respectively, when using water as solvent at room temperature. Finally, the optical technique is applied to study Rayleigh-Bénard convection in a horizontal cubical cavity. Since natural convection can be studied in more depth by solving the Navier-Stokes equations, a bifurcation analysis is proposed to conduct a thorough study of natural convection in three-dimensional parallelepiped cavities. Here, a continuation method is developed from a three-dimensional spectral finite element code. The proposed numerical method is particularly well suited for the studies involving complex bifurcation diagrams of three-dimensional convection in rectangular parallelepiped cavities. This continuation method allows the calculation of solution branches, the stability analysis of the solutions along these branches, the detection and precise direct calculation of the bifurcation points, and the jump to newly detected stable or unstable branches, all this being managed by a simple continuation algorithm. This can be used to calculate the bifurcation diagrams describing the convection in tilted cavities. [...]

As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered.

Philosophiae Doctor - PhD / Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009.