Functional differential equations with delay have long been studied due to their practical applications. For the delay term is not a constant number, many researches study the case when this deviating argument depends on the state variable. So we deal with the differential and functional equations involving with the compositions of the unknown function, i.e. the iterative functional differential equations (IFDEs) and iterative functional equations (IFEs) without derivative. The main purpose of this dissertation is to investigate the solutions of such equations, including their analytic solutions, numerical solutions and qualitative behaviors. First, we survey some well known differential equations of this type which possess analytic solutions. Then the classical method of undetermined coefficients is used to compute these power series solutions for the first order IFDEs in Chapter 1, the second order IFDEs in Chapter 2 and FDEs in Chapter 3. Taylor series method is also used to get these analytic solutions in Chapter 4. Systematical method is found to locate the fixed point in generalized sense, so we can use these methods to calculate the coefficients of their analytic solutions. Furthermore, we also establish the existence and uniqueness theorem for analytic solution in Chapter 5. Second, we survey the known existence and uniqueness theorems of solutions for these IFDEs and FDEs in Chapter 6. Then we apply Schauder fixed point theorem to establish new existence theorems of local solutions for general IFDEs. Under certain conditions, these local solutions can be extended to global solutions. Chapter 7 deals with the simplest IFDEs the Eder's equation. We extend the qualitative properties of this case and find its solution is not unique. In Chapter 8, we use Euler method to get the numerical solution of IFDEs. Under some conditions, we have the error analysis on these equations. In Chapter 9, we employ the method of undetermined coefficients, Taylor series, Picard's iteration and Si's methods to get their analytic solutions. Their comparisons, the advantage and disadvantage of these methods are also discussed.

O objetivo deste trabalho é desenvolver um método numérico capaz de simular escoamentos viscoelásticos com superfícies livres governados pela equação constitutiva não-linear PTT (Phan-Thien-Tanner). Neste trabalho foram apresentados três métodos numéricos para simular escoamentos viscoelásticos modelados pela equação PTT. Dois desses métodos foram desenvolvidos para simular escoamentos viscoelásticos bidimensionais enquanto o terceiro método foi desenvolvido para simular escoamentos viscoelásticos tridimensionais. Estes métodos numéricos foram incorporados aos ambientes de simulação FreeDow2D e FreeDow3D, extendendo estes ambientes para escoamentos viscoelásticos descritos por uma equação constitutiva não-linear. Inicialmente, uma descrição de FreeDow2D e FreeDow3D é apresentada. As equações governantes para escoamentos descritos pelo modelo PTT são dadas na forma de tensorial e as formulações matemáticas para obtenção dos métodos numéricos são apresentadas. As equações que descrevem os métodos numéricos são resolvidas pela técnica de diferenças finitas numa malha deslocada e o fluido é representado por partículas arcadoras usando o método Marker-and-Cell. As condições de contorno para cada tipo de contorno são descritas em detalhes e o cálculo do tensor extra-tensão no contorno rígido é obtido utilizando as idéias de Tomé et al. [84] para o modelo Oldroyd-B. Seguindo a metodologia de Alves et al. [2], a solução analítica do modelo PTT para escoamentos totalmente desenvolvidos em um canal bidimensional é apresentada em detalhes. Esta solução analítica é então usada para validar o método numérico desenvolvido neste trabalho. Os métodos numéricos desenvolvidos nesse trabalho foram aplicados para simular os seguintes problemas: um jato de fluido viscoelástico incidindo numa placa rígida; o inchamento do extrudado e o problema de uma gota esférica de fluido viscoelástico incidindo perpendicularmente contra uma superfície rígida plana / The aim of this work is to develop a numerical method capable of simulating viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien-Tanner). In this work three numerical methods to simulate vicoelastic Flows of fluids modelled by the PTT equation are presented. Two of these methods were developed to simulate two-dimensional viscoelastic flows while the third method was developed to simulate three-dimensional viscoelastic flows. These numerical methods were incorporated into the codes FreeFlow2D and FreeFlow3D, extending these codes to viscoelastic flows described by the non-linear constitutive equation PTT. Initially, a description of FreeFlow2D and FreeFlow3D is presented. The governing equations for flows described by the PTT model are given in index form and the mathematical formulations for obtaining the numerical methods are presented. The equations describing the numerical methods are solved by the finite difference method on a staggered grid and the fluid is modelled using a Marker-and-Cell type method. The boundary conditions for each type of boundary are described in details and the calculation of the extra-stress tensor on rigid boundaries is performed using the ideas of Tomé et al. [84] for an Oldroyd-B Fluid. Following the methodology presented by Alves et al. [2], the analytic solution of the PTT model for fully developed flows in a two-dimensional channel is presented in details. This analytic solution is then used to validate the numerical method developed in this work. Finally, the numerical methods developed in this work were applied to simulate viscoelastic flows such as a viscoelastic jet flowing onto a rigid plate, the extrudate swell of viscoelastic fluids and the simulation of a viscoelastic drop hitting a rigid plate

O objetivo deste trabalho é desenvolver um método numérico capaz de simular escoamentos viscoelásticos com superfícies livres governados pela equação constitutiva não-linear PTT (Phan-Thien-Tanner). Neste trabalho foram apresentados três métodos numéricos para simular escoamentos viscoelásticos modelados pela equação PTT. Dois desses métodos foram desenvolvidos para simular escoamentos viscoelásticos bidimensionais enquanto o terceiro método foi desenvolvido para simular escoamentos viscoelásticos tridimensionais. Estes métodos numéricos foram incorporados aos ambientes de simulação FreeDow2D e FreeDow3D, extendendo estes ambientes para escoamentos viscoelásticos descritos por uma equação constitutiva não-linear. Inicialmente, uma descrição de FreeDow2D e FreeDow3D é apresentada. As equações governantes para escoamentos descritos pelo modelo PTT são dadas na forma de tensorial e as formulações matemáticas para obtenção dos métodos numéricos são apresentadas. As equações que descrevem os métodos numéricos são resolvidas pela técnica de diferenças finitas numa malha deslocada e o fluido é representado por partículas arcadoras usando o método Marker-and-Cell. As condições de contorno para cada tipo de contorno são descritas em detalhes e o cálculo do tensor extra-tensão no contorno rígido é obtido utilizando as idéias de Tomé et al. [84] para o modelo Oldroyd-B. Seguindo a metodologia de Alves et al. [2], a solução analítica do modelo PTT para escoamentos totalmente desenvolvidos em um canal bidimensional é apresentada em detalhes. Esta solução analítica é então usada para validar o método numérico desenvolvido neste trabalho. Os métodos numéricos desenvolvidos nesse trabalho foram aplicados para simular os seguintes problemas: um jato de fluido viscoelástico incidindo numa placa rígida; o inchamento do extrudado e o problema de uma gota esférica de fluido viscoelástico incidindo perpendicularmente contra uma superfície rígida plana / The aim of this work is to develop a numerical method capable of simulating viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien-Tanner). In this work three numerical methods to simulate vicoelastic Flows of fluids modelled by the PTT equation are presented. Two of these methods were developed to simulate two-dimensional viscoelastic flows while the third method was developed to simulate three-dimensional viscoelastic flows. These numerical methods were incorporated into the codes FreeFlow2D and FreeFlow3D, extending these codes to viscoelastic flows described by the non-linear constitutive equation PTT. Initially, a description of FreeFlow2D and FreeFlow3D is presented. The governing equations for flows described by the PTT model are given in index form and the mathematical formulations for obtaining the numerical methods are presented. The equations describing the numerical methods are solved by the finite difference method on a staggered grid and the fluid is modelled using a Marker-and-Cell type method. The boundary conditions for each type of boundary are described in details and the calculation of the extra-stress tensor on rigid boundaries is performed using the ideas of Tomé et al. [84] for an Oldroyd-B Fluid. Following the methodology presented by Alves et al. [2], the analytic solution of the PTT model for fully developed flows in a two-dimensional channel is presented in details. This analytic solution is then used to validate the numerical method developed in this work. Finally, the numerical methods developed in this work were applied to simulate viscoelastic flows such as a viscoelastic jet flowing onto a rigid plate, the extrudate swell of viscoelastic fluids and the simulation of a viscoelastic drop hitting a rigid plate

In this study, the development and testing of a finite-difference model for wind-induced flow in shallow lakes, and, in particular, a new technique for improving the land--water boundary representation, are documented. The model solves nonlinear, as well as linear, versions of the two-dimensional depth-integrated shallow water equations. Finite-difference methods on rectangular grids are widely used in numerical models of environmental flows. In these models, land--water boundaries are usually approximated by a series of perpendicular line segments, which enable the impermeability condition to be easily implemented. A disadvantage of this approach is that the actual boundary is often poorly approximated, particularly in regions which have complicated coastlines, and, as a result, currents in these regions cannot be accurately predicted. A technique for improving the land--water boundary representation in finite-difference models is introduced. This technique permits the model boundary to contain diagonal line segments, in addition to the vertical and horizontal line segments used in traditional models. The new technique is based on a simple concept and can easily be included in existing finite-difference models. In order to test the new method, the linearised shallow water equations are solved numerically for oscillatory wind-driven flow in lakes with simple geometry. Predictions obtained using the new approach are compared with predictions from the traditional stepped boundary and known analytic solutions. A significant improvement in the accuracy of results is noticed when the new approach is used, particularly in currents close to shore. The increased accuracy obtained using the improved boundary representation can lead to a significant computational saving, when compared with running the rectangular grid model with smaller grid spacings. A second-order analytic solution to the nonlinear shallow water equations is developed for oscillatory wind-driven flow in a rectangular lake. Comparisons between this solution and numerical results, obtained using the traditional stepped boundary and the improved boundary, verify the finite-difference formulae used in these models, including the approximations used for the cross-advective terms close to shore. Once more, currents are predicted with greater accuracy when the new technique for representing the land--water boundary is implemented. The lake circulation model is applied to the Lower Murray Lakes, South Australia, and predicted water levels at Tauwitchere Barrage are shown to agree very well with observations. The model is then used to examine the effectiveness of two schemes that have been proposed to increase wind-induced circulation, and therefore potentially decrease salinity, in Lake Albert, demonstrating the model's use as an efficient and effective tool for analysing flow behaviour in lakes. / Thesis (Ph.D.)--Mathematical Sciences (Applied Mathematics), 2005.

In this study, the development and testing of a finite-difference model for wind-induced flow in shallow lakes, and, in particular, a new technique for improving the land--water boundary representation, are documented. The model solves nonlinear, as well as linear, versions of the two-dimensional depth-integrated shallow water equations. Finite-difference methods on rectangular grids are widely used in numerical models of environmental flows. In these models, land--water boundaries are usually approximated by a series of perpendicular line segments, which enable the impermeability condition to be easily implemented. A disadvantage of this approach is that the actual boundary is often poorly approximated, particularly in regions which have complicated coastlines, and, as a result, currents in these regions cannot be accurately predicted. A technique for improving the land--water boundary representation in finite-difference models is introduced. This technique permits the model boundary to contain diagonal line segments, in addition to the vertical and horizontal line segments used in traditional models. The new technique is based on a simple concept and can easily be included in existing finite-difference models. In order to test the new method, the linearised shallow water equations are solved numerically for oscillatory wind-driven flow in lakes with simple geometry. Predictions obtained using the new approach are compared with predictions from the traditional stepped boundary and known analytic solutions. A significant improvement in the accuracy of results is noticed when the new approach is used, particularly in currents close to shore. The increased accuracy obtained using the improved boundary representation can lead to a significant computational saving, when compared with running the rectangular grid model with smaller grid spacings. A second-order analytic solution to the nonlinear shallow water equations is developed for oscillatory wind-driven flow in a rectangular lake. Comparisons between this solution and numerical results, obtained using the traditional stepped boundary and the improved boundary, verify the finite-difference formulae used in these models, including the approximations used for the cross-advective terms close to shore. Once more, currents are predicted with greater accuracy when the new technique for representing the land--water boundary is implemented. The lake circulation model is applied to the Lower Murray Lakes, South Australia, and predicted water levels at Tauwitchere Barrage are shown to agree very well with observations. The model is then used to examine the effectiveness of two schemes that have been proposed to increase wind-induced circulation, and therefore potentially decrease salinity, in Lake Albert, demonstrating the model's use as an efficient and effective tool for analysing flow behaviour in lakes. / Thesis (Ph.D.)--Mathematical Sciences (Applied Mathematics), 2005.

Die aus magnetisierbaren Partikeln und einer elastischen Matrix bestehenden magnetorheologischen Elastomere sind ein Verbundwerkstoff mit magnetisch steuerbaren Eigenschaften. In der vorliegenden Arbeit wird ein kontinuumsmechanisches Modell zur Beschreibung der relevanten physikalischen Phänomene bereitgestellt. Die Lösung zugehöriger Randwertaufgaben basiert auf der erweiterten Finiten Elemente Methode. Zur Verifikation und Validierung des Modells werden analytische Referenzlösungen zweidimensionaler Problemstellungen herangezogen. Die Homogenisierung des magnetomechanischen Feldproblems erfolgt mit kleinen Deformationen. Aus einer Volumenmittelung der lokal inhomogenen Feldverteilungen ergeben sich makroskopische Variablen. Auf Basis dieser Größen lassen sich Aussagen über das effektive Verhalten ableiten. Somit ist neben den rein magnetischen und mechanischen Materialeigenschaften das gekoppelte magnetomechanische Verhalten analysierbar. Darunter sind aktuatorische Spannungen, magnetostriktive Dehnungen und der magnetorheologische Effekt zu verstehen. / Magnetorheological elastomers are composite materials consisting of magnetizable particles embedded in an elastic matrix. Their properties can be altered by an external magnetic field. In this work a continuum based formulation is applied to model relevant physical phenomena. Boundary value problems are solved by the extended Finite Element Method. For the purposes of verification and validation analytic solutions are provided. The homogenization of the magnetomechanical field problem is limited to small deformations. Macroscopic variables are obtained by volume averaging. In addition to macroscopic magnetic and mechanical properties the effective behavior is analyzed in terms of actuatoric stresses, magnetostrictive strains and the magnetorheological effect.

Ce travail de doctorat porte sur l'utilisation des mathématiques analytiques dans le cadre de méthodes inverses appliquées à l'industrie. Ces travaux tiennent au développement de capteurs inverses en temps réel adaptés au laminage industriel. Le producteur d'acier ArcelorMittal dirige un projet européen, qui vise à montrer la faisabilité de capteurs mesurant les champs (température, contraintes) dans le contact entre le produit et l'outil sans altérer les conditions de ce contact. Les travaux de thèse présentés dans ce mémoire ont été réalisés au sein du laboratoire Navier et financé par l'école des Ponts ParisTech. Cependant un contrat sur trois ans signé avec ArcelorMittal a permis à l'auteur d'être partie prenante de ce projet européen, et ainsi de voir ses recherches concrétisées par une demande industrielle réelle. L'enjeu de cette thèse est double, académique et industriel. Académique en ce sens que les travaux cherchent à recenser et utiliser efficacement les méthodes de résolution analytiques existantes, pour des problèmes inverses en thermoélasticité, dans le cadre d'une métrologie en temps réel. En effet, les solutions analytiques sont souvent exploitées comme des cas purement théoriques, trouvant à bien des égards un certain succès dans la culture de l'ingénieur (comme c'est le cas par exemple du calcul en élasticité linéaire des facteurs d'intensité de contraintes), mais qui par bien des aspects sont reléguées à des cas d'écoles anciens servant au mieux à valider des codes de calculs numériques sur des exemples particulièrement simples. Ces solutions et méthodes de résolution analytiques ne font d'ailleurs guère plus l'objet de recherches en mathématiques pures. Cependant les problématiques propres, liées au caractère inverse des problèmes à traiter, pénalisent les méthodes de résolution numériques, en ce sens que les problèmes inverses sont mal posés, et qu'une stabilisation des algorithmes numériques est nécessaire mais souvent délicate si l'on considère les conditions extrêmes (champs très singuliers) appliquées aux outils industriels de laminage. Par ailleurs la métrologie en temps réel exclut l'utilisation de codes numériques trop coûteux en temps de calcul (méthodes itératives etc...). Ces deux aspects contribuent à renouveler assez largement l'intérêt pour les solutions analytiques. Il convient alors d'en regrouper (dans la mesure du possible) les méthodes les plus efficaces (en termes de précision et de temps de calcul notamment) et les plus adaptées pour la métrologie. Nous verrons notamment différents développements en séries élémentaires permettant non seulement de donner à une suite de points mesurés une forme analytique, mais également de simplifier les équations aux dérivées partielles à résoudre. D'autre part l'enjeu de cette thèse est également industriel, car ces travaux s'inscrivent dans une démarche de développement de capteurs adaptés à la mise en forme de l'acier par laminage. Ainsi l'étude de la robustesse au bruit de mesure, les contraintes technologiques liées à l'insertion des capteurs, les limitations en terme de fréquence d'acquisition et les problématiques de calibrage sont au coeur des développements. Ainsi, l'ensemble des travaux présentés, peut constituer une sorte de réhabilitation des méthodes analytiques, dont la supériorité sur les méthodes numériques (en termes de temps de calcul et parfois aussi de précision) est mise en lumière, dans le contexte précis de la métrologie en temps réel sur des géométries simples. Trois méthodes inverses en deux dimensions, adaptées au laminage industriel, ont été menées à bien (élastique isotherme, thermique et couplage thermoélastique), ainsi qu'une série d'applications expérimentales réalisées sur le laminoir de laboratoire d'ArcelorMittal. Par ailleurs, des extensions en trois dimensions des méthodes inverses élastiques et thermiques sont également détaillées / This thesis is about the use of analytical mathematics within the framework of inverse methods applied to industry. These works are devoted to the development of sensors using real-time inverse methods adapted for rolling process. Steel producer ArcelorMittal leads a European project that aims to demonstrate the feasibility of sensors measuring fields (temperature, stress) in the contact between the product and the tool without altering physical conditions of this contact. The thesis has been funded by l'école des Ponts ParisTech. However, a three-year contract signed with ArcelorMittal has enabled the author to be part of the European project, and thus his research has been motivated by a real industrial demand. The aim of this thesis is twofold, academic and industrial; academic in the sense that these works seek to identify and use efficiently existing analytical methods for inverse problems occurring in thermo-elasticity in the context of real-time metrology. Indeed, analytical solutions are often exploited as purely theoretical cases, finding in many ways some success in engineering (for example in linear elasticity with stress intensity factors), but are most of the time relegated to the validation of numerical codes under simple assumptions. As a matter of fact, there is no research any more in pure mathematics concerning these solutions and analytical methods. However, the specific complications related to the inverse nature of problems under consideration, penalize numerical algorithms because inverse problems are ill-posed and stabilization is needed. But it remains often difficult if we consider the extreme loads (very sharp gradients) applied to industrial tools during rolling. Moreover, the real-time metrology excludes the use of numerical codes too costly in terms of computation time (iterative methods etc...). Both aspects contribute to renew widely interest for analytical solutions. It is then necessary to collect most effective and efficient (in terms of computation time and precision) methods and emphasis the most suitable for metrology. We will see various series expansions, not only to give a sequence of measured points an analytical form, but also to simplify the partial differential equations to solve. On the other hand, the goal of this thesis is also industrial, as these works are part of a process of development of sensors adapted for steel rolling industry. Thus, the robustness to measurement noise, technological constraints related to the local measurement systems (such as limitations in terms of frequency of acquisition) and calibration issues are central in the developments. Thus, the whole work can be a kind of rehabilitation of analytical methods. Their superiority over numerical methods (in terms of computation time and sometime accuracy) is highlighted, in the specific context of metrology in real-time on simple geometries. Three inverse methods in two-dimensions suitable for rolling process were developed successfully (isothermal elastic, thermal and thermoelastic coupling) and a series of experimental tests were made on the laboratory mill of ArcelorMittal. In addition, three-dimensional extensions of elastic and thermal inverse methods are also detailed

Die aus magnetisierbaren Partikeln und einer elastischen Matrix bestehenden magnetorheologischen Elastomere sind ein Verbundwerkstoff mit magnetisch steuerbaren Eigenschaften. In der vorliegenden Arbeit wird ein kontinuumsmechanisches Modell zur Beschreibung der relevanten physikalischen Phänomene bereitgestellt. Die Lösung zugehöriger Randwertaufgaben basiert auf der erweiterten Finiten Elemente Methode. Zur Verifikation und Validierung des Modells werden analytische Referenzlösungen zweidimensionaler Problemstellungen herangezogen. Die Homogenisierung des magnetomechanischen Feldproblems erfolgt mit kleinen Deformationen. Aus einer Volumenmittelung der lokal inhomogenen Feldverteilungen ergeben sich makroskopische Variablen. Auf Basis dieser Größen lassen sich Aussagen über das effektive Verhalten ableiten. Somit ist neben den rein magnetischen und mechanischen Materialeigenschaften das gekoppelte magnetomechanische Verhalten analysierbar. Darunter sind aktuatorische Spannungen, magnetostriktive Dehnungen und der magnetorheologische Effekt zu verstehen. / Magnetorheological elastomers are composite materials consisting of magnetizable particles embedded in an elastic matrix. Their properties can be altered by an external magnetic field. In this work a continuum based formulation is applied to model relevant physical phenomena. Boundary value problems are solved by the extended Finite Element Method. For the purposes of verification and validation analytic solutions are provided. The homogenization of the magnetomechanical field problem is limited to small deformations. Macroscopic variables are obtained by volume averaging. In addition to macroscopic magnetic and mechanical properties the effective behavior is analyzed in terms of actuatoric stresses, magnetostrictive strains and the magnetorheological effect.

O objetivo desse trabalho é desenvolver métodos numéricos para simular escoamentos de cristais líquidos nemáticos governados pelas equações dinâmicas de Ericksen-Leslie. São apresentados dois métodos numéricos para a simulação de escoamentos de cristais líquidos nemáticos. O primeiro método foi desenvolvido para simular escoamentos tridimensionais de cristais líquidos nemáticos sob efeito de forte campo magnético enquanto que o segundo método foi desenvolvido para a simulação de escoamentos bidimensionais. Utilizando a notação de Einstein, as equações dinâmicas de Ericksen-Leslie são apresentadas. Empregando variáveis primitivas e coordenadas cartesianas, as equações governantes para escoamentos de cristais líquidos nemáticos são derivadas e as formulações matemáticas para a obtenção dos métodos numéricos são apresentadas. As equações descrevendo os métodos numéricos são resolvidas por um método numérico baseado na metodologia GENSMAC3D para o caso tridimensional enquanto que o método bidimensional é baseado na metodologia GENSMAC (GENeralized-Simplified-Marker-And-Cell). Em ambos os métodos, a técnica de diferenças finitas em uma malha deslocada é utilizada. As equações que descrevem as técnicas numéricas desenvolvidas foram incorporadas aos ambientes de simulação Freeflow2D e Freeflow3D. As condições de contorno para cada tipo de contorno são descritas em detalhes. A solução analítica apresentada por Stewart para o escoamento entre duas placas paralelas é utilizada para a validação do método numérico tridimensional. Empregando as hipóteses de escoamento desenvolvido e que o ângulo de orientação do diretor é pequeno, uma solução analítica para o escoamento em um canal bidimensional é encontrada. O método numérico bidimensional é então validado utilizando a solução analítica obtida. Utilizando refinamento de malha, resultados de convergência dos métodos numéricos são apresentados. Os métodos numéricos desenvolvidos nesse trabalho são aplicados para a simulação dos seguintes problemas: escoamento de um cristal líquido nemático em um canal tridimensional; investigação numérica do escoamento em L-canais e escoamento através de uma contração 4:1 e de uma expansão planar 4:1 / The aim of this work is to develop numerical methods capable of simulating nematic liquid crystal flows described by the dynamic Ericksen-Leslie equations. Two numerical techniques have been proposed: a numerical method for solving three-dimensional flows of nematic liquid crystals under strong magnetic fields and a method for simulating flows of nematic liquid crystals in two-dimensional complex geometries. Both methods employ Cartesian coordinates using primitive variables of pressure and velocity. These techniques are based on the GENSMAC (GENeralized-Simplified-Marker-And-Cell) methodology and a detailed description of the equations involved is presented. The resulting governing equations are solved by the finite difference method on a staggered grid. The three-dimensional technique was applied to solve fully developed flow between two paralel plates for which an analytic solution exists. By using this analytic solution, validation and convergence results of the developed numerical technique were obtained. To validate the two-dimensional method developed herein, an analytic solution for steady state flow in a 2D-channel was found which was used to obtain validation and convergence results. The method was then applied to simulate the flow in L-shaped channels, flow through a 4:1 contraction and flow in a 4:1 expansion

O objetivo desse trabalho é desenvolver métodos numéricos para simular escoamentos de cristais líquidos nemáticos governados pelas equações dinâmicas de Ericksen-Leslie. São apresentados dois métodos numéricos para a simulação de escoamentos de cristais líquidos nemáticos. O primeiro método foi desenvolvido para simular escoamentos tridimensionais de cristais líquidos nemáticos sob efeito de forte campo magnético enquanto que o segundo método foi desenvolvido para a simulação de escoamentos bidimensionais. Utilizando a notação de Einstein, as equações dinâmicas de Ericksen-Leslie são apresentadas. Empregando variáveis primitivas e coordenadas cartesianas, as equações governantes para escoamentos de cristais líquidos nemáticos são derivadas e as formulações matemáticas para a obtenção dos métodos numéricos são apresentadas. As equações descrevendo os métodos numéricos são resolvidas por um método numérico baseado na metodologia GENSMAC3D para o caso tridimensional enquanto que o método bidimensional é baseado na metodologia GENSMAC (GENeralized-Simplified-Marker-And-Cell). Em ambos os métodos, a técnica de diferenças finitas em uma malha deslocada é utilizada. As equações que descrevem as técnicas numéricas desenvolvidas foram incorporadas aos ambientes de simulação Freeflow2D e Freeflow3D. As condições de contorno para cada tipo de contorno são descritas em detalhes. A solução analítica apresentada por Stewart para o escoamento entre duas placas paralelas é utilizada para a validação do método numérico tridimensional. Empregando as hipóteses de escoamento desenvolvido e que o ângulo de orientação do diretor é pequeno, uma solução analítica para o escoamento em um canal bidimensional é encontrada. O método numérico bidimensional é então validado utilizando a solução analítica obtida. Utilizando refinamento de malha, resultados de convergência dos métodos numéricos são apresentados. Os métodos numéricos desenvolvidos nesse trabalho são aplicados para a simulação dos seguintes problemas: escoamento de um cristal líquido nemático em um canal tridimensional; investigação numérica do escoamento em L-canais e escoamento através de uma contração 4:1 e de uma expansão planar 4:1 / The aim of this work is to develop numerical methods capable of simulating nematic liquid crystal flows described by the dynamic Ericksen-Leslie equations. Two numerical techniques have been proposed: a numerical method for solving three-dimensional flows of nematic liquid crystals under strong magnetic fields and a method for simulating flows of nematic liquid crystals in two-dimensional complex geometries. Both methods employ Cartesian coordinates using primitive variables of pressure and velocity. These techniques are based on the GENSMAC (GENeralized-Simplified-Marker-And-Cell) methodology and a detailed description of the equations involved is presented. The resulting governing equations are solved by the finite difference method on a staggered grid. The three-dimensional technique was applied to solve fully developed flow between two paralel plates for which an analytic solution exists. By using this analytic solution, validation and convergence results of the developed numerical technique were obtained. To validate the two-dimensional method developed herein, an analytic solution for steady state flow in a 2D-channel was found which was used to obtain validation and convergence results. The method was then applied to simulate the flow in L-shaped channels, flow through a 4:1 contraction and flow in a 4:1 expansion