Le mélange de fluides représente un élément important du domaine de la dynamique des fluides, ce qui rend la compréhension de ce sujet si significative du point de vue fondamental et appliqué (p. ex., les processus industriels). Dans les géométries miniaturisées (dans des conditions typiques) le mélange est un processus lent, difficile et inefficace. Cela en raison du caractère naturellement laminaire de ces écoulements, qui oblige l'homogénéisation de différents éléments fluides à se produire par diffusion moléculaire au lieu d'un transport advectif, à l'action plus rapide. Cependant, des études expérimentales récentes sur les écoulements viscoélastiques à faible nombre de Reynolds ont montré qu'un mélange efficace peut être déclenché dans plusieurs configurations géométriques (y compris les dispositifs à l'échelle microscopique), par le phénomène de la turbulence élastique. La première partie de cette thèse est consacrée à la compréhension et à l'investigation des défis numériques présents dans le domaine de la dynamique des fluides non newtonienne, en se concentrant plus particulièrement au problème du haut nombre de Weissenberg. Ce dernier se manifeste par une rupture du schéma numérique, lorsque les équations d'évolution d'extra-contraintes polymériques sont évaluées de façon directe. Ceci pose des limites importantes à la possibilité de simuler avec précision des écoulements turbulents-élastiques. Nous fournissons des preuves numériques de l'effet bénéfique (en termes de gain en stabilité) de la décomposition en racine carrée de l'extra-contrainte dans une implémentation en volumes finis des équations régissant l'écoulement dans un canal bidimensionnel. La deuxième partie de la thèse traite de l’émergence et de la caractérisation d’instabilités purement élastiques dans des simulations numériques de fluides Oldroyd-B à nombre de Reynolds zéro dans une géométrie du type cross-slot bidimensionnel. Grâce à un travail numérique approfondi, nous présentons une caractérisation détaillée des instabilités purement élastiques. Ces instabilités apparaissant dans le système pour de larges plages d'élasticité du fluide et de concentration des polymères. Pour les solutions concentrées et des nombres de Weissenberg assez grands, nos simulations indiquent l’apparition d’un écoulement désordonné pointant vers la turbulence élastique. Nous analysons le passage à une dynamique irrégulière et caractérisons les propriétés statistiques de tels écoulements très élastiques, en discutant des similitudes et des différences avec les résultats expérimentaux de la littérature. / Fluid mixing represents an important component of the field of fluid dynamics, what makes the understanding of this subject so meaningful from both the fundamental and applied (e.g. industrial processes) point of view. In miniaturised geometries, under typical conditions, mixing is a slow, difficult and inefficient process due to the naturally laminar character of these flows, which forces the homogenisation of different fluid elements to occur via molecular diffusion instead of faster-acting advective transport. However, recent experimental studies on low-Reynolds-number viscoelastic flows have shown that efficient mixing can be triggered in several geometrical configurations (including micro-scale devices), by the phenomenon of elastic turbulence. The first part of this thesis is devoted to the understanding and investigation of numerical challenges present in the domain of non-Newtonian fluid dynamics, focusing in particular on the high-Weissenberg number problem. The latter manifests as a breakdown of the numerical scheme when the polymeric extra-stress evolution equations are implemented in a direct way, which poses severe limits to the possibility to accurately simulate elastic turbulent flows. We provide numerical evidence of the beneficial effect (in terms of increased stability) of the square-root decomposition of the extra-stress in a finite-volume-based implementation of the governing equations in a two-dimensional channel. The second part of the thesis reports about the emergence and characterisation of purely-elastic instabilities in numerical simulations of zero-Reynolds-number Oldroyd-B fluids in a two-dimensional cross-slot geometry. By means of extensive numerical work, we provide a detailed characterisation of the purely-elastic instabilities arising in the system for wide ranges of both the fluid elasticity and the polymer concentration. For concentrated solutions and large enough Weissenberg numbers, our simulations indicate the emergence of disordered flow pointing to elastic turbulence. We analyse the transition to irregular dynamics and characterise the statistical properties of such highly elastic flows, discussing the similarities and differences with experimental results from the literature.

Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of infinite Weissenberg and Reynolds numbers. Specifically, we study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette flows, the hyperbolic-tangent shear layer and the Bickley jet. The equation for stability is derived and solved numerically using the Chebyshev collocation spectral method. This algorithm is computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and ultimately suppress the inviscid instability. The numerical solutions are compared with the analysis of the long wave limit and excellent agreement is shown between the analytical and the numerical solutions. We found flows which are long wave stable, but nevertheless unstable to wave numbers in a certain finite range. While elasticity is ultimately stabilizing, this effect is not monotone; there are instances where a small amount of elasticity actually destabilizes the flow. The linear stability in the short wave limit of shear flows bounded by two parallel free surfaces is investigated. Unlike the plane Couette flow which has no short wave instability, we show that plane Poiseuille flow has two unstable eigenmodes localized near the free surfaces which can be combined into an even and an odd eigenfunctions. The derivation of the asymptotics of these modes shows that our numerical eigenvalues are in agreement with the analytic formula and that the difference between the two eigenvalues tends to zero exponentially with the wavenumber. / Ph. D.

Aplicações industriais envolvendo escoamentos multifásicos são inúmeras, sendo que, o aprimoramento de alguns desses processos pode resultar em um grande salto tecnológico com significativo impacto econômico. O estudo numérico dessas aplicações é imprescindível, pois fornece informações precisas e mais detalhadas do que a realização de testes experimentais. Um grande desafio é o estudo numérico de escoamentos viscoelásticos multifásicos envolvendo altas taxa de elasticidade, devido às instabilidades causadas por altas tensões elásticas, grandes deformações, e até mudanças topológicas na interface. Assim, a investigação numérica desse tipo de problema exige uma formulação precisa e robusta. No presente trabalho, um novo resolvedor de escoamentos bifásicos envolvendo fluidos complexos é apresentado, com particular interesse em escoamentos com altas taxas de elasticidade. A formulação proposta é baseada no método Volume-of-fluid (VOF) para representação da interface e no algoritmo Continuum Surface Force (CSF) para o balanço de forças na interface. A curvatura e advecção da interface são calculados via métodos geométricos para garantir a precisão dos resultados. Métodos de estabilização são utilizados quando números críticos de Weissenberg (Wi) são encontrados, devido ao famoso problema do alto número de Weissenberg (HWNP). O método da projeção, combinado com um método implícito para solução da equação da quantidade de movimento, são discretizados por um esquema de diferenças finitas em uma malha deslocada. Problemas de benchmarks foram resolvidos para acessar a precisão numérica da formulação em diferentes níveis de complexidade física, tal como representação e advecção da interface, influência das forças interfaciais, e características reológicas do fluido. A fim de demonstrar a capacidade do novo resolvedor, dois problemas bifásicos transientes, envolvendo fluidos viscoelásticos, foram resolvidos: o efeito de Weissenberg e o reômetro extensional (CaBER). O efeito de Weissenberg ou rod-climbing effect consiste em um bastão que gira dentro de um recipiente com fluido viscoelástico e, devido às forças elásticas, o fluido escala o bastão. Os resultados foram comparados com dados teóricos, numéricos e experimentais, encontrados na literatura para pequenas velocidades angulares. Além disso, resultados obtidos com altas velocidades angulares (alta elasticidade) são apresentados com o modelo Oldroyd-B, em que escaladas muito elevadas foram observadas. Valores críticos da velocidade angular foram identificados, e para valores acima foi observada a ocorrência de instabilidades elásticas, originadas pela combinação de tensões elásticas, curvatura interfacial, e escoamentos secundários. Até onde sabemos, numericamente, essas instabilidades nunca foram capturadas antes. O CaBER consiste no comportamento e colapso de um filamento de fluido viscoelástico, formado entre duas placas paralelas devido às forças capilares. Esse experimento envolve consideráveis dificuldades, dentre as quais podemos destacar a grande influência das forças capilares e a diferença de escalas de comprimento no escoamento. Em grande parte dos resultados encontrados na literatura, o CaBER é resolvido por modelos simplificados em uma dimensão. Resultados obtidos foram comparados com tais resultados da literatura e com soluções teóricas, apresentando admirável precisão. / Industrial applications involving multiphase flow are numerous. The improvement of some of these processes can result in a major technological leap with significant economic impact. The numerical study of these applications is essential because it provides accurate and more detailed information than conducting experiments. A challenge is the numerical study of high viscoelastic multiphase flows due to instabilities caused by the high elastic tension, large deformations and even topological changes in the interface. Thus the numerical investigation of this problem requires a robust formulation. In this study a new two-phase solver involving complex fluids is presented, with particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed formulation is based on the volume-of-fluid method (VOF) to interface representation and continuum surface force algorithm (CSF) for the balance of forces in the interface. The curvature and interface advection are calculated via geometric methods to ensure the accuracy of the results. Stabilization methods are used when critical Weissenberg numbers are found due to the famous high Weissenberg number problem (HWNP). The projection method combined with an implicit method for the solution of the momentum equation are discretized by a finite difference scheme in a staggered grid. Benchmark test problems are solved in order to access the numerical accuracy of different levels of physical complexities, such as the dynamic of the interface and the role of fluid rheology. In order to demonstrate the ability of the new resolver, two-phase transient problems involving viscoelastic fluids have been solved, theWeissenberg effect problem and the extensional rheometer (CaBER). The Weissenberg effect problem or rod-climbing effect consists of a rod that spins inside of a container with viscoelastic fluid and due to the elastic forces the fluid climbs the rod. The results were compared with numerical and experimental data from the literature for small angular velocities. Moreover results obtained for high angular velocities are presented using the Oldroyd-B model, which showed high climbing heights. Critical values of the angular speed have been identified. For values above a critical level were observed the occurrence of elastic instabilities caused by the combination of elastic tension, interfacial curvature and secondary flows. To our knowledge, numerically these instabilities were never captured before. The CaBER consists of the behavior and collapse of a viscoelastic fluid filament formed between two parallel plates due to capillary forces. This experiment involves considerable difficulties, among which we can highlight the great influence of the capillary forces and the difference of the length scales in the flow. In much of the results found in the literature, the CaBER is solved by simplified models. The results were compared with results reported in the literature and theoretical solutions, which showed remarkable accuracy.

Aplicações industriais envolvendo escoamentos multifásicos são inúmeras, sendo que, o aprimoramento de alguns desses processos pode resultar em um grande salto tecnológico com significativo impacto econômico. O estudo numérico dessas aplicações é imprescindível, pois fornece informações precisas e mais detalhadas do que a realização de testes experimentais. Um grande desafio é o estudo numérico de escoamentos viscoelásticos multifásicos envolvendo altas taxa de elasticidade, devido às instabilidades causadas por altas tensões elásticas, grandes deformações, e até mudanças topológicas na interface. Assim, a investigação numérica desse tipo de problema exige uma formulação precisa e robusta. No presente trabalho, um novo resolvedor de escoamentos bifásicos envolvendo fluidos complexos é apresentado, com particular interesse em escoamentos com altas taxas de elasticidade. A formulação proposta é baseada no método Volume-of-fluid (VOF) para representação da interface e no algoritmo Continuum Surface Force (CSF) para o balanço de forças na interface. A curvatura e advecção da interface são calculados via métodos geométricos para garantir a precisão dos resultados. Métodos de estabilização são utilizados quando números críticos de Weissenberg (Wi) são encontrados, devido ao famoso problema do alto número de Weissenberg (HWNP). O método da projeção, combinado com um método implícito para solução da equação da quantidade de movimento, são discretizados por um esquema de diferenças finitas em uma malha deslocada. Problemas de benchmarks foram resolvidos para acessar a precisão numérica da formulação em diferentes níveis de complexidade física, tal como representação e advecção da interface, influência das forças interfaciais, e características reológicas do fluido. A fim de demonstrar a capacidade do novo resolvedor, dois problemas bifásicos transientes, envolvendo fluidos viscoelásticos, foram resolvidos: o efeito de Weissenberg e o reômetro extensional (CaBER). O efeito de Weissenberg ou rod-climbing effect consiste em um bastão que gira dentro de um recipiente com fluido viscoelástico e, devido às forças elásticas, o fluido escala o bastão. Os resultados foram comparados com dados teóricos, numéricos e experimentais, encontrados na literatura para pequenas velocidades angulares. Além disso, resultados obtidos com altas velocidades angulares (alta elasticidade) são apresentados com o modelo Oldroyd-B, em que escaladas muito elevadas foram observadas. Valores críticos da velocidade angular foram identificados, e para valores acima foi observada a ocorrência de instabilidades elásticas, originadas pela combinação de tensões elásticas, curvatura interfacial, e escoamentos secundários. Até onde sabemos, numericamente, essas instabilidades nunca foram capturadas antes. O CaBER consiste no comportamento e colapso de um filamento de fluido viscoelástico, formado entre duas placas paralelas devido às forças capilares. Esse experimento envolve consideráveis dificuldades, dentre as quais podemos destacar a grande influência das forças capilares e a diferença de escalas de comprimento no escoamento. Em grande parte dos resultados encontrados na literatura, o CaBER é resolvido por modelos simplificados em uma dimensão. Resultados obtidos foram comparados com tais resultados da literatura e com soluções teóricas, apresentando admirável precisão. / Industrial applications involving multiphase flow are numerous. The improvement of some of these processes can result in a major technological leap with significant economic impact. The numerical study of these applications is essential because it provides accurate and more detailed information than conducting experiments. A challenge is the numerical study of high viscoelastic multiphase flows due to instabilities caused by the high elastic tension, large deformations and even topological changes in the interface. Thus the numerical investigation of this problem requires a robust formulation. In this study a new two-phase solver involving complex fluids is presented, with particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed formulation is based on the volume-of-fluid method (VOF) to interface representation and continuum surface force algorithm (CSF) for the balance of forces in the interface. The curvature and interface advection are calculated via geometric methods to ensure the accuracy of the results. Stabilization methods are used when critical Weissenberg numbers are found due to the famous high Weissenberg number problem (HWNP). The projection method combined with an implicit method for the solution of the momentum equation are discretized by a finite difference scheme in a staggered grid. Benchmark test problems are solved in order to access the numerical accuracy of different levels of physical complexities, such as the dynamic of the interface and the role of fluid rheology. In order to demonstrate the ability of the new resolver, two-phase transient problems involving viscoelastic fluids have been solved, theWeissenberg effect problem and the extensional rheometer (CaBER). The Weissenberg effect problem or rod-climbing effect consists of a rod that spins inside of a container with viscoelastic fluid and due to the elastic forces the fluid climbs the rod. The results were compared with numerical and experimental data from the literature for small angular velocities. Moreover results obtained for high angular velocities are presented using the Oldroyd-B model, which showed high climbing heights. Critical values of the angular speed have been identified. For values above a critical level were observed the occurrence of elastic instabilities caused by the combination of elastic tension, interfacial curvature and secondary flows. To our knowledge, numerically these instabilities were never captured before. The CaBER consists of the behavior and collapse of a viscoelastic fluid filament formed between two parallel plates due to capillary forces. This experiment involves considerable difficulties, among which we can highlight the great influence of the capillary forces and the difference of the length scales in the flow. In much of the results found in the literature, the CaBER is solved by simplified models. The results were compared with results reported in the literature and theoretical solutions, which showed remarkable accuracy.

Uma das mais recentes abordagens propostas na literatura para tratar o problema do alto número de Weissenberg (We) é a Formulação Log-Conformação (FLC). Nesta formulação, a equação constitutiva viscoelástica utilizada é reescrita em termos de uma variável Ψ, que é o logaritmo do tensor conformação. Apesar do potencial de aplicação da FLC, pouca atenção tem sido dirigida para análise da acurácia da solução obtida para o campo de tensões quando se utiliza esta formulação. Assim, o objetivo do presente trabalho foi estudar a acurácia da solução obtida pela FLC na análise de escoamentos de fluidos viscoelásticos usando duas geometrias padrão de estudo: placas paralelas e cavidade quadrada com tampa móvel. Primeiramente, a FLC foi implementada no pacote de CFD OpenFOAM. Em seguida foram verificados os limites do número de Weissenberg na formulação numérica padrão (Welim,P), onde para a geometria de placas paralelas foi encontrado Welim,P = 0,3 e para a geometria da cavidade quadrada com tampa móvel foi encontrado Welim,P = 0,8. Depois o código implementado foi aplicado em ambas as geometrias, comparando-se a solução obtida pela FLC com aquela da formulação padrão na faixa de We < Welim,P. Os resultados obtidos na geometria de placas paralelas apresentaram boa concordância com a solução padrão e solução analítica. Para a geometria da cavidade quadrada com tampa móvel, que não possui solução analítica, boa concordância dos resultados também foi observada em comparação com a solução padrão. Posteriormente foram comparados os resultados obtidos pela FLC na faixa de We > Welim,P. Na geometria de placas paralelas, além da boa concordância com a solução analítica, obteve-se convergência em todos os casos estudados neste trabalho, com o maior número de Weissenberg utilizado sendo igual a 8 Os resultados da geometria da cavidade quadrada com tampa móvel também apresentaram boa concordância em comparação com dados da literatura, porém a convergência foi obtida até para We = 2. Com respeito à comparação das formulações numéricas com a solução analítica, feita apenas na geometria de placas paralelas, foi observado um erro máximo de 7,57% na solução padrão e de 12,33% na FLC. Em relação à análise da qualidade das tensões usando os resíduos da equação constitutiva viscoelástica como critério de acurácia, foi verificado nas duas geometrias que os valores de tensão obtidos usando a FLC são menos acurados que aqueles obtidos pela formulação explícita no tensor das tensões nos casos em que esta última converge. Também foi observado que a acurácia diminui com o aumento do We. Esse efeito pôde ser melhor notado na geometria de placas paralelas. Uma razão para a perda de acurácia da tensão provavelmente ocorre devido à natureza matemática da transformação algébrica inversa de Ψxx para τxx. O novo solver implementado neste trabalho apresentou convergência e soluções corretas para as duas geometrias, logo foi implementado corretamente. Ele também potencializa o solver de partida viscoelastiFluidFoam ao estender simulações para uma faixa maior do número de Weissenberg. / A recent approach proposed in the literature to deal with the High Weissenberg Number Problem is the Log-Conformation formulation (LCF). In this formulation the viscoelastic constitutive equation is rewritten in terms of the logarithm of the conformation tensor Ψ. Despite the great potential application of the LCF, little attention has been given in the literature to the accuracy of the obtained stress fields. The purpose of this work was to study the solution obtained by LCF in the analysis of viscoelastic flows using two benchmark geometries: parallel plates and lid driven cavity. Firstly, the LCF was implemented in the OpenFOAM CFD package. Then, the limits of Weissenberg number for the standard numerical formulation (Welim,P) were verified, obtaining Welim,P = 0.3 for the parallel plates and Welim,P = 0.8 for the lid driven cavity. When comparing the solution obtained by the LCF with that of the standard formulation in a range of We < Welim,P, the results obtained for the parallel plates geometry showed good agreement with the standard solution and the analytical solution. For the lid driven cavity geometry, for which there is not analytical solution, good agreement with the standard solution was also observed. For We > Welim,P in the parallel plates geometry, in addition to the good agreement with the analytical solution, it was possible to obtain convergence in all the cases studied in this work, with the largest number of Weissenberg used being equal to 8 The results of the lid driven cavity geometry also presented good agreement in comparison with literature data, but convergence was obtained up to We = 2. With respect to the comparison of the numerical formulations with the analytical solution for the parallel plates geometry, a maximum error of 7.57% was observed in the standard solution and of 12.33% in the LCF. When using the residues of the viscoelastic constitutive equation as a criterion of accuracy, it was verified that for the two geometries the stress values obtained using the LCF were less accurate than those obtained by the explicit formulation in the stress tensor. It has also been observed that accuracy decreases with increasing of We. One reason for the loss of stress accuracy probably occurs because of the mathematical nature of the inverse algebraic transformation from Ψxx to τxx. The new solver implemented in this work presented convergence and correct solutions for the two geometries, so it was implemented correctly. It also potentiates the viscoelastiFluidFoam starting solver by extending simulations to a larger range of Weissenberg number.

Uma das mais recentes abordagens propostas na literatura para tratar o problema do alto número de Weissenberg (We) é a Formulação Log-Conformação (FLC). Nesta formulação, a equação constitutiva viscoelástica utilizada é reescrita em termos de uma variável Ψ, que é o logaritmo do tensor conformação. Apesar do potencial de aplicação da FLC, pouca atenção tem sido dirigida para análise da acurácia da solução obtida para o campo de tensões quando se utiliza esta formulação. Assim, o objetivo do presente trabalho foi estudar a acurácia da solução obtida pela FLC na análise de escoamentos de fluidos viscoelásticos usando duas geometrias padrão de estudo: placas paralelas e cavidade quadrada com tampa móvel. Primeiramente, a FLC foi implementada no pacote de CFD OpenFOAM. Em seguida foram verificados os limites do número de Weissenberg na formulação numérica padrão (Welim,P), onde para a geometria de placas paralelas foi encontrado Welim,P = 0,3 e para a geometria da cavidade quadrada com tampa móvel foi encontrado Welim,P = 0,8. Depois o código implementado foi aplicado em ambas as geometrias, comparando-se a solução obtida pela FLC com aquela da formulação padrão na faixa de We < Welim,P. Os resultados obtidos na geometria de placas paralelas apresentaram boa concordância com a solução padrão e solução analítica. Para a geometria da cavidade quadrada com tampa móvel, que não possui solução analítica, boa concordância dos resultados também foi observada em comparação com a solução padrão. Posteriormente foram comparados os resultados obtidos pela FLC na faixa de We > Welim,P. Na geometria de placas paralelas, além da boa concordância com a solução analítica, obteve-se convergência em todos os casos estudados neste trabalho, com o maior número de Weissenberg utilizado sendo igual a 8 Os resultados da geometria da cavidade quadrada com tampa móvel também apresentaram boa concordância em comparação com dados da literatura, porém a convergência foi obtida até para We = 2. Com respeito à comparação das formulações numéricas com a solução analítica, feita apenas na geometria de placas paralelas, foi observado um erro máximo de 7,57% na solução padrão e de 12,33% na FLC. Em relação à análise da qualidade das tensões usando os resíduos da equação constitutiva viscoelástica como critério de acurácia, foi verificado nas duas geometrias que os valores de tensão obtidos usando a FLC são menos acurados que aqueles obtidos pela formulação explícita no tensor das tensões nos casos em que esta última converge. Também foi observado que a acurácia diminui com o aumento do We. Esse efeito pôde ser melhor notado na geometria de placas paralelas. Uma razão para a perda de acurácia da tensão provavelmente ocorre devido à natureza matemática da transformação algébrica inversa de Ψxx para τxx. O novo solver implementado neste trabalho apresentou convergência e soluções corretas para as duas geometrias, logo foi implementado corretamente. Ele também potencializa o solver de partida viscoelastiFluidFoam ao estender simulações para uma faixa maior do número de Weissenberg. / A recent approach proposed in the literature to deal with the High Weissenberg Number Problem is the Log-Conformation formulation (LCF). In this formulation the viscoelastic constitutive equation is rewritten in terms of the logarithm of the conformation tensor Ψ. Despite the great potential application of the LCF, little attention has been given in the literature to the accuracy of the obtained stress fields. The purpose of this work was to study the solution obtained by LCF in the analysis of viscoelastic flows using two benchmark geometries: parallel plates and lid driven cavity. Firstly, the LCF was implemented in the OpenFOAM CFD package. Then, the limits of Weissenberg number for the standard numerical formulation (Welim,P) were verified, obtaining Welim,P = 0.3 for the parallel plates and Welim,P = 0.8 for the lid driven cavity. When comparing the solution obtained by the LCF with that of the standard formulation in a range of We < Welim,P, the results obtained for the parallel plates geometry showed good agreement with the standard solution and the analytical solution. For the lid driven cavity geometry, for which there is not analytical solution, good agreement with the standard solution was also observed. For We > Welim,P in the parallel plates geometry, in addition to the good agreement with the analytical solution, it was possible to obtain convergence in all the cases studied in this work, with the largest number of Weissenberg used being equal to 8 The results of the lid driven cavity geometry also presented good agreement in comparison with literature data, but convergence was obtained up to We = 2. With respect to the comparison of the numerical formulations with the analytical solution for the parallel plates geometry, a maximum error of 7.57% was observed in the standard solution and of 12.33% in the LCF. When using the residues of the viscoelastic constitutive equation as a criterion of accuracy, it was verified that for the two geometries the stress values obtained using the LCF were less accurate than those obtained by the explicit formulation in the stress tensor. It has also been observed that accuracy decreases with increasing of We. One reason for the loss of stress accuracy probably occurs because of the mathematical nature of the inverse algebraic transformation from Ψxx to τxx. The new solver implemented in this work presented convergence and correct solutions for the two geometries, so it was implemented correctly. It also potentiates the viscoelastiFluidFoam starting solver by extending simulations to a larger range of Weissenberg number.

Uma das mais recentes abordagens propostas na literatura para tratar o problema do alto número de Weissenberg (We) é a Formulação Log-Conformação (FLC). Nesta formulação, a equação constitutiva viscoelástica utilizada é reescrita em termos de uma variável Ψ, que é o logaritmo do tensor conformação. Apesar do potencial de aplicação da FLC, pouca atenção tem sido dirigida para análise da acurácia da solução obtida para o campo de tensões quando se utiliza esta formulação. Assim, o objetivo do presente trabalho foi estudar a acurácia da solução obtida pela FLC na análise de escoamentos de fluidos viscoelásticos usando duas geometrias padrão de estudo: placas paralelas e cavidade quadrada com tampa móvel. Primeiramente, a FLC foi implementada no pacote de CFD OpenFOAM. Em seguida foram verificados os limites do número de Weissenberg na formulação numérica padrão (Welim,P), onde para a geometria de placas paralelas foi encontrado Welim,P = 0,3 e para a geometria da cavidade quadrada com tampa móvel foi encontrado Welim,P = 0,8. Depois o código implementado foi aplicado em ambas as geometrias, comparando-se a solução obtida pela FLC com aquela da formulação padrão na faixa de We < Welim,P. Os resultados obtidos na geometria de placas paralelas apresentaram boa concordância com a solução padrão e solução analítica. Para a geometria da cavidade quadrada com tampa móvel, que não possui solução analítica, boa concordância dos resultados também foi observada em comparação com a solução padrão. Posteriormente foram comparados os resultados obtidos pela FLC na faixa de We > Welim,P. Na geometria de placas paralelas, além da boa concordância com a solução analítica, obteve-se convergência em todos os casos estudados neste trabalho, com o maior número de Weissenberg utilizado sendo igual a 8 Os resultados da geometria da cavidade quadrada com tampa móvel também apresentaram boa concordância em comparação com dados da literatura, porém a convergência foi obtida até para We = 2. Com respeito à comparação das formulações numéricas com a solução analítica, feita apenas na geometria de placas paralelas, foi observado um erro máximo de 7,57% na solução padrão e de 12,33% na FLC. Em relação à análise da qualidade das tensões usando os resíduos da equação constitutiva viscoelástica como critério de acurácia, foi verificado nas duas geometrias que os valores de tensão obtidos usando a FLC são menos acurados que aqueles obtidos pela formulação explícita no tensor das tensões nos casos em que esta última converge. Também foi observado que a acurácia diminui com o aumento do We. Esse efeito pôde ser melhor notado na geometria de placas paralelas. Uma razão para a perda de acurácia da tensão provavelmente ocorre devido à natureza matemática da transformação algébrica inversa de Ψxx para τxx. O novo solver implementado neste trabalho apresentou convergência e soluções corretas para as duas geometrias, logo foi implementado corretamente. Ele também potencializa o solver de partida viscoelastiFluidFoam ao estender simulações para uma faixa maior do número de Weissenberg. / A recent approach proposed in the literature to deal with the High Weissenberg Number Problem is the Log-Conformation formulation (LCF). In this formulation the viscoelastic constitutive equation is rewritten in terms of the logarithm of the conformation tensor Ψ. Despite the great potential application of the LCF, little attention has been given in the literature to the accuracy of the obtained stress fields. The purpose of this work was to study the solution obtained by LCF in the analysis of viscoelastic flows using two benchmark geometries: parallel plates and lid driven cavity. Firstly, the LCF was implemented in the OpenFOAM CFD package. Then, the limits of Weissenberg number for the standard numerical formulation (Welim,P) were verified, obtaining Welim,P = 0.3 for the parallel plates and Welim,P = 0.8 for the lid driven cavity. When comparing the solution obtained by the LCF with that of the standard formulation in a range of We < Welim,P, the results obtained for the parallel plates geometry showed good agreement with the standard solution and the analytical solution. For the lid driven cavity geometry, for which there is not analytical solution, good agreement with the standard solution was also observed. For We > Welim,P in the parallel plates geometry, in addition to the good agreement with the analytical solution, it was possible to obtain convergence in all the cases studied in this work, with the largest number of Weissenberg used being equal to 8 The results of the lid driven cavity geometry also presented good agreement in comparison with literature data, but convergence was obtained up to We = 2. With respect to the comparison of the numerical formulations with the analytical solution for the parallel plates geometry, a maximum error of 7.57% was observed in the standard solution and of 12.33% in the LCF. When using the residues of the viscoelastic constitutive equation as a criterion of accuracy, it was verified that for the two geometries the stress values obtained using the LCF were less accurate than those obtained by the explicit formulation in the stress tensor. It has also been observed that accuracy decreases with increasing of We. One reason for the loss of stress accuracy probably occurs because of the mathematical nature of the inverse algebraic transformation from Ψxx to τxx. The new solver implemented in this work presented convergence and correct solutions for the two geometries, so it was implemented correctly. It also potentiates the viscoelastiFluidFoam starting solver by extending simulations to a larger range of Weissenberg number.

It is well known that when a flexible or flexibly-mounted structure is placed perpendicular to the flow of a Newtonian fluid, it can oscillate due to the shedding of separated vortices at high Reynolds numbers. If the same flexible object is placed in non-Newtonian flows, however, the structure's response is still unknown. The main objective of this thesis is to introduce a new field of viscoelastic fluid-structure interactions by showing that the elastic instabilities that occur in the flow of viscoelastic fluids can drive the motion of a flexible structure placed in its path. Unlike Newtonian fluids, the flow of viscoelastic fluids can become unstable at infinitesimal Reynolds numbers due to the onset of a purely elastic flow instability. This instability occurs in the absence of nonlinear effects of fluid inertia and the Reynolds number of the flows studied here are in the order of 10-4. When such an elastic flow instability occurs in the vicinity of a flexible structure, the fluctuating fluid forces exerted on the structure grow large enough to cause a structural instability which in turn feeds back into the fluid resulting in a flow instability. Nonlinear periodic oscillations of the flexible structure are observed which have been found to be coupled to the time-dependent growth and decay of viscoelastic stresses in the wake of the structure. Presented in this thesis are the results of an investigation of the interaction occurring in the flow of a viscoelastic wormlike micelle solution past a flexible rectangular sheet. The structural geometries studied include: flexible sheet inclinations at 20°, 45° and 90° and flexible sheet widths of 5mm and 2.5mm. By varying the flow velocity, the response of the flexible sheet has been characterized in terms of amplitude and frequency of oscillations. Steady and dynamic shear rheology and filament stretching extensional rheology measurements are conducted in order to characterize the viscoelastic wormlike micelle solution. Bright field images show the deformation of the flexible sheet during an unstable oscillation while flow-induced birefringence images highlight the viscoleastic fluid stresses produced in the wake of the flexible sheet.

Recently, a new kind of turbulence has been discovered in the flow of concentrated polymer melts and solutions. These flows, known as purely elastic flows, become unstable when the elastic forces are stronger than the viscous forces. This contrasts with Newtonian turbulence, a more familiar regime where the fluid inertia dominates. While there is little understanding of purely elastic turbulence, there is a well-established dynamical systems approach to the transition from laminar flow to Newtonian turbulence. In this project, I apply this approach to purely elastic flows. Laminar flows are characterised by ordered, locally-parallel streamlines of fluid, with only diffusive mixing perpendicular to the flow direction. In contrast, turbulent flows are in a state of continuous instability: tiny differences in the location of fluid elements upstream make a large difference to their later locations downstream. The emerging understanding of the transition from a laminar to turbulent flow is in terms of exact coherent structures (ECS) — patterns of the flow that occur near to the transition to turbulence. The problem I address in this thesis is how to predict when a purely elastic flow will become unstable and when it will transition to turbulence. I consider a variety of flows and examine the purely elastic instabilities that arise. This prepares the ground for the identification of a three-dimensional steady state solution to the equations, corresponding to an exact coherent structure. I have organised my research primarily around obtaining a purely elastic exact coherent structure, however, solving this problem requires a very accurate prediction of the exact solution to the equations of motion. In Chapter 2 I start from a Newtonian ECS (travelling wave solutions in two-dimensional flow) and attempt to connect it to the purely elastic regime. Although I found no such connection, the results corroborate other evidence on the effect of elasticity on travelling waves in Poiseuille flow. The Newtonian plane Couette ECS is sustained by the Kelvin-Helmholtz instability. I discover a purely elastic counterpart of this mechanism in Chapter 3, and explore the non-linear evolution of this instability in Chapter 4. In Chapter 5 I turn to a slightly different problem, a (previously unexplained) instability in a purely elastic oscillatory shear flow. My numerical analysis supports the experimental evidence for instability of this flow, and relates it to the instability described in Chapter 3. In Chapter 6 I discover a self-sustaining flow, and discuss how it may lead to a purely elastic 3D exact coherent structure.

The polymers are integral part of vast number of products used in day to day life due to their anomalous viscoelastic behaviour. The remarkable flow behaviour exhibited by the polymeric fluids including rod climbing, extrudate swell, tube-less siphon, viscoelastic jet, elastic recoil and sharkskin instability is attributed to the complex microstructures in the polymeric liquids that arise due to the interactions of long chain polymer molecules with each other and with the surrounding fluid particles. The significance of polymer in transportation, packaging, pharmaceutical, chemical, biomedical, textiles, food and polymer processing industries highlights the requirement to comprehend the complex rheology of polymeric fluids. First, we investigate the flow features exhibited by different shear thinning vis-coelastic fluids in rectangular cavities over a wide range of depth to width ratio. We have developed a viscoelastic flow solver in order to perform numerical simulations of highly elastic flow of viscoelastic fluids. In particular, we discuss the simulations of flows of constant viscosity Boger and shear thinning viscoelastic fluids in the complex flow problems using different constitutive equations. The effects of elasticity and inertia on the flow behaviour of two shear thinning vis-coelastic fluids modeled using Giesekus and linear PTT constitutive equations in rectangular cavities is studied. The size of the primary eddies and critical aspect ratio over which the corner eddies merge to yield a second primary eddy in deep cavities is discussed. We demonstrate that the flow in the shallow and deep cavities can be characterized using Weissenberg number, defined based on the shear rate, and Deborah number, specified based on the convective time scale, respectively. The study of flow in driven cavities is important in understanding of the mixing process during synthesis of blends and composites. Next, we study two phase polymeric flow in confined geometries. Nowadays, polymer processing industries prefer to develop newer polymer with the desired material properties mechanically by mixing and blending of different polymer components instead of chemically synthesizing fresh polymer. The microstructure of blends and emulsions following drop deformation, breakup and coalescence during mixing determines its macroscopic interfacial rheology. We developed a two phase viscoelastic flow solver using volume conserving sharp interface volume-of-fluid (VOF) method for studying the dynamics of single droplet subjected to the complex flow fields. We investigated the effects of drop and matrix viscoelasticity on the motion and deformation of a droplet suspended in a fully developed channel flow. The flow behaviour exhibited by Newtonian-Newtonian, viscoelastic-Newtonian, Newtonian-viscoelastic and viscoelastic-viscoelastic drop-matrix systems is presented. The difference in the drop dynamics due to presence of constant viscosity Boger fluid and shear thinning viscoelastic fluid is represented using FENE-CR and linear PTT constitutive equations, respectively. The presence of shear thinning viscoelastic fluid either in the drop or the matrix phase suppresses the drop deformation due to stronger influence of matrix viscoelasticity as compared to the drop elasticity. The shear thinning viscoelastic drop-matrix system further restricts the drop deformation and it displays non-monotonic de-formation. The constant viscosity Boger fluid droplet curbs the drop deformation and exhibits flow dynamics identical to the shear thinning viscoelastic droplet, thus indicating that the nature of the drop viscoelasticity has little influence on the flow behaviour. The matrix viscoelasticity due to Boger fluid increases drop deformation and displays non-monotonic deformation. The drop deformation is further enhanced in the case of Boger fluid in viscoelastic drop-matrix system. Interestingly, the pressure drop due to the presence of viscoelastic drop in a Newtonian matrix is lower than the single phase flow of Newtonian fluid. We also discuss the effects of inertia, surface tension, drop to matrix viscosity ratio and the drop size on these drop-matrix systems. Finally, we investigate the emulsion rheology by studying the motion of a droplet in the square lid driven cavity flow. The viscoelastic effects due to constant viscosity Boger fluid and shear thinning viscoelastic fluid are illustrated using FENECR and Giesekus rheological relations, respectively. The presence of viscoelasticity either in drop or matrix phase boosts the drop deformation with the drop viscoelasticity displaying intense deformation. The drop dynamics due to the droplet viscoelasticity is observed to be independent of the nature of vis-coelastic fluid. The shear thinning viscoelastic matrix has a stronger influence on the drop deformation and orientation compared to the Boger fluid matrix. The different blood components, cells and many materials of industrial importance are viscoelastic in nature. Thus, the present study has significant applications in medical diagnostics, drug delivery, manufacturing and processing industries, study of biological flows, pharmaceutical research and development of lab-on-chip devices.