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Este trabalho apresenta o desenvolvimento de um método numérico para simular escoamentos viscoelásticos com superfícies livres de um fluido governado pelo modelo de Oldroyd-B. As equações governantes para um fluido Oldroyd-B são consideradas. A derivada temporal é aproximada por um método de segunda ordem. Uma formulação para o cálculo do tensor de tensão extra nos contornos rígidos é apresentada. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada utilizando variáveis primitivas. O método numérico descrito neste trabalho foi implementado no ambiente de simulação Freeflow-2D. Resultados numéricos demonstrando que o método numérico empregado neste trabalho aplicado a vários escoamentos bidimensionais de um fluido Oldroyd-B são apresentados.

In the recent years, researchers have discovered a multitude of applications using nanofibers in fields like composites, biotechnology, environmental engineering, defense, optics and electronics. This increase in nanofiber applications needs a higher rate of nanofiber production. Electrospinning has proven to be the best nanofiber manufacturing process because of simplicity and material compatibility. Study of effects of various electrospinning parameters is important to improve the rate of nanofiber processing. In addition, several applications demand well-oriented nanofibers. Researchers have experimentally tried to control the nanofibers using secondary external electric field. In the first study, the electrospinning process is modeled and the bending instability of a viscoelastic jet is simulated. For this, the existing discrete bead model is modified and the results are compared, qualitatively, with previous works in literature. In this study, an attempt is also made to simulate the effect of secondary electric field on electrospinning process and whipping instability. It is observed that the external secondary field unwinds the jet spirals, reduces the whipping instability and increases the tension in the fiber. Lattice Boltzmann method (LBM) has gained popularity in the past decade as the method is easy implement and can also be parallelized. In the second part of this thesis, a hybrid numerical scheme which couples lattice Boltzmann method with finite difference method for a Oldroyd-B viscoelastic solution is proposed. In this scheme, the polymer viscoelastic stress tensor is included in the equilibrium distribution function and the distribution function is updated using SRT-LBE model. Then, the local velocities from the distribution function are evaluated. These local velocities are used to evaluate local velocity gradients using a central difference method in space. Next, a forward difference scheme in time is used on the Maxwell Upper Convected model and the viscoelastic stress tensor is updated. Finally, using the proposed numerical method start-up Couette flow problem for Re = 0.5 and We = 1.1, is simulated. The velocity and stress results from these simulations agree very well with the analytical solutions.

Este trabalho apresenta o desenvolvimento de um método numérico para simular escoamentos viscoelásticos com superfícies livres de um fluido governado pelo modelo de Oldroyd-B. As equações governantes para um fluido Oldroyd-B são consideradas. A derivada temporal é aproximada por um método de segunda ordem. Uma formulação para o cálculo do tensor de tensão extra nos contornos rígidos é apresentada. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada utilizando variáveis primitivas. O método numérico descrito neste trabalho foi implementado no ambiente de simulação Freeflow-2D. Resultados numéricos demonstrando que o método numérico empregado neste trabalho aplicado a vários escoamentos bidimensionais de um fluido Oldroyd-B são apresentados.

Cette thèse est consacrée à l’analyse mathématique du problème du mouvement d’un nombre fini de corps rigides homogènes au sein d’un fluide visqueux incompressible homogène. Les fluides visqueux sont classés en deux catégories: les fluides newtoniens et les fluides non newtoniens. En premier lieu, nous considérons le système formé par les équations de Navier Stokes incompressible couplées aux lois de Newton pour décrire le mouvement de plusieurs disques rigides dans un fluide newtonien visqueux homogène dans l’ensemble de l’espace R^2. Nous montrons que ce problème est bien posé jusqu’à l’apparition de la première collision. Ensuite, nous éliminons tous les types de contacts pouvant survenir si le domaine fluide reste connexe à tout moment. Avec cette hypothèse, le système considéré est globalement bien posé. Dans la deuxième partie de cette thèse, nous montrons la non-unicité des solutions faibles au problème d’interaction fluide-solide 3D, dans le cas d’un fluide newtonien, après collision. Nous montrons qu’il existe des conditions initiales telles que nous pouvons étendre les solutions faibles après le temps pour lequel le contact a eu lieu de deux manières différentes. Enfin, dans la dernière partie, nous étudions le mouvement bidimensionnel d’un nombre fini de disques immergés dans une cavité remplie d’un fluide viscoélastique tel que des solutions polymériques. Les équations de Navier Stokes incompressible sont utilisées pour modéliser le solvant, dans lesquelles un tenseur de contrainte élastique supplémentaire apparaît comme un terme source. Dans cette partie, nous supposons que le tenseur de contrainte supplémentaire satisfait la loi différentielle d’Oldroyd ou sa version régularisée. Dans les deux cas, nous prouvons l’existence et l’unicité des solutions fortes locales en temps du problème considéré. / This thesis is devoted to the mathematical analysis of the problem of motion of afinite number of homogeneous rigid bodies within a homogeneous incompressible viscous fluid. Viscous fluids are classified into two categories: Newtonian fluids, and non-Newtonian fluids. First, we consider the system formed by the incompressible Navier-Stokes equations coupled with Newton’s laws to describe the movement of several rigid disks within a homogeneous viscous Newtonian fluid in the whole space R^2. We show the well-posedness of this system up to the occurrence of the first collision. Then we eliminate all type of contacts that may occur if the fluid domain remains connected at any time. With this assumption, the considered system is well-posed globally in time. In the second part of this thesis, we prove the non-uniqueness of weak solutions to the fluid-rigid body interaction problem in 3D in Newtonian fluid after collision. We show that there exist some initial conditions such that we can extend weak solutions after the time for which contact has taken place by two different ways. Finally, in the last part, we study the two-dimensional motion of a finite number of disks immersed in a cavity filled with a viscoelastic fluid such as polymeric solutions. The incompressible Navier–Stokes equations are used to model the flow of the solvent, in which the elastic extra stress tensor appears as a source term. In this part, we suppose that the extra stress tensor satisfies either the Oldroyd or the regularized Oldroyd constitutive differential law. In both cases, we prove the existence and uniqueness of local-in-time strongsolutions of the considered moving-boundary problem.

Submitted by RODRIGO KOITI ISHIZAKA (rodrigo_ishizaka@hotmail.com) on 2018-04-09T20:10:27Z No. of bitstreams: 1 Dissertação Final PDF.pdf: 10461391 bytes, checksum: a78974d2db3d166540e03d4cee7def37 (MD5) / Rejected by ALESSANDRA KUBA OSHIRO ASSUNÇÃO (alessandra@fct.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: - Número de página das referências no sumário está incorreto (consta p.86 quando é p.87); - No sumário consta Apêndices, porém no trabalho não. É finalizado nas referências. Agradecemos a compreensão. on 2018-04-10T12:44:45Z (GMT) / Submitted by RODRIGO KOITI ISHIZAKA (rodrigo_ishizaka@hotmail.com) on 2018-04-10T16:09:35Z No. of bitstreams: 1 Dissertação Final PDF.pdf: 10461372 bytes, checksum: 679a6e28fa80deb9079156b9b6b70c36 (MD5) / Approved for entry into archive by ALESSANDRA KUBA OSHIRO ASSUNÇÃO (alessandra@fct.unesp.br) on 2018-04-10T20:02:48Z (GMT) No. of bitstreams: 1 ishizaka_rk_me_prud.pdf: 10461372 bytes, checksum: 679a6e28fa80deb9079156b9b6b70c36 (MD5) / Made available in DSpace on 2018-04-10T20:02:48Z (GMT). No. of bitstreams: 1 ishizaka_rk_me_prud.pdf: 10461372 bytes, checksum: 679a6e28fa80deb9079156b9b6b70c36 (MD5) Previous issue date: 2018-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O presente trabalho de mestrado consiste em apresentar uma modelagem para escoamentos isotérmicos viscoelásticos em que a viscosidade varia de acordo com a pressão e estudos numéricos para alguns problemas bidimensionais, como escoamentos entre placas paralelas e expansão planar 1:4. Para este trabalho é utilizado o modelo viscoelástico Oldroyd-B e uma modelagem linear da viscosidade com relação a pressão. O método numérico desenvolvido é baseado no método da projeção para desacoplar velocidade e pressão nas equações de Navier-Stokes e depois calcula os tensores pela equação constitutiva com a informação da viscosidade variando com a pressão. As equações são discretizadas em uma malha deslocada pelo método de diferenças finitas. Os resultados numéricos são obtidos das simulações de escoamentos em um canal e expansão planar 1:4 bidimensional, cujo foco é destacar algumas diferenças entre o modelo Oldroyd-B Standard e o modelo Oldroyd-B sob influência da viscosidade que varia linearmente com a pressão. Este trabalho de mestrado também tem por objetivo resolver problemas, com fluidos viscoelásticos, em que haja regiões com variações da pressão e estudar estes resultados numéricos comparando-os com o modelo Oldroyd-B Standard. / The present master's work consists in presenting a model for viscoelastic isothermal flows with pressure-dependent viscosity and numerical studies for some two-dimensional problems, such as channel flow and planar expansion 1:4. In this work will be considered the Oldroyd-B viscoelastic model and a linear modeling viscosity with a linear relationship between viscosity and pressure. The numerical method developed is based on the projection method to decouple velocity and pressure in the Navier-Stokes equations and then calculates the tensor from the constitutive equation taking account the pressure-dependent viscosity. The equations are discretized on a staggered grid by using the finite diference method. The numerical results are obtained for two problems: two-dimensional channel and planar expansion 1:4 flows, whose focus is to highlight some diferences between the standard Oldroyd-B model and the Oldroyd-B model under the influence of viscosity that varies linearly with pressure. This master's work aims to solve problems of viscoelastic fluids in which there exist regions where pressure varies and to study these numerical results comparing them with the standard Oldroyd-B model.

Numerical simulations of the complex flows of complex (viscoelastic) fluids are investigated. The primary fluid investigated in this thesis is human blood, a complex fluid which can be modelled via viscoelastic constitutive models. The most commonly used constitutive models for viscoelastic fluids include the OldroydB, Giesekus, Johnson-Segalman, Finitely Extensible Non-Linear Elastic (FENE), Phan-Thein-Tanner (PTT) models etc. Our Numerical approach is based on the finite volume methods implemented on the OpenFOAM platform. We employ the Giesekus, Oldroyd-B, and Generalized Oldroyd-B viscoelastic constitutive models in this thesis, depending on the underlying context. Numerical validation of our results is conducted via the most used benchmark flow problems for viscoelastic fluid flow. The robust and efficient numerical methodologies are then deployed to investigate the flow characteristics, and hence illustrate various novel behavior, for blood flow in stenosed and bifurcated arteries. The present work took advantage of the availability of a reasonable set of viscoelastic constitutive model solvers within OpenFOAM, specifically the viscoelasticFluidFoam solver which we modified and developed to suit our focused needs for blood flow computations. The modified computational algorithms were successfully validated against well-known benchmark flow problems in the literature. Noting that the Giesekus viscoelastic constitutive model is a generalization of both the Oldroyd-B and Generalized Oldroyd-B models, the validation of results is carried out via the Giesekus model enabling us to develop a general-purpose code capable of simulating several viscoelastic constitutive models. The main results were otherwise presented for the Oldroyd-B and Generalized Oldroyd-B models as these are the most applicable to blood flow modelling. The results demonstrate that the velocity spurt through the stenosis is directly proportional to the constriction caused by the stenosis. The higher the blockage from the constriction, the higher the corresponding velocity spurt through the constriction. This velocity behavior, as the constriction blockage increases, correspondingly increase the wall shear stresses. High wall shear stresses significantly increase the possibility of rupture of the stenosis/blockage. This can lead to catastrophic consequences in the usual case where the stenosis is caused by tumor growth.

[pt] Filmes finos de líquidos estão presentes em uma gama de aplicações industriais, como processos de atomização e revestimento de substrato. O processo de quebra pode ser divido em duas etapas: o estágio de ruptura, e o estágio de retração. O primeiro, movido pelas forças de van der Waals, ocorre quando uma pequena perturbação cresce e provoca o aparecimento de um pequeno furo no filme. O segundo, movido por forças capilares, provoca o crescimento desse furo levando à desintegração do filme de líquido. A estabilidade de uma cortina de líquido depende das características da perturbação, da espessura do filme e das propriedades do fluido. Análises experimentais mostraram que uma cortina super fina pode ser obtida pela utilização de fluidos viscoelásticos. Os mecanismos físicos associados à esta estabilidade, contudo, não são totalmente compreendidos. Este trabalho apresenta um estudo numérico e teórico dos efeitos das propriedades viscoelásticas na estabilidade de uma cortina de fluido, englobando ambos os estágio do processo. As análises numéricas foram desenvolvidas através da expansão assintótica das variáveis do escoamento com aplicação de um esquema de integração no tempo totalmente implícito. A partir da análise teórica da dinâmica de ruptura foi possível obter um critério de estabilidade linear para perturbações planares e axissimétricas em fluidos Newtonianos e não-Newtonianos. O tempo de ruptura e a velocidade de retração do filme foram calculados numericamente como função das propriedades viscoelásticas do líquido. Resultados mostraram que as forças elásticas atuam de forma a dificultar o processo de quebra e retração. Análises da evolução da espessura mostraram que as propriedades reológicas do fluído também interferem no formato que o filme de fluido assume durante o processo de retração. Para regimes de baixa viscosidade, as forças elásticas atuaram evitando a formação de ondas capilares observadas em fluidos Newtonianos. / [en] Thin free liquid sheets are ubiquitous in many industrial processes, such as atomization and curtain coating. Liquid sheets are susceptible to instabilities at the interface, which can grow, triggering a breakup process. This process can be divided into two different stages: the rupture stage and retraction. The first, driven by van der Waals force, occurs when a small instability grows until it pinches-off the sheet. The second, driven by capillary forces, induces the growth of the hole caused by the pinch-off, leading to the full disintegration of the liquid sheet. The stability of a liquid sheet depends on disturbance characteristics, sheet thickness, and fluid properties. Experimental analyses have shown that thinner stable liquid curtain can be obtained with viscoelastic liquids. The underlyning physical mechanisms associated with increased stability are, however, not fully understood. This work presents a theoretical and numerical analysis of the effect of viscoelasticity on the stability of a thin liquid sheet during both stages of the breakup process. We first analyze the rupture dynamics, deriving linear stability criteria for both planar and axisymmetric perturbations of Newtonian and Oldroyd-B liquids. The time evolution of planar and axisymmetric perturbations in an Oldroyd-B liquid sheet is evaluated using the asymptotic expansion of the flow variables and a fully-implicit time integration scheme. The rupture time and retraction velocity are calculated as a function of the viscoelastic properties. The results show that the liquid rheological behavior does not influence the linear stability criterion. Nevertheless, it has a strong effect on the growth rate of the disturbance and retraction velocity, increasing, thus, the breakup time. The results show that elastic forces act to hinder the rupture and retraction stages. Analysis of the temporal evolution of the thickness profile reveals that liquid rheological behavior also affects the shape of the liquid sheet. For low viscosity regime, the elastic forces damp the capillary waves that arise during the retraction of Newtonian sheets.

In this thesis we provide an existence result for a regularized model of viscoelastic non- newtonian fluid. We consider incompressible fluid with shear rate dependent viscosity and with Cauchy stress tensor capable to describe stress relaxation. An elastic part of the Cauchy stress tensor is governed by Oldroyd-type differential equation. In particular, we are interested in fluids with strong shear thinning effect. We prove that if the viscosity function µ (D) is such that tensor µ (D) D is p-coercive, monotone and has (p − 1)-growth for p > 6 5 and some other additional assumptions are satisfied, then there exists a solution to the system of PDEs describing the flow in a bounded domain. The proof is not simple because the convective term is not integrable with a high power. The problem is solved using Lipschitz truncation method for evolution PDEs. 1

Our aim was to aid the viscoelasticity into the model for thermal convection by developing our own code in Fortran 90 and to study the role of viscoelasticity in this model. We should have included the viscoelasticity by Maxwell model; however, due to numerical instability we changed it for Oldroyd-B model. We were adding the terms of objective derivative into our code step by step and we were observing how they influence the behaviour of thermal convection. Partial time derivative and advective terms were included in whole complexity, the corrotational terms need more numerical testing. Our work suggest that the influence of viscoelasticity on thermal convection is noticeable. Powered by TCPDF (www.tcpdf.org)