CAPES; Petrobras / Fluidos de perfuração apresentam comportamento viscoplástico, que é definido por uma tensão limite de escoamento. Se a tensão imposta não supera a tensão limite de escoamento, o material se comporta como um sólido elástico, e quando esta tensão é excedida, o fluido escoa como um líquido viscoso. Durante o processo de perfuração de poços de petróleo, válvulas posicionadas na extremidade da coluna de perfuração são atuadas por diferenças de pressão no fluido. Argumenta-se, entretanto, que pressões aplicadas na superfície não são totalmente transmitidas até a válvula, impedindo sua operação. Logo, a transmissão de pressão em fluidos viscoplásticos demanda mais estudo. Para suprir a falta de estudos experimentais sobre o assunto, uma avaliação experimental da transmissão de pressão em um fluido viscoplástico é realizada neste trabalho. O aparato experimental construído para este propósito consiste de uma longa tubulação em formato helicoidal (serpentina) mantida sob temperatura controlada, ao longo da qual estão instalados transdutores de pressão relativa. O fluido é bombeado a uma vazão controlada por uma bomba do tipo helicoidal. Durante os testes, o fluido de trabalho é pressurizado na serpentina até um certo patamar ao mesmo tempo em que as pressões são medidas. Os resultados dos testes com fluidos viscoplásticos corroboram a literatura, que afirma que um fluido com tensão limite de escoamento confinado em uma tubulação fechada não transmite totalmente a pressão imposta na entrada ao longo do restante da tubulação. Além disso, foi observado que a diferença de pressão entre dois pontos da tubulação quando o fluido está em repouso é proporcional à tensão limite de escoamento deste fluido. Os resultados experimentais foram comparados com resultados de simulações numéricas de dois modelos matemáticos desenvolvidos no Centro de Pesquisas em Reologia e Fluidos Não newtonianos (CERNN), com boa concordância. / Drilling fluids exhibit a viscoplastic behavior, which is defined by a yield stress. If the stress imposed to the fluid does not surpass the yield stress, the material behaves like an elastic solid; if the yield stress is exceeded, it flows like a viscous liquid. In well drilling operations, some valves installed on the drillpipe near the bottom of the hole are actuated by pressure differences in the drilling fluid. However, it has been argued that the pressure applied at the surface is not fully transmitted to the valve’s position, preventing its actuation. Therefore, the pressure transmission in viscoplastic fluids demands further investigation. In order to compensate for the lack of experimental studies about the problem, an experimental analysis of the pressure transmission in yield stress fluids has been performed in this work. The experimental rig consists of a long thermally-controlled helical pipe, on which are installed relative pressure transducers. Fluid is displaced by a helical pump at a controlled flow rate. During the experiments, the fluid is pressurized inside the closed pipeline while the pressures are measured and recorded. The results showed that, in agreement with literature, the pressure at one end of a closed pipeline filled with a yield stress fluid is not fully transmitted to the other end. Moreover, it was observed that the pressure gradient in the pressurized fluid is proportional to its yield stress, which indicates a relation between pressure transmission and the presence and magnitude of the yield stress. The experiments were compared to simulation work developed at the Research Center for Rheology and Non-Newtonian Fluids with good agreement.

CERNN / Neste trabalho, propõe-se o estudo numérico do escoamento de fluidos de lei de potência e Bingham junto à interface entre uma região livre e outra porosa (interface fluido-porosa) utilizando o método lattice Boltzmann. Para tanto, considera-se o escoamento entre placas planas e paralelas entre as quais se faz presente um meio poroso abordado de forma heterogênea (resolução espacial da ordem de grandeza dos poros), representado através de obstáculos sólidos quadrados uniformemente distribuídos na parte inferior do canal. As análises realizadas mostram o efeito dos diversos parâmetros adimensionais que descrevem o problema sobre o fator de atrito na região livre do canal. De um modo geral, constata-se que a discrepância entre os fatores de atrito na região livre do canal e para o escoamento entre placas planas e paralelas cresce com o aumento da porosidade e do número de Bingham e com as reduções do número de obstáculos que compõem o meio poroso, número de Reynolds e índice de lei de potência. Ademais, propõe-se a adaptação do modelo analítico para a representação da interface fluido- porosa para escoamento de fluido newtoniano proposto por Ochoa-Tapia e Whitaker (1995b) ao escoamento de fluido de lei de potência, verificando-se a possibilidade de incorporar o comportamento não newtoniano do fluido ao parâmetro empírico do modelo. / The goal of this work is to numerically investigate the flow of power law and Bingham fluids next to the interface between a free and a porous region (fluid-porous interface) using the lattice Boltzmann method. For this, the flow between parallel plates partially filled by a porous material is studied, with the porous medium being represented by a set of solid square obstacles uniformly distributed in lower half of the channel. Results show the influence of non-dimensional parameters in the free region friction factor. In geral, it is observed that the friction factor decreases when porosity or Bingham number are increased and number of obstacles, Reynolds number or power law index are lowered. Moreover, it is porposed the application of the fluid-porous interface model proposed by Ochoa-Tapia e Whitaker (1995b) to the flow of power law fluids by varying the stress jump coefficient with the power law index.

CERNN / Neste trabalho, propõe-se o estudo numérico do escoamento de fluidos de lei de potência e Bingham junto à interface entre uma região livre e outra porosa (interface fluido-porosa) utilizando o método lattice Boltzmann. Para tanto, considera-se o escoamento entre placas planas e paralelas entre as quais se faz presente um meio poroso abordado de forma heterogênea (resolução espacial da ordem de grandeza dos poros), representado através de obstáculos sólidos quadrados uniformemente distribuídos na parte inferior do canal. As análises realizadas mostram o efeito dos diversos parâmetros adimensionais que descrevem o problema sobre o fator de atrito na região livre do canal. De um modo geral, constata-se que a discrepância entre os fatores de atrito na região livre do canal e para o escoamento entre placas planas e paralelas cresce com o aumento da porosidade e do número de Bingham e com as reduções do número de obstáculos que compõem o meio poroso, número de Reynolds e índice de lei de potência. Ademais, propõe-se a adaptação do modelo analítico para a representação da interface fluido- porosa para escoamento de fluido newtoniano proposto por Ochoa-Tapia e Whitaker (1995b) ao escoamento de fluido de lei de potência, verificando-se a possibilidade de incorporar o comportamento não newtoniano do fluido ao parâmetro empírico do modelo. / The goal of this work is to numerically investigate the flow of power law and Bingham fluids next to the interface between a free and a porous region (fluid-porous interface) using the lattice Boltzmann method. For this, the flow between parallel plates partially filled by a porous material is studied, with the porous medium being represented by a set of solid square obstacles uniformly distributed in lower half of the channel. Results show the influence of non-dimensional parameters in the free region friction factor. In geral, it is observed that the friction factor decreases when porosity or Bingham number are increased and number of obstacles, Reynolds number or power law index are lowered. Moreover, it is porposed the application of the fluid-porous interface model proposed by Ochoa-Tapia e Whitaker (1995b) to the flow of power law fluids by varying the stress jump coefficient with the power law index.

Nesse trabalho, foi desenvolvido um simulador numérico (C/C++) para a resolução de escoamentos de fluidos newtonianos incompressíveis, baseado no método de partículas Lagrangiano, livre de malhas, Smoothed Particle Hydrodynamics (SPH). Tradicionalmente, duas estratégias são utilizadas na determinação do campo de pressões de forma a garantir-se a condição de incompressibilidade do fluido. A primeira delas é a formulação chamada Weak Compressible Smoothed Particle Hydrodynamics (WCSPH), onde uma equação de estado para um fluido quase-incompressível é utilizada na determinação do campo de pressões. A segunda, emprega o Método da Projeção e o campo de pressões é obtido mediante a resolução de uma equação de Poisson. No estudo aqui desenvolvido, propõe-se três métodos iterativos, baseados noMétodo da Projeção, para o cálculo do campo de pressões, Incompressible Smoothed Particle Hydrodynamics (ISPH). A fim de validar os métodos iterativos e o código computacional, foram simulados dois problemas unidimensionais: os escoamentos de Couette entre duas placas planas paralelas infinitas e de Poiseuille em um duto infinito e foram usadas condições de contorno do tipo periódicas e partículas fantasmas. Um problema bidimensional, o escoamento no interior de uma cavidade com a parede superior posta em movimento, também foi considerado. Na resolução deste problema foi utilizado o reposicionamento periódico de partículas e partículas fantasmas. / In this work, we have developed a numerical simulator (C/C++) to solve incompressible Newtonian fluid flows, based on the meshfree Lagrangian Smoothed Particle Hydrodynamics (SPH) Method. Traditionally, two methods have been used to determine the pressure field to ensure the incompressibility of the fluid flow. The first is calledWeak Compressible Smoothed Particle Hydrodynamics (WCSPH) Method, in which an equation of state for a quasi-incompressible fluid is used to determine the pressure field. The second employs the Projection Method and the pressure field is obtained by solving a Poissons equation. In the study developed here, we have proposed three iterative methods based on the Projection Method to calculate the pressure field, Incompressible Smoothed Particle Hydrodynamics (ISPH) Method. In order to validate the iterative methods and the computational code we have simulated two one-dimensional problems: the Couette flow between two infinite parallel flat plates and the Poiseuille flow in a infinite duct, and periodic boundary conditions and ghost particles have been used. A two-dimensional problem, the lid-driven cavity flow, has also been considered. In solving this problem we have used a periodic repositioning technique and ghost particles.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-05-15T19:23:47Z No. of bitstreams: 1 tiberiodepaulanetto.pdf: 1926871 bytes, checksum: 17bceffda5c85de37a0d50a14f4f3f04 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-05-22T14:37:09Z (GMT) No. of bitstreams: 1 tiberiodepaulanetto.pdf: 1926871 bytes, checksum: 17bceffda5c85de37a0d50a14f4f3f04 (MD5) / Made available in DSpace on 2018-05-22T14:37:09Z (GMT). No. of bitstreams: 1 tiberiodepaulanetto.pdf: 1926871 bytes, checksum: 17bceffda5c85de37a0d50a14f4f3f04 (MD5) Previous issue date: 2017-02-22 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta tese, exploram-se diferentes aspectos e aplicações das teorias gravitacionais com correções quânticas. O texto é dividido em três partes principais. Na primeira parte, são consideradas as soluções linearizadas em diferentes teorias de gravitação com derivadas superiores. O potencial Newtoniano é calculado nos modelos locais, super-renormalizáveis no nível quântico, e mostra-se que a singularidade Newtoniana é cancelada devido a contribuição dos modos massivos extras. Logo depois, o colapso gravitacional de uma pequena massa é estudado na gravitação não-local livre de fantasmas, sendo o principal resultado a ausência da singularidade na solução do campo gravitacional e a possibilidade da não formação do miniburaco negro como resultado do colapso. Na segunda parte, algumas questões sobre a inflação induzida pela anomalia conforme são estudadas. É discutida a possibilidade da transição entre os períodos de inflação estável para instável. É mostrado que esta transição é automática se as correções quânticas nesse período forem desprezadas. Em seguida, considera-se o efeito de termos que violam as simetrias de CPT e Lorentz na inflação induzida pela anomalia conforme. É demonstrado que os novos termos responsáveis por violar essas simetrias não afetam a dinâmica do fator de escala da métrica. Por fim, na terceira parte as correções quânticas para o modelo dos Galileons e para as teorias dos campos massivos tensoriais antissimétricos são obtidas. É mostrado que o propagador da teoria dos Galileons recebe correções quânticas com derivadas superiores e que o teorema de não-renormalização do modelo dos Galileons permanece, de uma maneira generalizada, válido na região das baixas energias. Depois, por meio de cálculos explícitos das correções quânticas semiclássicas não-locais é confirmada a equivalência quântica entre os modelos dos campos tensoriais antissimétricos massivos com a teoria de Proca e com o modelo do campo escalar massivo mínimo. / In this thesis, different aspects and applications of gravitational theories with quan-tum corrections are explored. The text is divided into three main parts. In the first part, the linearized solutions in different gravity theories with higher derivatives are considered. The Newtonian potential is calculated in the local models, super-renormalizable at the quantum level, and it is shown that the Newtonian singularity is cancelled due to the contributions of the extra massive modes. Then the gravitational collapse of a small mass is studied in non-local ghost-free gravity, being the main result the absence of singularity in the gravitational field solution and the possibility of non-mini black hole formation as the collapse result. In the second part, some issues about anomaly-induced inflation are studied. It is discussed the possibility of the transition between stable to unstable periods of inflation. It is shown that this transition is automatic if the quantum corrections in this period are neglected. In the following, we consider the effect of CPT and Lorentz-violating terms in the conformal anomaly-induced inflation. It is shown the new terms responsible to violate these symmetries do not affect the dynamics of the metric scale factor. Finally, in the third part, the quantum corrections for the Galileon model and for the theory of the massive antisymmetric tensor fields are obtained. It is shown that the propaga-tor of Galilean theory receives quantum corrections with higher derivatives and that the non-renormalization theorem for Galileon models remains, in a generalized way, valid in the low-energy region. Then, by means of explicit calculations of non-local semiclassical quantum corrections, the quantum equivalence between the massive antisymmetric tensor field models with the Proca theory and minimal massive scalar field model is confirmed.

L’objectif de ce mémoire est d’introduire une nouvelle méthode smoothed particle hydrodynamics (SPH) implicite purement lagrangienne, pour la résolution des équations de Navier- Stokes incompressibles bidimensionnelles en présence d’une surface libre. Notre schéma de discrétisation est basé sur celui de Kéou Noutcheuwa et Owens [19]. Nous avons traité la surface libre en combinant la méthode multiple boundary tangent (MBT) de Yildiz et al. [43] et les conditions aux limites sur les champs auxiliaires de Yang et Prosperetti [42]. Ce faisant, nous obtenons un schéma de discrétisation d’ordre $\mathcal{O}(\Delta t ^2)$ et $\mathcal{O}(\Delta x ^2)$, selon certaines contraintes sur la longueur de lissage $h$. Dans un premier temps, nous avons testé notre schéma avec un écoulement de Poiseuille bidimensionnel à l’aide duquel nous analysons l’erreur de discrétisation de la méthode SPH. Ensuite, nous avons tenté de simuler un problème d’extrusion newtonien bidimensionnel. Malheureusement, bien que le comportement de la surface libre soit satisfaisant, nous avons rencontré des problèmes numériques sur la singularité à la sortie du moule. / The objective of this thesis is to introduce a new implicit purely lagrangian smoothed particle hydrodynamics (SPH) method, for the resolution of the two-dimensional incompressible Navier-Stokes equations in the presence of a free surface. Our discretization scheme is based on that of Kéou Noutcheuwa et Owens [19]. We have treated the free surface by combining Yildiz et al. [43] multiple boundary tangent (MBT) method and boundary conditions on the auxiliary fields of Yang et Prosperetti [42]. In this way, we obtain a discretization scheme of order $\mathcal{O}(\Delta t ^2)$ and $\mathcal{O}(\Delta x ^2)$, according to certain constraints on the smoothing length $h$. First, we tested our scheme with a two-dimensional Poiseuille flow by means of which we analyze the discretization error of the SPH method. Then, we tried to simulate a two-dimensional Newtonian extrusion problem. Unfortunately, although the behavior of the free surface is satisfactory, we have encountered numerical problems on the singularity at the output of the die.

A mecânica celular jaz nas propriedades materiais da membrana plasmática, fundamentalmente uma bicamada fosfolipídica com espessura de dimensões moleculares. Além de forças elásticas, tal material bidimensional também experimenta tensões viscosas devido ao seu comportamento fluido (presumivelmente newtoniano) na direção tangencial. A despeito da notável concordância entre teoria e experimentos biofísicos sobre a geometria de membranas celulares, ainda não se faz presente um método computacional para simulação de sua (real) dinâmica viscosa governada pela lei de Boussinesq-Scriven. Assim sendo, introduzimos uma formulação variacional mista de três campos para escoamentos viscosos de superfícies fechadas curvas. Nela, o fluido circundante é levado em conta considerando-se uma restrição de volume interior, ao passo que uma restrição de área corresponde à inextensibilidade. As incógnitas são a velocidade, o vetor curvatura e a pressão superficial, todas estas interpoladas com elementos finitos lineares contínuos via estabilização baseada na projeção do gradiente de pressão. O método é semi-implícito e requer a solução de apenas um único sistema linear por passo de tempo. Outro ingrediente numérico proposto é uma força que mimetiza a ação de uma pinça óptica, permitindo interação virtual com a membrana, onde a qualidade e o refinamento de malha são mantidos por remalhagem adaptativa automática. Extensivos experimentos numéricos de dinâmica de relaxação são apresentados e comparados com soluções quasi-analíticas. É observada estabilidade temporal condicional com uma restrição de passo de tempo que escala como o quadrado do tamanho de malha. Reportamos a convergência e os limites de estabilidade de nosso método e sua habilidade em predizer corretamente o equilíbrio dinâmico de compridas e finas elongações cilíndricas (tethers) que surgem a partir de pinçamentos membranais. A dependência de forma membranal com respeito a uma velocidade imposta de pinçamento também é discutida, sendo que há um valor limiar de velocidade abaixo do qual um tether não se forma de início. Testes adicionais ilustram a robustez do método e a relevância dos efeitos viscosos membranais quando sob a ação de forças externas. Sem dúvida, ainda há um longo caminho a ser trilhado para o entendimento completo da mecânica celular (há de serem consideradas outras estruturas tais como citoesqueleto, canais iônicos, proteínas transmembranares, etc). O primeiro passo, porém, foi dado: a construção de um esquema numérico variacional capaz de simular a intrigante dinâmica das membranas celulares. / Cell mechanics lies on the material properties of the plasmatic membrane, fundamentally a two-molecule-thick phospholipid bilayer. Other than bending elastic forces, such a two-dimensional interfacial material also experiences viscous stresses due to its (presumably Newtonian) surface fluid tangential behaviour. Despite the remarkable agreement on membrane shapes between theory and biophysical experiments, there is no computational method for simulating its (actual) viscous dynamics governed by the Boussinesq- Scriven law. Accordingly, we introduce a mixed three-field variational formulation for viscous flows of closed curved surfaces. In it, the bulk fluid is taken into account by means of an enclosed-volume constraint, whereas an area constraint stands for the membranes inextensible character. The unknowns are the velocity, vector curvature and surface pressure fields, all of which are interpolated with linear continuous finite elements by means of a pressure-gradient-projection scheme. The method is semi-implicit and it requires the solution of a single linear system per time step. Another proposed ingredient is a numerical force that emulates the action of an optical tweezer, allowing for virtual interaction with the membrane, where mesh quality and refinement are maintained by adaptively remeshing. Extensive relaxation experiments are reported and compared with quasi-analytical solutions. Conditional time stability is observed, with a time step restriction that scales as the square of the mesh size. We discuss both convergence and the stability limits of our method, its ability to correctly predict the dynamical equilibrium of the tether due to tweezing. The dependence of the membrane shape on imposed tweezing velocities is also addressed. Basically, there is a threshold velocity value below which the tethers shape does not arise at first. Further tests illustrate the robustness of the method and show the significance of viscous effects on membranes deformation under external forces. Undoubtedly, there is still a long way to track toward the understanding of celullar mechanics (one still has to account other structures such as cytoskeleton, ion channels, transmembrane proteins, etc). The first step has given, though: the design of a numerically robust variational scheme capable of simulating the engrossing dynamics of fluid cell membranes.

<p>Particle suspensions in viscoelastic fluids (e.g., polymeric fluids, liquid crystalline solutions, gels) are ubiquitous in industrial processes and in biology. In such fluids, particles often acquire lift forces that push them to preferential streamlines in the flow domain. This lift force depends greatly on the fluid’s rheology, and plays a vital role in many applications such as particle separations in microfluidic devices, particle rinsing on silicon wafers, and particle resuspension in enhanced oil recovery. Previous studies have provided understanding on how fluid rheology affects the motion of spherical particles in simple viscoelastic fluid flows such as shear flows. However, the combined effect of more complex flow profiles and particle shape is still under-explored. The main contribution of this thesis is to: (a) provide understanding on the migration and rotation dynamics of an arbitrary-shaped particle in complex flows of a viscoelastic fluid, and (b) develop guidelines for designing such suspensions for general applications.</p> <p><br></p> <p>In the first part of the thesis, we develop theories based on the second-order fluid (SOF) constitutive model to provide solutions for the polymeric force and torque on an arbitrary-shaped solid particle under a general quadratic flow field. When the first and second normal stress coefficients satisfy  <strong>Ψ</strong>₁  = −2 <strong>Ψ</strong> ₂ (corotational limit), the fluid viscoelasticity modifies only the fluid pressure and we provide exact solutions to the polymer force and torque on the particle. For a general SOF with  <strong>Ψ</strong> ₁ ≠  −2 <strong>Ψ</strong> ₂, fluid viscoelasticity modifies the shear stresses, and we provide a procedure for numerical solutions. General scaling laws are also identified to quantify the polymeric lift based on different particle shapes and orientation. We find that the particle migration speed is directly proportional to the length the particle spans in the shear gradient direction (L_sg), and that polymeric torques lead to unique orientation behavior under flow.</p> <p><br></p> <p>Secondly, we investigate the migration and rotational behavior of prolate and oblate spheroids in various viscoelastic, pressure-driven flows. In a 2-D slit flow, fluid viscoelasticity causes prolate particles to transition to a log-rolling motion where the particles orient perpendicular to the flow-flow gradient plane. This behavior leads to a slower overall migration speed (i.e., lift) of prolate particles towards the flow centerline compared to spherical particles of the same volume. In a circular tube flow, prolate particles align their long axis along the flow direction due to the extra polymer torque generated by the velocity curvature in all radial directions. Again, this effect causes prolate particles to migrate slower to the flow centerline than spheres of the same volume. For oblate particles, we quantify their long-time orientation and find that they migrate slower than spheres of the same volume, but exhibit larger migration speeds than prolate particles. Lastly, we examine the effect of normal stress ratio ? <strong>α</strong>  = <strong>Ψ</strong> ₂ /<strong>Ψ</strong>₁on the particle motion and find that this parameter only quantitatively impacts the particle migration velocity but has negligible effect on the rotational dynamics. We therefore can utilize the exact solution derived under the corotational limit (?<strong>α</strong> = −1/2) for a quick and reasonable prediction on the particle dynamics.</p> <p><br></p> <p>We next experimentally investigate the migration behavior of spheroidal particles in microfluidic systems and draw comparisons to our theoretical predictions. A dilute suspension of prolate/oblate microparticles in a density-matched 8% aqueous polyvinylpyrrolidone (PVP) solution is used as the model suspension system. Using brightfield microscopy, we qualitatively confirm our theoretical predictions for flow Deborah numbers 0 < De < 0.1 – i.e., that spherical particles show faster migration speed than prolate and oblate particles of the same volume in tube flows.</p> <p><br></p> <p>We finally design a holographic imaging method to capture the 3-D position and orientation of dynamic microparticles in microfluidic flow. We adopt in-line holography setup and propose a straightforward hologram reconstruction method to extract the 3-D position and orientation of a non-spherical particle. The method utilizes image moment to locate the particle and localize the detection region. We detect the particle position in the depth direction by quantifying the image sharpness at different depth position, and uses principal component analysis (PCA) to detect the orientation of the particle. For a semi-transparent particle that produces complex diffraction patterns, a mask based on the image moment information can be utilized during the image sharpness process to better resolve the particle position.</p> <p><br></p> <p>In the last part of this thesis, we conclude our work and discuss the future research perspectives. We also comment on the possible application of current work to various fields of research and industrial processes.</p> <p><br></p>

[pt] Este trabalho apresenta aplicações da técnica de otimização topológica para problemas de escoamento com fluidos não Newtonianos, usando o método dos elementos virtuais (VEM) em domínios bidimensionais arbitrários. O objetivo é projetar a trajetória ótima, a partir da minimização da energia dissipativa, de um escoamento governado pelas equações de Navier-Stokes-Brinkman e do modelo não Newtoniano de Carreau-Yasuda. A abordagem de porosidade proposta por (Borrvall e Petersson, 2003) [1] é usada na formulação do problema de otimização topológica. Para resolver este problema numericamente é usado o método VEM, recentemente proposto. A principal característica que diferencia o VEM do método dos elementos finitos (FEM) é que as funções de interpolação no interior dos elementos não precisam ser computadas explicitamente. Isso ocorre porque a integração é feita em funções polinomiais e bases de ordem inferior, permitindo assim uma grande flexibilidade no que diz respeito ao uso de elementos não convexos. Portanto, o cálculo das matrizes e vetores elementares se reduz à avaliação de grandezas geométricas nos contornos desses elementos. Finalmente, são apresentados exemplos numéricos representativos para demonstrar a eficiência do VEM em comparação com o FEM e a aplicabilidade da otimização topológica para esta classe de problemas de escoamento. / [en] This work presents selected applications of topology optimization for non-Newtonian fluid flow problems using the virtual element method (VEM) in arbitrary two-dimensional domains. The objective is to design an optimal layout into a fluid flow domain to minimize dissipative energy governed by the Navier-Stokes-Brinkman and non-Newtonian Carreau-Yasuda model equations. The porosity approach proposed by (Borrvall and Petersson, 2003) [1] is used in the topology optimization formulation. To solve this problem numerically, the recently proposed VEM method is used. The key feature that distinguishes VEM from the standard finite element method (FEM) is that the interpolation functions in the interior of the elements do not need to be computed explicitly. This is because the integration is on lower-order polynomial and basis functions, and there is great flexibility by using a non-convex element. Therefore, the computation of the main element matrices and vectors are reduced to the evaluation of geometric quantities on the boundary of the elements. Finally, several numerical examples are provided to demonstrate the efficiency of the VEM compared to FEM and the applicability of the topology optimization to fluid flow problems.

<p>Free surface flows, characterized by a deformable interface between two immiscible fluids or between a liquid and a gas, play a pivotal role in numerous natural phenomena and industrial processes. The fluid-fluid interface dynamics, governed by the complex interplay of forces such as inertia, capillary force, viscous force, and possibly elastic force, significantly influence the behavior of the fluids involved. Examples of free surface flows can be observed in everyday situations, such as droplet formation from a faucet, propagation and breaking of ocean waves, and tear films that coat the eye. An in-depth understanding of free surface flows and fluid-fluid interface dynamics has extensive implications for optimizing applications like inkjet printing, coating, spraying, and droplet formation while providing insights into the intricate behavior of natural fluid systems. Most of these applications, except for coating, involve abrupt and catastrophic topological changes of interfaces present in processes such as drop breakup, sheet rupture, and drop impact, where small droplets form from liquid sheets or filaments.</p> <p>This thesis examines the dynamics of contracting liquid filaments through computational means. Previous computational simulations have assumed that initially the fluid within the filament is quiescent which, however, may not typically be the case in practical applications. Here, the effect of a realistic, non-zero initial velocity profile is considered with the hypothesis that the fact that the fluid is already in motion when it starts to contract may result in significant alterations in the filament’s final fate vis-a-vis whether it breaks up into multiple small droplets or contracts into a sphere as its ends retract toward each other. The transient system of governing equations, the three-dimensional but axisymmetric (3DA) Navier-Stokes and continuity equations subjected to interfacial boundary conditions, are solved using rigorous and robust numerical algorithms in both fully 3DA and one-dimensional (1D) settings using the Galerkin finite element (GFEM) method. The simulation results are then used to construct comprehensive phase diagrams to delineate regions where filaments break up into smaller droplets from those where filaments contract to spheres without breakup.</p> <p>Polymer additives are often present in practical applications involving filament contraction and breakup. The presence of polymer molecules in an otherwise Newtonian solvent gives rise to non-Newtonian rheology. In this thesis, the dynamics of filaments containing polymer additives are analyzed using a 1D algorithm that is developed specifically for simulating viscoelastic free surface flows where the fluid’s rheology is described by the oft-used Oldroyd-B model. In real-world applications, filaments produced from nozzles are expected to be prestressed at the instant when they are created and begin to contract. It is demonstrated that the retraction velocity of tips of highly viscous, prestressed filaments is significantly increased compared to filaments in which the polymer molecules are initially relaxed and Newtonian filaments. This enhancement is explained by examining the value of f σ: D (σ: Elastic stress; D: Rate-of-strain tensor), which can be positive or negative. This quantity is positive when the flow does work on the polymer molecules but negative when the molecules do work on the flow, i.e., when elastic recoiling or unloading takes place. In prestressed filaments, elastic unloading takes place because σ: D < 0. The elastic stresses work by pulling the fluid in axially and pushing it out radially, thereby drastically increasing the tip velocity.  However, this enhancement in contraction velocity is not observed in low to intermediate viscosity prestressed filaments and whose Newtonian counterparts typically experience end-pinching. It has been established by others that end-pinching can be precluded in either filaments of intermediate viscosity or surfactant-laden filaments of low viscosity through a process known as escape from end-pinching. In this study, we demonstrate that a similar escape can also occur in prestressed viscoelastic filaments of low-to-intermediate viscosity, as revealed by one-dimensional numerical simulations and rationalized by examining when and where the elastic recoil takes place.</p> <p>Beyond cylindrical filaments, thin liquid films or planar liquid sheets are also prevalent in atomization, curtain coating, and other processes where liquid sheet stability has been a subject of extensive research. Numerous authors have examined wave formation and growth leading to sheet breakup. Free liquid films or sheets without edges or caps at their two ends, which typically have two free surfaces and are surrounded by air or sometimes another liquid, can destabilize and rupture due to intermolecular van der Waals attractive forces, despite the stabilizing influence of surface tension. In this thesis, the dynamics of contracting free films or sheets with caps---two-dimensional (2D) drops---of Newtonian fluids is examined without considering van der Waals forces to confirm or refute the hypothesis that such systems can rupture due to finite-amplitude perturbations even in the absence of intermolecular forces. In particular, both two-dimensional and one-dimensional high-accuracy simulations are employed to demonstrate that unlike inviscid 2D drops that can rupture in the absence of van der Waals forces, 2D drops or sheets can escape from pinch-off due to the action of viscous forces which are present in real systems no matter how small their viscosity. The reopening of the interface and escape from pinch-off in 2D drops and sheets are explained by demonstrating the key role played by vorticity. New power-law relations or scaling laws are obtained as a function of Ohnesorge number (ratio of viscous to the square root of the product of inertial and capillary forces) for the value of the minimum film thickness for which 2D drops or sheets stop thinning and after which the interface begins to reopen. Simple yet powerful arguments are presented rationalizing these scaling laws. It is expected that these power-law relations should be of great interest to experimentalists who study such phenomena by high-speed visualization experiments.</p> <p>Some of the motivation for this thesis research comes from crop spraying applications in which achieving zero or negligible drift is highly desirable. To further the understanding of fluid mechanics underpinning current and future drift reduction technologies, a simplified experimental setup is adopted to generate liquid sheets and analyze their disintegration into droplets. This new setup is both simpler and more universal than commonly utilized experimental systems that use single or multiple nozzles to generate liquid sheets and spray droplets from the disintegration of free liquid films. In the current experiments, droplets of test fluids are made to collide with or impact the top planar surface of a solid cylinder or rod. A series of MATLAB codes are developed and employed to extract droplet size distributions from images that are obtained from high-speed visualization experiments. The experimental setup and the means of data analysis are then used to probe the effect of fluid properties on the dynamics of sheet disintegration and droplet size distributions. It is hoped that future researchers will be able to combine what has been done in this thesis by simulations and in this chapter via experimental observations to develop an improved mechanistic understanding of spray formation.</p>