O presente trabalho destina-se ao estudo numérico da geometria de tubos de seção elíptica que facilite a transferência de calor adimensional e diminua a queda de pressão adimensional (Δ̃) sofrida pelo escoamento. O método aplicado é o Design Construtal, que visa determinar a geometria que apresentará a menor resistência ao escoamento, ou seja, busca-se determinar a razão de aspecto da elipse (=⁄) que favorece a transferência de calor e diminui a queda de pressão do escoamento. O fluido empregado neste estudo apresenta características de viscoplasticidade. A relação entre a tensão cisalhante e a taxa de deformação obedece ao modelo de Herschel-Bulkley modificado. Considera-se que o escoamento é incompressível, laminar, bidimensional, externo e ocorre em regime permanente. A solução numérica do problema proposto é realizada com um código comercial baseado no método dos volumes finitos. É investigada a influência do índice de potência, , sobre a seção elíptica que facilita o escoamento e, para isso, este índice é variado de 0,4 a 1. A influência dos números de Reynolds (√), Herschel-Bulkley (√) e Prandtl (√) sobre o comportamento do escoamento também é avaliada. √ é variado de 1 a 40, √ é variado de 1 a 100 e √ é variado de 0,1 a 100 Os resultados mostram que, para um escoamento com √=1, √=1 e √=1, o aumento do índice de potência influencia negativamente na transferência de calor adimensional e a seção elíptica, que maximiza esta transferência de calor adimensional, tende a ser mais alongada na direção do escoamento. Já e influenciam positivamente na transferência de calor adimensional. Para um escoamento com √=1, √=1, =0,4 conclui-se que com o aumento de a razão de aspecto ótima (q,opt), do ponto de vista térmico, diminui. Quando é considerado um escoamento com √=1, √=1, =0,4 conclui-se que q,opt diminui com o aumento de , ou seja, a elipse torna-se mais alongada no sentido do escoamento. A variação de √ em um escoamento com √=1, √=1, =0,4 mostra que o aumento deste parâmetro acarreta em aumento da taxa de transferência adimensional e de Δ̃. / The present work is aimed at the numerical study of the geometry of elliptic section tubes that facilitates the dimensionless heat transfer and decreases the dimensionless pressure drop (Δ̃) suffered by the flow. The applied method is the Construtal Design, which aims to determine the geometry that will present the least resistance to the flow, that is, to determine the aspect ratio of the ellipse (=⁄) that favors heat transfer and decreases the flow pressure drop. The fluid used in this study has viscoplasticity characteristics. The relationship between shear stress and strain rate follows the modified Herschel-Bulkley model. It is considered that the flow is incompressible, laminar, two-dimensional, external and occurs in steady state. The numerical solution of the proposed problem is carried out with a commercial code based on the finite volume method. The influence of the power index, n, on the elliptical section facilitating the flow is investigated, and for this, the index is varied from 0.4 to 1. The influence of the Reynolds number (√), Herschel-Bulkley number (√) and Prandtl number (√) on the flow behavior is also evaluated √ is varied from 1 to 40, √ is varied from 1 to 100 and √ is varied from 0.1 to 100. The results show that for a flow with √=1, √=1 and √=1, the increase of the power index negatively influences the dimensionless heat transfer and the elliptic section, which maximizes this dimensionless heat transfer, tends to be more elongated in the direction of flow. Already √ and √ influence positively the dimensionless heat transfer. For a flow with √=1, √=1, =0.4 it is concluded that with the increase of √ the optimum aspect ratio (q,opt), from the thermal point of view, decreases. When a flow is considered with √=1, √=1, =0.4 it is concluded that q,opt decreases with the increase of √, that is, ellipse becomes more elongated in the flow direction. The variation of √ in a flow with √=1, √=1, =0.4 shows that the increase of this parameter causes an increase of the dimensionless transfer rate and Δ̃.

La résolution analytique du problème de la stabilité d’une suspension solide (particules rigides de forme sphérique) dans le système de Taylor-Couette cylindrique a été menée. On s’est basé sur les travaux de Ali and Lueptow (2002) pour formuler les équations régissant la stabilité de l’écoulement dans le cadre d’une théorie linéaire. Ces équations sont valables dans l’approximation du faible espace annulaire et ont pour but la prévision de l’instabilité primaire. A cet effet, nous avons utilisé une méthode variationnelle telle que la méthode de Galerkin pour résoudre le problème aux valeurs propres conduisant à établir le diagramme de stabilité liée au nombre d’onde au voisinage de l’état critique du développement de la première instabilité. Tout d’abord, on a cherché à mettre au point les calculs dans le cas de l’écoulement de Taylor-Couette classique en se référant aux travaux de Ali and Lueptow (2002). Ensuite on a procédé à la résolution systématique des équations du mouvement et l’on cherche à déterminer le critère d’apparition des instabilités en présence de particules en suspension et l’on détermine simultanément les paramètres de couplage entre forces d’interaction liquide-solide. L’ensemble des travaux ainsi réalisés permettront de lever la contradiction fondamentale entre la théorie et l’expérience. L’étude expérimentale a permis d’analyser les effets de la concentration des particules en suspension (disques) et du rapport d’aspect radial ’ sur l’apparition des instabilités dans le système de Taylor-Dean. Le dispositif expérimental utilisé est constitué d’un cylindre intérieur tournant et le cylindre extérieur est maintenu fixe. Le comportement rhéologique du fluide utilisé est viscoplastique obéissant au modèle de Herschel Bulkley. L’apparition des instabilités est examinée en utilisant une technique de visualisation. Pour une concentration donnée dans l’intervalle étudié, la nature des structures apparaissant dans le système d’écoulement dépend ’, alors que pour une valeur donnée de ’ dans l’intervalle étudié, la valeur du nombre de Taylor critique Tac dépend de la concentration des particules. Nous avons obtenu que le nombre de Taylor critique Tac correspondant au déclenchement de la première instabilité évolue non linéairement en fonction de ’. De plus, nous avons examiné expérimentalement les effets de limitation axiale (effet de bords) sur le déclenchement des instabilités dans le système de Taylor-Dean. Les résultats obtenus montrent que les bords tournants n’affectent pas le type de structures qui apparaissent dans le système d’écoulement de Taylor-Dean. Cependant, ils influencent le seuil critique d’apparition des instabilités qui est marquée par des valeurs élevées du nombre de Taylor critique pour des bords tournants ce qui indique un effet stabilisant des bords mobiles. / The analytical solution of the stability problem of a solid suspension (rigid spherical particles) in the system of cylindrical Taylor-Couette was conducted. We are based on the work of Ali and Lueptow (2002) to formulate the equations governing the stability of the flow in a linear theory. These equations are valid in the approximation of small gap configuration and aim to predict the primary instability. For this purpose, we used a variational method such as the Galerkin method to solve the eigenvalue problem leading to establish the stability diagram related to the wave number in the vicinity of the critical state of development of the first instability. First, we develop the calculations in the case of Taylor-Couette flow with reference to classic work of Ali and Lueptow (2002). Then, we carried out a systematic solution of the equations of motion and we search to determine the criterion of onset of instabilities in the presence of suspended particles and coupling parameters are simultaneously determined from liquid-solid interaction force. All work carried out and will remove the fundamental contradiction between theory and experiment. The experimental study has analyzed the effect of the concentration of suspended particles (disks) and radial aspect ratio ' on the occurrence of instabilities in the Taylor-Dean flow system. The experimental device used consists of a rotating inner cylinder and the outer cylinder is stationary. The rheological behavior of the fluid is viscoplastic obeying to Herschel Bulkley model. The onset of instability is examined using a visualization technique. For a given concentration in the range studied, the nature of the structure appearing in the flow system depends on ', while for a given value of ' in the range studied, the value of the critical Taylor number Tac depends on the particle concentration. We obtain that the critical Taylor number Tac corresponding to the onset of the first instability evolves nonlinearly versus '. In addition, we examined experimentally the effect of axial limitation (endwall effects) on the onset of instabilities in the Taylor-Dean flow system. The results show that the rotating endwalls do not affect the type of structures that appear in the Taylor-Dean flow system. However, they influence the threshold of appearance of instabilities which is characterized by high values of the critical Taylor number for rotating endwalls indicating a stabilizing effect of the rotating endwalls.

[pt] Uma abordagem experimental foi utilizada para o estudo da eficiência de varredura de fluidos viscoplásticos na célula de Hele-Shaw , que representa um meio poroso com porosidade constante. A análise baseia-se em um fluido empurrando outro onde, pela razão de viscosidades acima do punto de referência , manifesta-se a instabilidade de Saffman-Taylor, que desencadeia em uma interface instável com divisões sucessivas conhecidas como viscous fingers. A formação da instabilidade é considerada uma condição indesejável, e no caso da indústria petrolífera, torna-se de interesse na invasão da formação pelo fluido de perfuração, pasta de cimento, e no deslocamento de petróleo pesado em reservatórios entre outros. Apesar de que o fenômeno é vastamente estudado utilizando a célula de Hele-Shaw, a maioria dos trabalhos concentram-se no deslocamento de líquidos newtonianos por ar, deixando uma ampla gama de possibilidades de estudo, particularmente no que diz respeito aos fluidos não newtonianos. Utilizando equipamentos que permitem assegurar uma vazão constante de injeção, testou-se três concentrações diferentes de solução aquosa de carbopol em duas configurações diferentes onde o líquido viscoplástico é o deslocador e vice-versa. Fazendo uso do processamento digital de imagens, avaliou-se a forma da interface em função do desvio padrão, além da velocidade e da eficiência de varredura no deslocamento. Determinaram-se parâmetros adimensionais em função da geometria da bancada e dos fatores dinâmicos e reológicos dos fluidos onde se obtiveram duas zonas vem marcadas de varredura. Observou-se que a razão de viscosidades e o número de capilaridade modificada crítica são os indicadores que governam o deslocamento. / [en] An experimental approach has been used to study the sweep efficiency of viscoplastic fluid in a Hele-Shaw cell, which represents a porous medium with constant porosity. The analysis is based on a lower viscosity fluid pushing a more viscous through a geometry, which manifests the instability of Saffman-Taylor, that result in an unstable interface with successive divisions known as viscous fingers. The occurrence of instability is considered an undesirable condition, and in the petroleum oil industry, it becomes of interest in the reservoir oil invasion and heavy oil displacement in reservoirs. Despite being widely studied, most studies focus on the displacement of Newtonian liquids by air, leaving the field of study of non-Newtonian fluids without being evaluated where none of viscosities of the fluids involved can be considered negligible. Using equipment that ensures a constant flow rate of injection three different aqueous concentrations of carbopol were tested in two different configurations: carbopol as the displacing fluid and vice-versa. Using digital image processing, the shape interface was evaluated according to their standard deviation, average velocity and efficiency sweep during the displacement. Determined dimensionless parameters was evaluated in function of the experimental geometry, dynamic and rheological factors of the fluids in order to get benchmark that allow separating interfaces zones that tend to 100 per cent or plugs and the zone where the viscous fingers reveal themselves. It was found that the viscosity ratio and modified critical capillary govern the displacement.

[pt] A superposição de um escoamento circular de Couette e um fluxo com gradiente de pressão axial, através de um espaço anular ocorre em muitas aplicações práticas, tais como: reatores químicos catalíticos, filtros, extratores líquido- líquido, mancais e o fluxo de retorno de lamas de perfuração entre a coluna de perfuração rotatória e a formação rochosa na perfuração de poços produtores de petróleo e gás. As linhas de corrente curvadas do fluxo circular de Couette podem causar uma instabilidade centrífuga que produz vórtices toroidais, conhecidos como vórtices de Taylor. A presença destes vórtices muda as características hidrodinâmicas e a transferência de calor no processo. Em conseqüência, é muito importante ser capaz de prever o aparecimento da instabilidade. A maioria das análises numéricas e experimentais disponíveis na literatura são para fluidos Newtonianos e viscoelásticos (soluções polimericas). Neste trabalho, o efeito das propriedades viscoplásticas de suspensões de altas concentrações neste tipo de escoamento e nas condições críticas para o aparecimento de vórtices são determinadas teoricamente através da solução das equações de conservação. As equações diferenciais foram integradas pelo método de elementos finitos-Galerkin e o sistema de equações algébricas não lineares resultante foi resolvido pelo método de Newton. / [en] The superposition of a circular Couette flow and a pressure- driven axial flow in an annulus occurs in many practical applications, such as catalytic chemical reactors, filtration devices, liquid-liquid extractors, journal bearings, and the return flow of drilling mud between the rotating drill string and the stationary wall in oil and gas well drilling. The curved streamlines of the circular Couette flow can cause a centrifugal instability leading to toroidal vortices, well known as Taylor vortices. The presence of these vortices changes the hydrodynamic and heat transfer characteristics of the process. Therefore, it is very important to be able to predict the onset of instability. Most of the available theoretical and experimental analyses are for Newtonian and viscoelastic (polymeric solutions) liquids. In this work, the effect of the viscoplastic properties of high concentration suspensions on the onset of the Taylor vortices are determined theoretically by solving the conservation equations and searching the critical conditions. The differential equations were solved by the Galerkin / finite element method and the resulting set of non-linear algebraic equations, by Newtons method.

Les écoulements rampants, recirculants et instationnaires d’un fluide viscoplastique autour d’un cylindre ont été étudiés.Numériquement, les morphologies des écoulements, la localisation des zones rigides, les champs de contraintes et pression autour du cylindre ainsi que le coefficient de traînée, ont été déterminés sur un large domaine des nombres de Reynolds et d’Oldroyd.Expérimentalement, les fluides étudiés sont des gels de polymère Carbopol®. Le comportement élastoviscoplastique de ces gels a été modélisé par une loi d’Herschel-Bulkley adaptée. Le montage expérimental conçu et réalisé a été validé par l'étude de l'écoulement d'un fluide newtonien autour d'un cylindre et la mise en place d’une procédure adaptée pour les fluides à seuil.On a pu constater l'influence des conditions d’interface avec l’apparition d’une morphologie de lâchers de tourbillons simultanés et symétriques. / Creeping, recirculating and unsteady flows around a cylinder for yield stress fluid were studied.Numerically, morphologies of the flows, the location of the unyielded zones, the pressure and stress fields around the cylinder and the drag coefficient were determined over a wide range of Reynolds and Oldroyd numbers.Experimentally, fluids studied are polymer gels Carbopol®. The elastoviscoplastic behavior of these gels was modeled by a Herschel-Bulkley adapted law. The experimental setup was designed and validated by studying the flow of a newtonian fluid around a cylinder. An appropriate procedure for the viscoplastic fluid was implemented.We observe the influence of interface conditions with the appearance of a morphology showing simultaneous and symmetrical vortex shedding.

[pt] Este trabalho investiga o impacto de suspensões de grafeno funcionalizado sobre reologia de uma dispersão aquosa de Carbopol. Os derivados de grafeno usados foram óxido de grafeno (GO) e óxido de grafeno amino funcionalizado (AFGO). A variação da concentração e do grau de oxidação do grafeno funcionalizado foram avaliadas e relacionadas à reologia das suspensões. O GO foi produzido a partir da síntese de óxido de grafite pelo método de Hummers modificado, e caracterizado pelas técnicas de DRX, Raman, TGA, FTIR, XPS, TEM e AFM. O GO foi funcionalizado com trietilenotetramina por reação assistida por microondas para produzir AFGO, caracterizado por TGA e XPS. As suspensões foram caracterizadas por experimentos de potencial zeta e testes reológicos de escoamento em estado estacionário e oscilatório. A caracterização do GO mostra que grupos funcionais oxigenados foram incorporados em sua superfície grafítica. O GO oxidado por 96 horas (GO 96 h) apresentou maior distância interplanar e também apresentou menos camadas quando comparado com GO oxidado por 2 horas (GO 2 h). A caracterização do AFGO aponta que grupos aminados foram covalentemente ligados às nanofolhas GO e o GO com maior grau de oxidação produziu AFGO com maior nível de aminação. Todas as nanofolhas sintetizadas são coloidalmente estáveis quando suspensas em meio neutro. Todas as suspensões foram bem modeladas pela equação de Hershel-Bulkley. O aumento da concentração de nanofolhas nas suspensões prejudica a microestrutura do fluido e leva a uma diminuição da viscosidade, tensão de escoamento e elasticidade. O GO 96 h promoveu menor decréscimo na viscosidade, tensão de escoamento e elasticidade do que a suspensão GO 2 h. No caso do AFGO, o maior grau de aminação pode levar a uma queda mais pronunciada nas propriedades reológicas da suspensão. Para a suspensão com maior concentração de GO 96 h, observou-se o aparecimento de histerese em baixas taxas de cisalhamento. Esses resultados mostram que pequenas mudanças na superfície das nanofolhas de grafeno funcionalizado podem influenciar as respostas reológicas de um fluido não newtoniano. / [en] This work investigates the impact of functionalized graphene suspensions on the rheology of a Carbopol aqueous dispersion. The graphene derivatives used were graphene oxide (GO) and amino-functionalized graphene oxide (AFGO). The variation of the functionalized graphene concentration and oxidation was evaluated and related with the suspensions rheology. GO nanosheets were produced from synthesis of graphite oxide by modified Hummers method, and they were characterized by XRD, Raman,TGA, FTIR, XPS, TEM and AFM techniques. The GO was functionalized with triethylenetetramine by microwave assisted reaction to produce the AFGO, which was characterized by TGA and XPS techiniques. The suspensions were characterized by zeta potential experiments and rheological tests through steady-state and oscillatory flow. The GO characterization shows that oxygenated functional groups were incorporated in its graphitical surface. GO oxidized for 96 hours (GO 96 h) showed higher interplanar distance and also presented fewer layers when compared with GO oxidized for 2 hours (GO 2 h). The AFGO characterization points out that aminated groups were covalently attached to the GO nanosheets and the GO with a higher oxidation degree produced an AFGO with higher amination level. All synthesized nanosheets are colloidally stable when suspended at neutral media. All suspensions were well modeled by the Hershel-Bulkley equation. The increase of the nanosheets concentration in the suspensions impairs the level of fluid structure and leads to a decrease in viscosity, yield stress, and elasticity. The GO 96 h promoted a lower decrease in viscosity, yield stress and elasticity than the GO 2 h suspension. In the case of AFGO, the greater amination degree can lead to a more pronounced drop in the suspension rheological properties. For the suspension with a higher concentration of GO 96 h, it was observed the appearance of hysteresis at low shear rates. These results show that small changes in the graphene functionalized nanosheets surface can influence the rheological responses of a non-Newtonian fluid.

Nous présentons une contribution autour de la modélisation des écoulements viscoplastiques. En vue d'applications réalistes telle que la simulation numérique des coulées de lave volcanique, le travail se concentre particulièrement sur les fluides complexes dont la rhéologie dépend fortement de grandeurs physiques telle que la température ou la concentration en particule. Nous développons un nouvel algorithme de résolution numérique des équations de Herschel-Bulkley combinant une méthode de Lagrangien augmenté à paramètre d'augmentation variable, une méthode des caractéristiques d'ordre 2 et une adaptation de maillage automatique. Sur des problèmes stationnaires ou en évolution tel que le problème test de la cavité entraînée, il apporte une solution efficace pour garantir à la fois une précision numérique élevée et un temps de calcul raisonnable. Cet algorithme est ensuite étendue et adapté au cas des rhéologies non-isothermes et aux suspensions. Concernant la simulation numérique des coulées de lave volcanique, nous détaillons une méthode de réduction par analyse asymptotique des équations de Herschel-Bulkley pour des écoulements de faible épaisseur sur une topographie arbitraire. Elle permet alors de décrire ces écoulements tridimensionnels de fluides viscoplastiques à surface libre par des équations bidimensionnelles surfaciques. Cette approche est ensuite étendue au cas non-isotherme en y ajoutant l'équation de la chaleur et des dépendances thermiques sur la rhéologie. Par intégration verticale de l'équation de la chaleur, on retrouve un modèle bidimensionnel. Le modèle non-isotherme est validé sur une expérience de dôme réalisée en laboratoire et une simulation numérique est réalisée autour d'une coulée qui a eu lieu sur le volcan du Piton de la Fournaise à la Réunion, en décembre 2010. La comparaison donne des résultats qui sont de notre point de vue satisfaisants et encourageants. / We present a contribution about modeling of viscoplastic flows. For realistic applications such as numerical simulation of volcanic lava flows, the work focuses particularly on complex fluids whose rheology strongly depends on physical quantities such as temperature or the particle concentration. We develop a new numerical resolution algorithm of Herschel-Bulkley's equations combining an augmented Lagrangian method with variable augmentation parameter, a second order characteristic method and an auto-adaptive mesh procedure. On stationary or evolving problems as the lid-driven cavity flow benchmark, it provides an effective solution to ensure both a high numerical accuracy within a reasonable computing time. This algorithm is then extended and adapted to the case of non-isothermal rheological and suspensions. On the numerical simulation of volcanic lava flows, we describe a method of reducing by asymptotic analysis of the Herschel-Bulkley's equations for thin flows on arbitrary topography. It allows to describe the three-dimensional flows of viscoplastic fluid with free surface by bidimensional surface equations. This approach is then extended to the non-isothermal case by adding the heat equation and thermal dependencies on rheology. By vertical integration of the heat equation, a two-dimensional model is maintained . The non-isothermal model is validated on a laboratory experiment of dome and a numerical simulation is performed on a December 2010 Piton de la Fournaise lava flow from La Réunion island. In our view, the comparison gives satisfactory and encouraging results.

Nous présentons une contribution autour de la modélisation des écoulements viscoplastiques. En vue d'applications réalistes telle que la simulation numérique des coulées de lave volcanique, le travail se concentre particulièrement sur les fluides complexes dont la rhéologie dépend fortement de grandeurs physiques telle que la température ou la concentration en particule. Nous développons un nouvel algorithme de résolution numérique des équations de Herschel-Bulkley combinant une méthode de Lagrangien augmenté à paramètre d'augmentation variable, une méthode des caractéristiques d'ordre 2 et une adaptation de maillage automatique. Sur des problèmes stationnaires ou en évolution tel que le problème test de la cavité entraînée, il apporte une solution efficace pour garantir à la fois une précision numérique élevée et un temps de calcul raisonnable. Cet algorithme est ensuite étendue et adapté au cas des rhéologies non-isothermes et aux suspensions. Concernant la simulation numérique des coulées de lave volcanique, nous détaillons une méthode de réduction par analyse asymptotique des équations de Herschel-Bulkley pour des écoulements de faible épaisseur sur une topographie arbitraire. Elle permet alors de décrire ces écoulements tridimensionnels de fluides viscoplastiques à surface libre par des équations bidimensionnelles surfaciques. Cette approche est ensuite étendue au cas non-isotherme en y ajoutant l'équation de la chaleur et des dépendances thermiques sur la rhéologie. Par intégration verticale de l'équation de la chaleur, on retrouve un modèle bidimensionnel. Le modèle non-isotherme est validé sur une expérience de dôme réalisée en laboratoire et une simulation numérique est réalisée autour d'une coulée qui a eu lieu sur le volcan du Piton de la Fournaise à la Réunion, en décembre 2010. La comparaison donne des résultats qui sont de notre point de vue satisfaisants et encourageants. / We present a contribution about modeling of viscoplastic flows. For realistic applications such as numerical simulation of volcanic lava flows, the work focuses particularly on complex fluids whose rheology strongly depends on physical quantities such as temperature or the particle concentration. We develop a new numerical resolution algorithm of Herschel-Bulkley's equations combining an augmented Lagrangian method with variable augmentation parameter, a second order characteristic method and an auto-adaptive mesh procedure. On stationary or evolving problems as the lid-driven cavity flow benchmark, it provides an effective solution to ensure both a high numerical accuracy within a reasonable computing time. This algorithm is then extended and adapted to the case of non-isothermal rheological and suspensions. On the numerical simulation of volcanic lava flows, we describe a method of reducing by asymptotic analysis of the Herschel-Bulkley's equations for thin flows on arbitrary topography. It allows to describe the three-dimensional flows of viscoplastic fluid with free surface by bidimensional surface equations. This approach is then extended to the non-isothermal case by adding the heat equation and thermal dependencies on rheology. By vertical integration of the heat equation, a two-dimensional model is maintained . The non-isothermal model is validated on a laboratory experiment of dome and a numerical simulation is performed on a December 2010 Piton de la Fournaise lava flow from La Réunion island. In our view, the comparison gives satisfactory and encouraging results.