This thesis examines how the non-Newtonian characteristics of liquid hog manure affect the flow conditions within a steady-flow anaerobic digester. There are three main parts to this thesis. In the first part of this thesis, the physical properties of liquid hog manure and their variation with temperature and solids concentration are experimentally determined. Naturally¬¬-settled manure sampled from an outdoor storage lagoon is studied, and density, viscosity, and particle size distribution are measured. Hog manure with total solids concentrations of less than 3.6% exhibits Newtonian behaviour; manure between 3.6% and 6.5% total solids is pseudoplastic, and fits the power law; manure with more than 6.5% total solids exhibits non-Newtonian and time-dependent characteristics. The second part of this thesis investigates the flow of Newtonian and non-Newtonian fluids—represented by tap water and xanthan gum solution, respectively—within four lab-scale reactor geometries, using residence time distribution (RTD) experiments. The effect of reactor geometry, flow rate, and fluid viscosity are evaluated. In the third part of this thesis, flow conditions within lab-scale and pilot-scale anaerobic digester reactors are simulated using three-dimensional modeling techniques. The RTDs of lab-scale reactors as predicted by the 3D numerical models compare well to the experimental results. The 3D models are also validated using data from particle image velocimetry (PIV) experiments. Finally, the viscous properties of liquid hog manure at 3% and 8% total solids are incorporated into the models, and the results are evaluated.

A research program was designed to investigate the characteristics of three-dimensional laminar wall jet flow of both Newtonian and two shear-thinning non-Newtonian fluids. The non-Newtonian fluids were prepared from xanthan gum solutions of various concentrations. Both experimental and numerical methodologies were employed in this study. The wall jet was created using a circular pipe of diameter 7 mm and flows into an open fluid tank. The initial Reynolds numbers based on the pipe diameter and jet exit velocity ranged from 250 to 800. The velocity measurements were conducted using a particle image velocimetry technique. The measurements were conducted at several streamwise locations to cover both the developing and self-similar regions. For the numerical study, the complete nonlinear Navier-Stokes equation was solved using an in-house colocated finite volume based CFD code. A Carreau model was employed for the non-Newtonian fluids. The viscosity in the governing equations was obtained explicitly. From the PIV measurements and CFD results, velocity profiles and jet half-widths were extracted at selected downstream locations to study the effects of Reynolds number and specific fluid type on the jet characteristics. It was observed that the numerical results are in reasonable agreement with the experimental data. The decay of maximum velocity, jet spread rates, skin friction coefficient, streamwise velocity profiles, and secondary flows depend strongly on the initial Reynolds number irrespective of the fluid. The results also show that the jet spreads more in the spanwise direction than in the transverse direction in the early flow development whereas the reverse is true in the downstream region. Important differences were observed when the results for the non-Newtonian fluids were compared with those for Newtonian fluid.

Este trabalho tem como propósito investigar como a variação geométrica de determinados parâmetros envolvidos na construção de uma geometria bifurcada de seção circular, em forma de Y, afeta a resistência ao escoamento, tanto de fluidos newtonianos como não newtonianos. As geometrias estudadas foram construídas utilizando-se o princípio do Design Construtal. Os parâmetros variados foram a relação entre os comprimentos dos dutos pais e filhos, a relação entre os diâmetros dos mesmos dutos, e o ângulo central da estrutura em forma de Y. Para as relações geométricas lineares foram utilizados os valores de 0,5; 0,6; 0,7; 0,8; 0,9 e 1, enquanto para os ângulos, foram utilizados os valores de 155°, 135°, 115°, 95°, 75°, 45°, 25° e 10°. Os fluidos utilizados foram do tipo newtoniano e não newtoniano, dentre estes últimos, foram estudados fluidos pseudoplásticos e dilatantes. O trabalho foi realizado através de simulações numéricas, implementadas com a utilização do software comercial Ansys Fluent, o qual resolve as equações governantes através do método dos volumes finitos. As malhas utilizadas foram do tipo poliédrica. Os resultados indicam que há uma diferença em relação ao que se espera da literatura para as relações entre os diâmetros e os comprimentos. A Lei Hess-Murray indica que estas relações ótimas seriam de 2-1/3 para as relações entre os diâmetros e comprimentos. No presente trabalho, foram determinadas relações entre os diâmetros próximas de 0,6, e entre os comprimentos, iguais a 1. Os ângulos ótimos ficaram localizados no intervalo entre 100° e 135°. / This work aims to investigate how the geometric variation of certain parameters involved in the construction of a bifurcated Y-shaped circular cross-section geometry affects the flow resistance of both Newtonian and non-Newtonian fluids. The geometries studied were constructed using the Constructal Design principle. The parameters were the relationship between the lengths of the daughter and parent ducts, the relationship between the diameters of the same ducts, and the central angle of the Y-shaped structure. For the linear geometric relations, values of 0.5; 0.6; 0.7; 0.8; 0.9 and 1 where used, for the angles, the values of 155 °, 135 °, 115 °, 95°, 75 °, 45 °, 25 ° and 10 ° were used. The fluids used were of the Newtonian and non-Newtonian type, among the latter, pseudo plastic and dilatant fluids were studied. The work was carried out through numerical simulations, implemented with the commercial software Ansys Fluent, which solves the governing equations through the finite volume method. The meshes used were of the polyhedral type. The results indicate that there is a difference in relation to what is expected from the literature for the relationships between diameters and lengths. The Hess-Murray Law indicates that these optimal relations would be 2-1/3 for the relationships between diameters and lengths. In the present work, relationships between the diameters close to 0,6 were found and s equal to 1 between the lengths. The optimum angles were located in the range between 100 ° and 135 °.

This dissertation is devoted to the finite element (FE) approximation of equations describing the motion of a class of non-Newtonian fluids. The main focus is on incompressible fluids whose viscosity nonlinearly depends on the shear rate and pressure. The equations of motion are discretized with equal-order d-linear finite elements, which fail to satisfy the inf-sup stability condition. In this thesis a stabilization technique for the pressure-gradient is proposed that is based on the well-known local projection stabilization (LPS) method. If the viscosity solely depends on the shear rate, the well-posedness of the stabilized discrete systems is shown and a priori error estimates quantifying the convergence of the method are proven. In the shear thinning case, the derived error estimates provide optimal rates of convergence with respect to the regularity of the solution. As is well-known, the Galerkin FE method may suffer from instabilities resulting not only from lacking inf-sup stability but also from dominating convection. The proposed LPS approach is then extended in order to cope with both instability phenomena. Finally, shear-rate- and pressure-dependent viscosities are considered. The Galerkin discretization of the governing equations is analyzed and the convergence of discrete solutions is...

[pt] Muitas indústrias usam em seus processos materiais viscoplásticos. Esses materiais possuem propriedades que dependem fortemente da temperatura. Não é incomum encontrar processos envolvendo escoamentos não isotérmicos de materiais viscoplásticos. Nesses casos, informações sobre a transferência de calor são extremamente necessárias para um bom atendimento e aperfeiçoamento das operações. Fluidos de perfuração são tipicamente suspensões aquosas, e, por consequência, de natureza viscoplástica.Tais fluidos devem possuir densidade correta para manter a integridade física dos poços e evitar a produção prematura de hidrocarbonetos. Além disso, suas propriedades reológicas devem garantir a capacidade de arraste das partículas de rocha geradas durante o processo de perfuração, com um mínimo de potência de bombeamento. Tais particularidades requerem fluidos com baixas viscosidades a altas taxas de cisalhamento, que ocorrem em regiões próximas à parede, e altas viscosidades quando as taxas de deformação são baixas, o que ocorre na vizinhança do cascalho. Materiais viscoplásticos apresentam este tipo de comportamento. Portanto, o sucesso do processo de extração do petróleo depende do conhecimento e controle das propriedades reológicas dos fluidos de perfuração, as quais são fortemente dependentes da temperatura. Por esse motivo, a determinação do campo de temperatura no fluido de perfuração em escoamento faz-se necessária ainda em nível de projeto, o que só é possível com o conhecimento dos coeficientes de troca de calor. Estuda-se neste trabalho o problema da transferência de calor na região de entrada de escoamentos laminares de fluidos viscoplásticos através de espaços anulares. O comportamento do material é representado pelo modelo do fluido Newtoniano generalizado, com a função viscosidade descrita pela equação de Herschel-Bulkley. As equações de conservação são resolvidas numericamente via o método de volumes finitos. Investigam-se os efeitos (no coeficiente de troca de calor) da tensão limite de escoamento, índice power-law, razão de aspecto e dos números adimensionais de Reynolds e Peclet. Dentre outras conclusões, mostra-se que o números de Nusselt é uma função muito fraca das propriedades reológicas, desviando-se muito pouco dos valores Newtonianos. Surpreendentemente, esta conclusão contrasta-se fortemente com o comportamento observando em escoamentos de materiais viscoplásticos através de tubos. Convém enfatizar a importância desse fato no que tange a projetos de processos. / [en] There are many industries that use in their processes viscoplastic materials. These materials have properties that strongly depend on temperature. It is not uncommon to find processes involving the non-isothermal flow of viscoplastic materials. For these cases, heat transfer information is needed to allow reliable process designs. Drilling muds are typically aqueous suspensions and, consequently, viscoplastic in nature. They must have the correct density to provide the pressure needed for well integrity, and for avoiding premature production of hydrocarbons. Their rheological properties must be such as to aloe carrying the drill chips with a minimum of pumping power. This requires a highly shear-thinning rheological behavior. Also, the success of a well cementing operation depends to a great extent on the knowledge and control of cement rheological properties, which are also temperature dependent. In this work, heat transfer in the entrance-region flow of viscoplastic materials through annular spaces is analyzed. The flow is laminar, and the material is assumed to behave as a Generalized Newtonian fluid, with a Herschel-Bulkley viscosity function. The conservation equations are solved numerically via a finite volume method. The effect on heat transfer of yield stress, power-law exponent, aspect ratio and dimensionless Peclet and Reynolds numbers is investigated. Among other findings, it is shown that the Nusselt number is a rather weak function of the rheological properties, deviating very little from the Newtonian values. Surprisingly, this stands in strong contrast to the behavior observed for flows of viscoplastic materials through tubes. It is worth noting that this finding has important consequences in process design.

Non-Newtonian fluids are of great scientific interest due to their range of physical properties, which arise from the characteristic shear stress-shear rate relation for each fluid. The applications of non-Newtonian fluids are widespread and occur in many industrial (e.g. lubricants, suspensions, paints, etc.) and environmental (e.g. mud, ice, blood, etc.) problems, often involving multiple fluids. In this study, the novel technique of Incompressible Smoothed Particle Hydrodynamics (ISPH) with shifting (Lind et al., J. Comput. Phys., 231(4):1499-1523, 2012), is extended beyond the state-of-the-art to model non-Newtonian and multi-phase flows. The method is used to investigate important problems of both environmental and industrial interest. The proposed methodology is based on a recent ISPH algorithm with shifting with the introduction of an appropriate stress formulation. The new method is validated both for Newtonian and non-Newtonian fluids, in closed-channel and free-surface flows. Applications in complex moulding flows are conducted and compared to previously published results. Validation includes comparison with other computational techniques such as weakly compressible SPH (WCSPH) and the Control Volume Finite Element method. Importantly, the proposed method offers improved pressure results over state-of-the-art WCSPH methods, while retaining accurate prediction of the flow patterns. Having validated the single-phase non-Newtonian ISPH algorithm, this develops a new extension to multi-phase flows. The method is applied to both Newtonian/Newtonian and Newtonian/non-Newtonian problems. Validations against a novel semi-analytical solution of a two-phase Poiseuille Newtonian/non-Newtonian flow, the Rayleigh-Taylor instability, and a submarine landslide are considered. It is shown that the proposed method can offer improvements in the description of interfaces and in the prediction of the flow fields of demanding multi-phase flows with both environmental and industrial application. Finally, the Lituya Bay landslide and tsunami is examined. The problem is approached initially on the real length-scales and compared with state-of-the-art computational techniques. Moreover, a detailed investigation is carried out aiming at the full reproduction of the experimental findings. With the introduction of a k-ε turbulence model, a simple saturation model and correct experimental initial conditions, significant improvements over the state-of-the-art are shown, managing an accurate representation of both the landslide as well as the wave run-up. The computational method proposed in this thesis is an entirely novel ISPH algorithm capable of modelling highly deforming non-Newtonian and multi-phase flows, and in many cases shows improved accuracy and experimental agreement compared with the current state-of-the-art WCSPH and ISPH methodologies. The variety of problems examined in this work show that the proposed method is robust and can be applied to a wide range of applications with potentially high societal and economical impact.

Current problems on the interaction of complex liquids (i.e. droplets or foams) with complex surfaces (i.e. soft deformable or porous surfaces) are addressed in the following areas: (1) wetting of deformable substrates and surface forces, (2) kinetics of wetting and spreading of non-Newtonian liquids over porous substrates, (3) kinetics of spreading of non-Newtonian solutions over hair, (4) free drainage of foams produced from non-Newtonian solutions, and (5) foam drainage placed on porous substrates. Equilibrium of liquid droplets on deformable substrates was investigated and the effect of disjoining pressure action in the vicinity of the apparent three phase contact line was taken into account. It was proven that the deformation of soft solids is determined by the action of surface forces inside the transition zone. Spreading/imbibition of blood, which is a power law shear thinning non-Newtonian liquid, over a dry porous layer was investigated from both theoretical and experimental points of view. It was found that blood droplet spreading/imbibition over porous substrates shows two different behaviours: (i) partial wetting case with three subsequent stages: initial fast spreading, constant maximum droplet base and the shrinkage of the droplet base; (ii) complete wetting case with only two stages: initial fast spreading and the shrinkage of the droplet base. The wetting of hair tresses by aqueous solutions of two commercially available polymers, AculynTM 22 (A22) and AculynTM 33 (A33) was investigated experimentally. Both A22 and A33 solutions demonstrate well pronounced shear thinning behaviour. Initial contact angle of the A22 and A33 solutions on hair tresses was about 100o. The A22 droplets remained on the hair tress after spreading for at least half an hour. However, a fast penetration of the A33 droplets inside the hair tresses was observed when advancing contact angle in the course of spreading reached a critical value of about 60o. This could be explained by Cassie-Wenzel wetting transition which is caused by filling the pores inside the porous media by liquid. The influence of non-Newtonian rheology of A22 and A33 solutions on foam drainage was also investigated experimentally and a new theory of foam drainage was presented for the case of free drainage. For lowly viscous polymeric solutions and under the assumption of rigid surface of the Plateau border, the predicted values of the time evolution of the foam height and liquid content were in good agreement with the experimental data. However, in the case of highly viscous solutions an interfacial mobility at the surface of the Plateau border has to be taken into account. A completely new theory of foam drainage placed on porous substrate was developed. It was found that there are three different regimes of the process: (i) a rapid imbibition, the imbibition into the porous substrate dominates as compared with the foam drainage; (ii) an intermediate imbibition, that is, the imbibition into the porous substrate and the rate of drainage are comparable; (iii) a slow imbibition, the rate of drainage inside the foam is higher than the imbibition into the porous substrate for a period of time and a free liquid layer is formed over the porous substrate.

O estudo do comportamento reológico de fluidos não Newtonianos tem grande importância em diversas áreas da engenharia. O aumento na demanda destes fluidos por exemplo, no uso doméstico, pessoal e em processos químicos acarreta em dificuldades que vão desde o processo de sua mistura ao seu manuseio. Dentre os fluidos não Newtonianos estão os viscoelásticos, os quais exibem deformação aparente quando os níveis de tensões são inferiores ao limite de escoamento do material e, dentro desta classe, alguns ainda podem apresentam comportamento elástico quando submetidos à baixa taxa de cisalhamento. Juntamente com os efeitos elasto-viscoplásticos, os materiais podem apresentar comportamento tixotrópico, onde devido as tensões sua reestruturação não é instantânea. Na presente Tese, fez-se um estudo numérico a fim de analisar o problema especifico de escoamentos de fluidos elasto-viscoplásticos com comportamento tixotrópico em uma expansão planar abrupta na razão de aspecto 4 : 1. O modelo mecânico aplicado consiste de uma equação viscoelástica para o campo de tensões, uma evolutiva para o parâmetro de estrutura do material, bem como as equações de conservação de massa e momento. O modelo mecânico aplicado mostrou-se capaz de prever o comportamento tixotrópico. A aproximação numérica do modelo aplicado foi feita através do método estabilizado de elementos finitos, especificamente o método Galerkin Mínimos-Quadrados (GLS), o qual foi implementado no código de elementos finitos para fluidos não Newtonianos em desenvolvimento no Laboratório de Mecânica dos Fluidos Aplicada e Computacional (LAMAC) da Universidade Federal do Rio Grande do Sul. Os fenômenos reológicos presentes no problema foram analisados a partir da influência da cinemática do escoamento, da elasticidade e da tixotropia, no nível de estruturação do material, na posição e tamanho das regiões não escoadas e na deformação elástica do material. Os resultados mostraram-se satisfatórios, pois condizem com os apresentados na literatura, apontando a boa predição do modelo mecânico aplicado bem como a robustez de sua implementação computacional. / The rheological behavior of non-Newtonian uids study is of great then importance in many areas of engineering. The increase in demand of these uids - for example, domestic use, personal and processes chemical - causes di culties ranging from the process of mixing it to handling. Among the non-Newtonian uids are viscoelastic, which exhibit apparent deformation when stress levels are lower than the yield limit of the material and, within this class, some still have elastic behavior when subjected to low shear rates. Together with the elastic-viscoplastic e ects, materials may exhibit thixotropic behavior, ie, due the restructuring strain is not instantaneous. In this thesis was made a numerical study to simulate the speci c problem of ow of elastic-viscoplastic uids with thixotropic behavior in abrupt planar expansion { common geometry in industrial systems associated with elastic-viscoplastic uids, whose ratio the aspect is 4:1. The mechanical behavior of most of these structured materials, are highly non-Newtonian, with this, there is need to make them more pseudoplastic, causing undesirable behaviors such as thixotropic which is a phenomenon somewhat characterized and modeled in the literature. The mechanical model applied is able to predict thixotropic behavior and is composed of a viscoelastic equation for the stress eld and an evolutionary to the material structure parameter in addition to the mass and momentum conservation equations. This mechanical model is approximated by a stabilized nite element model, called the Galerkin method of least squares (GLS). In order to study the rheological phenomena present in the problem is analyzed the in uence of the ow kinematics, elasticity and thixotropy in the level of structure of the material, in position and size of unyielded regions and the elastic deformation of the material. The results were satisfactory, since the the study of intensity the U the results agree with those reported in the literature pointing to good prediction of the mechanical model applied well as the hardiness of their computational implementation.The results of the elasticity showed quite spectacular behavior of unyielded regions of the material since for high values of relaxation time, the unyielded region of channel larger has the form of " ngers"because of the high exibility of the material along the line of symmetry of the channel. The results of the analysis of the elastic deformation show, that model correctly dosing the elasticity in the unyielded regions. The thixotropic e ects reported a slower structuring of the material with increasing characteristic time, in response to strain change caused by the expansion, results indicate that for high values of relaxation time, higher the distance between the unyielded regions of smaller channel and larger.

[pt] Este trabalho descreve experimentos relacionamentos com o movimento relativo entre partículas sólidas e fluido não-newtoniano, confinados no interior de um duto circular. Medições da pressão dinâmica adicional, devida unicamente à presença da partícula (fonte de perturbação no escoamento) e do arrasto viscoso foram conduzidas de forma a se verificar a validade da Teoria de Brenner (1962). Esta teoria, já confirmada para fluidos newtonianos, permite que parâmetros característicos do escoamento perturbado sejam determinados, convenientemente, através de parâmetros do escoamento não-perturbado (ausência de partícula). Para o caso de fluido não-newtoniano, denominados puramente viscosos, do tipo Power-law, a teoria se mostrou perfeitamente aplicável. O valor da razão Delta P mais A/D descrito por Brenner foi confirmado com uma precisão de 3 por cento, num total de 70 experimentos realizados. Para fluidos não-newtonianos, viscoelásticos, com função viscosidade tipo Power-law, a validade da teoria parece, entretanto, depender de um parâmetro capaz de descrever na natureza constitutiva do fluido utilizado. Experimentos realizados com três diferentes fluidos viscoelásticos (expoentes power-law n igual 0,303; 0,343; 0,483) conduziram à identificação deste parâmetro, o Segundo Número Elástico, El2. Para valores de El2, inferiores a 14, caracterizando um escoamento predominantemente viscoso, o valor da razão Delta P mais A/D novamente é confirmado com precisão inferior a 4 por cento. Para valores de El2 superiores a 40 a razão Delta P mais A/D não mais pode ser avaliada com base em parâmetros do escoamento perturbado, analogamente ao que havia sido proposto por Brenner para o caso de fluidos newtonianos. Neste trabalho incluem-se também registros contínuos dos experimentos enfatizando os efeitos viscoelásticos envolvidos, bem como uma análise dos efeitos de parede associados ao movimento relativo entre fluidos não-newtonianos e partículas sólidas. Todos os experimentos foram realizados num regime de Reynolds variando de 0,1 a 90 e num regime de Weissenberg (calculando com base no modelo de Powell-Eyring) variando de 850 a 3800. / [en] This work describes experiments related to relative motion between solid particles and mon-newtonian fluid, inside a circular duct. Measurements of the aditional dynamic pressure, due to the presence of the particle (a source of disturbance in the flow) ando f the viscous drag, were conducted to verify the validity of Brenner’s Theory (1962). This theory, already confirmed for newtonian fluids, allows the determination of the characteristic parameters of the disturbed flow using parameters of non-disturbed flow (without particle). In the case of purely viscous non-newtonian fluids, of the power-law type, the theory was confirmed. The value of the ratio Delta P plus A/D, described by Brenner, was confirmed. The value an accuracy of 3 per cent, in a total of 70 experiments. For viscoelastic fluids, with Power-law viscosity function, it appears that the validity of the theory depends on the Second Elastic Number, El2. Experiments conducted with three different viscoelastic fluids (power-law exponents, n equal 0,303; 0,343 and 0,483) shows that for values of El2 bellow 14, which characterizes a predominantly viscous flow, the value of of the ratio Delta P plus A/D is agair confirmed, with na accuracy of 4 per cent. For values of the El2, parameter above 40, the ratio Delta P plus A/D cannot be determined using parameters of the non-disturbed flow, as proposed by Brenner for newtonian fluids. In this work are also included graphic registers of the experiment, showing the complex viscoelastic effects, as well as na analysis of the wall effects associated with the relative motion between non- newtonian fluid and solid particles. All the experiments were conducted with Reynolds number between 0,1 and 90 and a Weissenberg number (based in Powell-Eyring model) between 850 and 38.00.

Este trabalho tem como propósito investigar como a variação geométrica de determinados parâmetros envolvidos na construção de uma geometria bifurcada de seção circular, em forma de Y, afeta a resistência ao escoamento, tanto de fluidos newtonianos como não newtonianos. As geometrias estudadas foram construídas utilizando-se o princípio do Design Construtal. Os parâmetros variados foram a relação entre os comprimentos dos dutos pais e filhos, a relação entre os diâmetros dos mesmos dutos, e o ângulo central da estrutura em forma de Y. Para as relações geométricas lineares foram utilizados os valores de 0,5; 0,6; 0,7; 0,8; 0,9 e 1, enquanto para os ângulos, foram utilizados os valores de 155°, 135°, 115°, 95°, 75°, 45°, 25° e 10°. Os fluidos utilizados foram do tipo newtoniano e não newtoniano, dentre estes últimos, foram estudados fluidos pseudoplásticos e dilatantes. O trabalho foi realizado através de simulações numéricas, implementadas com a utilização do software comercial Ansys Fluent, o qual resolve as equações governantes através do método dos volumes finitos. As malhas utilizadas foram do tipo poliédrica. Os resultados indicam que há uma diferença em relação ao que se espera da literatura para as relações entre os diâmetros e os comprimentos. A Lei Hess-Murray indica que estas relações ótimas seriam de 2-1/3 para as relações entre os diâmetros e comprimentos. No presente trabalho, foram determinadas relações entre os diâmetros próximas de 0,6, e entre os comprimentos, iguais a 1. Os ângulos ótimos ficaram localizados no intervalo entre 100° e 135°. / This work aims to investigate how the geometric variation of certain parameters involved in the construction of a bifurcated Y-shaped circular cross-section geometry affects the flow resistance of both Newtonian and non-Newtonian fluids. The geometries studied were constructed using the Constructal Design principle. The parameters were the relationship between the lengths of the daughter and parent ducts, the relationship between the diameters of the same ducts, and the central angle of the Y-shaped structure. For the linear geometric relations, values of 0.5; 0.6; 0.7; 0.8; 0.9 and 1 where used, for the angles, the values of 155 °, 135 °, 115 °, 95°, 75 °, 45 °, 25 ° and 10 ° were used. The fluids used were of the Newtonian and non-Newtonian type, among the latter, pseudo plastic and dilatant fluids were studied. The work was carried out through numerical simulations, implemented with the commercial software Ansys Fluent, which solves the governing equations through the finite volume method. The meshes used were of the polyhedral type. The results indicate that there is a difference in relation to what is expected from the literature for the relationships between diameters and lengths. The Hess-Murray Law indicates that these optimal relations would be 2-1/3 for the relationships between diameters and lengths. In the present work, relationships between the diameters close to 0,6 were found and s equal to 1 between the lengths. The optimum angles were located in the range between 100 ° and 135 °.