Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006. / When considering flow through porous media, different flow regimes may be identified. At very small Reynolds numbers the relation between the pressure gradient and the velocity of the fluid is linear. This flow regime ...

Made available in DSpace on 2014-06-11T19:23:39Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-12-18Bitstream added on 2014-06-13T18:10:05Z : No. of bitstreams: 1 kiryu_hs_me_ilha.pdf: 2419349 bytes, checksum: 258056fdacb79f386b2e6b238dc26028 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Esta dissertação de mestrado traz à discussão o comportamento reológico de misturas formadas por água+colóides+detritos (areia fina), visando entender e esclarecer os processos físicos e mecânicos, tais como sedimentação e ressuspensão de materiais inertes no seio da massa fluida não-newtoniana (água+colóides), bem como verificar a validade ou adeqüabilidade do modelo reológico de Herschel-Bulkley (modelo previamente investigado e validado para misturas compostas de água+colóides) para misturas viscoplásticas com presença de grãos. A variação das propriedades reológicas das misturas, em função das características físicas dos grãos (diâmetro, massa específica e área superficial), é investigada, e um modelo de estimativa de tensão crítica é apresentado. Ademais foram realizados ensaios preliminares de escoamento de fluidos hiperconcentrados em canais inclinados, na tentativa de calibrar uma lei de atrito. Dentro dessas perspectivas, a dissertação é composta de 6 Capítulos com um denso Estado da Arte que descreve os fenômenos e mecanismos que regem os escoamentos desse tipo de compósito. Com base na literatura estudada e, a partir da análise dos resultados experimentais, pôde-se concluir que, para misturas compostas de água+colóides+detritos, o comportamento reológico das misturas é o mesmo que aquele do fluido intersticial (água+colóides), desde que a homogeneidade da mistura seja garantida (não ocorrência de sedimentação e ressuspensão sucessivas). Neste caso, o modelo reológico de Herschel-Bulkley continua sendo válido para explicar as curvas de escoamento ou de fluxo das misturas viscoplásticas com grãos. Para misturas que apresentem os fenômenos de sedimentação e ressuspensão, o modelo de Bagnold, adaptado a fluidos hiperconcentrados... / This work retakes the discussion about the rheological behavior of mixtures composed by water+ kaolinitic clay+fine sand in order to investigate the physical and mechanical processes such as sedimentation and suspension of inert materials into the non-Newtonian or interstitial fluid (water+colloids), as well as verify the adaptability of the Herschel-Bulkley rheological model (model previously investigated and validated for composed mixtures of water+ kaolinitic clay) for explain the viscoplastic+coarse materials rheological properties. The variation of the rheological properties of the mixtures in function of the coarse material characteristics (diameter, specific mass and superficial area) was investigated and a model predicting yield stress was proposed. Furthermore, some tests were performed in an inclined canal to determine a friction law for this kind of fluids. Inside of these perspectives, this dissertation is composed of 6 Chapters whit a dense State of the Art describing the phenomena and their mechanisms were pointed up. Based on literature and from the experimental results, one could concluded that the viscoplastic + coarse material mixtures behavior is the same of the interstitial fluid one, since that the homogeneity of the mixture is guaranteed (not occurrence of successive sedimentation and resuspension). In this case, Herschel-Bulkley rheological model is still valid to explain the curves of flow of the viscoplastic + coarse material. For mixtures that present the phenomena of sedimentation and resuspension, Bagnoldþs model, adapted to the hyperconcentrated fluids, describes well the variations of rheological parameters in function of the shear rates applied. Finally, it could be concluded that the experiments of free surface in canals, despite partial, can... (Complete abstract click electronic access below)

Thesis submitted in fulfilment of the requirements for the degree Doctor of Technology: Mechanical Engineering in the Faculty of Engineering at the Cape Peninsula University of Technology 2014 / Open channels, flumes or launders are used in the mining industry to transport slurries during processing and to disposal sites. Water plays a major part in the makeup of these slurries, its usage and availability is critical in countries where there are strict water usage management programs. The optimisation of flume design involves the maximisation of solids transport efficiency whilst, at the same time reduces water usage. The design of open channels is complex as it is dependent on both the slurry rheology and the channel shape. Very little has been reported in the literature for predicting non-Newtonian laminar flow in open channels of arbitrary cross-section. The only method available was that proposed by Kozicki and Tiu (1967, 1986). The shape factors they used were those evaluated from analytical solutions for flow of Newtonian fluids in open channels of the same cross-section. However, they carried out no experimental work to validate their model. Few experimental studies have been made on the effect of shape on non-Newtonian flow in open channels. Naik (1983) tested kaolin in water suspensions in a rectangular channel. Coussot (1994) provided some data for the flow of a Herschel-Bulkley fluid in rectangular and trapezoidal channels. Fitton (2007; 2008) obtained data for flow of three different non-Newtonian fluids (carboxymethylcellulose, carbopol and thickened tailings) in a semi-circular channel. A large experimental database for non-Newtonian flow in rectangular open channels was published by Haldenwang (2003) at the Flow Process Research Centre, Cape Peninsula University of Technology. Guang et al. (2011) performed Direct Numerical Simulations of turbulent flow of a yield- pseudoplastic fluid in a semi-circular channel. They compared their simulations with actual field measurements and found them to over-predict the flow velocity by approximately 40%. The source for this discrepancy was difficult to ascertain. A comprehensive database was compiled during this research of the flow of three non–Newtonian fluids in rectangular, trapezoidal, semi-circular and triangular channels. The flow of carboxymethylcellulose solutions and aqueous kaolin and bentonite suspensions was investigated in a 10 meter long flume at angles ranging from 1° to 5° from the horizontal plane. The effect of channel shape on the friction factor-Reynolds number relationship for laminar and turbulent open channel flow of these three fluids was investigated. New models for the prediction of laminar and turbulent flow of non-Newtonian fluids in open channels of different cross-sectional shapes are proposed. The new laminar and turbulent velocity models are compared with three previously-published velocity models for laminar flow and five previously-published velocity models for turbulent flow using average velocity as comparison criteria. For each channel shape, the laminar flow data can be described by a general relationship, f = K/Re where f is the Fanning friction factor and Re is the appropriate Haldenwang et al. (2002) Reynolds number. The K values were found to be 14.6 for triangular channels with a vertex angle of 90°, 16.2 for semi-circular channels, 16.4 for rectangular channels and 17.6 for trapezoidal channels with 60 degree sides. These K values were found to be in line with those reported by Straub et al. (1958) and Chow (1969) for open channel laminar flow of Newtonian fluids as opposed to the assumption made by Haldenwang et al. (2002; 2004) of using a constant value of 16 based on the pipe flow paradigm for all channel shapes. This new laminar model gave a closer fit to the laminar flow data than those from the three previously-published models. However, the presence of the yield stress still presents a problem, which makes the flow prediction in laminar flow for such fluids not very accurate. The investigation on non-Newtonian turbulent flow of the three fluids in the four different shaped open channels revealed that the data was described by the modified Blasius equation f = a Re b where a and b are constant values determined for each channel shape and Re is the Haldenwang et al. (2002) Reynolds number. Values of a and b for a rectangular channel were found to be 0.12 and -0.330, for a semi- circular channel 0.048 and -0.205, for a trapezoidal channel with 60° sides, 0.085 and -0.266 and for a triangular channel with vertex angle of 90°, 0.042 and -0.202. New laminar and turbulent velocity models were derived from using the new laminar f = K/Re and turbulent f = a Re b, friction factor-Reynolds number relationship. The laminar velocity model did not always give the best result, but the majority of the time it did, compared to the three previously published models. The new turbulent velocity model yielded the best results when compared to the five previously published models using average velocity as comparison criteria. The composite power law modelling procedure of Garcia et al. (2003) used for pipe flow predictions was extended to the present work on non-Newtonian flow in open channels of various cross-sections. The results show that the modelling technique used by Garcia et al. (2003) for pipe flow can be used to adequately predict flow in an open channel of a given cross-sectional shape provided that an appropriate Reynolds number is used to take into account the non-Newtonian behaviour of the test fluid. It was found that the results using the Haldenwang et al. (2002) Reynolds number yielded better results than those based on the adapted Metzner-Reed Reynolds number. The correlations and models developed and experimentally validated during this research can be used to further improve the design of rectangular, semi-circular, trapezoidal and triangular open channels to transport non-Newtonian fluids.

Cette thèse est consacrée à l’étude mathématique des écoulements complexes dans des tubes minces. Les difficultés ne sont pas seulement liées à la rhéologie complexe, mais aussi aux conditions au bord sur la pression en entrée et en sortie (qui sont moins habituelles, mais réalistes du point de vue physique). Dans une première partie, des écoulements quasi-newtoniens stationnaires sont étudiés. D’abord, on utilise la petitesse du domaine pour montrer l’existence de la solution. Ensuite, on écrit un développement asymptotique de cette solution et on calcule formellement ses coefficients. Finalement, on justifie rigoureusement la validité de ce développement en démontrant des estimations d’erreur. Dans une deuxième partie, on considère des écoulements de fluides visco-élastiques décrits par la loi d’Oldroyd en régime stationnaire. Le modèle que nous avons choisi contient un terme diffusif en contrainte, dont l’ordre de grandeur est lié à la petitesse du domaine. Similairement à la première partie, un développement asymptotique est complètement justifié du point de vue mathématique. Dans le cas particulier de domaines axisymétriques une solution numérique est cherchée afin de la comparer à la solution obtenue via la technique asymptotique. Dans une dernière partie, on étudie les équations de Navier-Stokes non stationnaires. Un résultat d’existence des solutions fortes pour des données petites est démontré. Malheureusement, la méthode directe ne nous a pas permis pas d’avoir suffisamment de contrôle par rapport à la petitesse du domaine. Pour obtenir le résultat désiré, on utilise l’approche à la Kato, basé sur la théorie de C0 semigroupes. / This thesis is devoted to the mathematical analysis of complex flows in thin pipes. The difficulties stem not only from the complex rheology, but also from the boundary conditions used involving the pressure (which are rather atypical, but realistic from a physical point of view).In the first part, we study stationary, quasi-newtonian flows. The existence of a solution is shown using the smallness of the domain as a key ingredient. Furthermore, an asymptotic expansion of this solution is sought and its coefficients are formally computed. Lastly, the validity of this expansion is rigorously justified by proving error estimates. In the second part, we consider visco-elastic flows represented by Oldroyd’s law in stationary regime. The model which we have chosen contains a diffusive stress term, whose order of magnitude is related to the smallness of the domain. Similarly to the first part, a complete asymptotic expansion in mathematically justified. For the special case of axisymmetric domains a numerical solution is sought in order to compare it against the one obtained via the asymptotic technique. In the last part we study the non stationary Navier-Stokes equations. An existence result of strong solutions for small initial data is proven. Unfortunately, the direct method – based on energy estimates – doesn’t give us an optimal control of the smallness constant with respect to the size of the domain. To obtain the desired result, we employ the method of C 0 semigroups of linear operators.

Cette thèse comporte trois chapitres indépendants, consacrés à l’étude mathématique de trois problèmes physiques distincts, ayant pour modèles trois équations aux dérivées partielles différentes. Ces équations relèvent plus précisément de la méthode des surfaces de niveau, de la théorie de l’écoulement incompressible des matériaux non newtoniens et de la théorie cinétique des gaz raréfiés. Le premier chapitre de la thèse porte sur la dynamique des frontières en mouvement et contient une justification mathématique de la procédure numérique dite de ré-initialisation, dont les applications sont nombreuses dans le contexte de la célèbre méthode des surfaces de niveau. Nous appliquons ces résultats pour une classe d’équations issues de la méthode des surfaces de niveau de premier ordre. Nous écrivons la procédure de ré-initialisation comme un algorithme de décomposition et nous étudions la convergence de l’algorithme en utilisant des techniques d’homogénéisation dans la variable temporelle. Grâce à cette analyse rigoureuse nous introduisons également une nouvelle méthode pour l’approximation de la fonction de distance dans le contexte de la méthode des surfaces de niveau. Dans le cas où l’on cherche seulement une fonction de l’ensemble de niveau avec un gradient minoré proche du niveau zéro, nous proposons une approximation plus simple. Dans le cas général, où le niveau zéro pourrait présenter des changements de topologie, nous introduisons une nouvelle notion de limites relâchées. Dans le deuxième chapitre de la thèse, nous étudions un problème de frontière libre résultant de l’étude de l’écoulement incompressible d’un matériau non-newtonien, avec limite d’élasticité de type Drucker-Prager, sur un plan incliné et sous l’effet de la pesanteur. Nous obtenons une équation sous-différentielle, que nous formulons comme un problème variationnel avec un terme à croissance linéaire de type gradient, et nous étudions le problème dans un domaine non borné. Nous montrons que les équations sont bien posées et satisfont certaines propriétés de régularité. Nous sommes alors capables de relier les paramètres physiques avec le problème abstrait et de prouver des propriétés quantitatives de la solution. En particulier, nous montrons que la solution a un support compact, la limite de ce que nous appelons la frontière libre. Nous construisons également des solutions explicites d’une équation différentielle ordinaire qui peut estimer la frontière libre. Enfin, le troisième et dernier chapitre de la thèse est dédié aux solutions de l’équation de Boltzmann homogène avec molécules maxwelliennes et énergie infinie. Nous obtenons de nouveaux résultats d’existence de solutions éternelles pour cette équation dans un espace de mesures de probabilité d’énergie infinie (i.e. de moment d’ordre deux infini). Elles permettent de décrire le comportement asymptotique en temps d’autres solutions d’énergie infinie, mais elles apparaissent aussi comme des états asymptotiques intermédiaires dans l’étude des solutions d’énergie finie, mais arbitrairement grande. Les méthodes issues de l’analyse harmonique sont utilisées pour étudier l’équation de Boltzmann, où la variable de vitesse est exprimée en Fourier. Enfin, un changement d’échelle logarithmique en la variable temporelle permet de déterminer le bon comportement asymptotique à l’infini des solutions / This thesis consists of three different and independent chapters, concerning the mathematical study of three distinctive physical problems, which are modelled by three non- linear partial differential equations. These equations concern the level set method, the theory of incompressible flow of non-Newtonian materials and the kinetic theory of rare- fied gases. The first chapter of the thesis concerns the dynamics of moving interfaces and contains a rigorous justification of a numerical procedure called re-initialization, for which there are several applications in the context of the level set method. We apply these results for first order level set equations. We write the re-initialization procedure as a splitting algorithm and study the convergence of the algorithm using homogenization techniques in the time variable. As a result of the rigorous analysis, we are also able to introduce a new method for the approximation of the distance function in the context of the level set method. In the case where one only looks for a level set function with gradient bounded from below near the zero level, we propose a simpler approximation. In the general case where the zero level might present changes of topology we introduce a new notion of relaxed limits. In the second chapter of the thesis, we study a free boundary problem arising in the study of the flow of an incompressible non-Newtonian material with Drucker-Prager plasticity on an inclined plane. We derive a subdifferential equation, which we reformulate as a variational problem containing a term with linear growth in the gradient variable, and we study the problem in an unbounded domain. We show that the equations are well posed and satisfy some regularity properties. We are then able to connect the physical parameters with the abstract problem and prove some quantitative properties of the solution. In particular, we show that the solution has compact support and the support is the free boundary. We also construct explicit solutions of an ordinary differential equation, which we use to estimate the free boundary. The last chapter of the thesis is dedicated to the study of infinite energy solutions of the homogeneous Boltzmann equation with Maxwellian molecules. We obtain new results concerning the existence of eternal solutions in the space of probability measure with infinite energy (i.e. the second order moment is infinite). These solutions describe the asymptotic behaviour of other infinite energy solutions but could also be useful in the study of intermediate asymptotic states of solutions with finite but arbitrarily large energy. We use harmonic analysis tools to study the equation, where the velocity variable is expressed in the Fourier space. Finally, a logarithmic scaling of the time variable allows to determine the correct asymptotic scaling of the solutions

The theory of homogenization allows to find for a given system of partial differential equations governing a model with a very complicated internal struc- ture a system governing a model without this structure, whose solution is in a certain sense an approximation of the solution of the original problem. In this thesis, methods of the theory of homogenization are applied to three sys- tems of partial differential equations. The first one governs a flow of a class of non-Newtonian fluid through a porous medium. The second system is utilized for modeling of a flow of a fluid through an electric field wherein the viscosity depends significantly on the intensity of the electric field. For the third system is considered an elliptic operator having growth and coercivity indicated by a general anisotropic inhomogeneous N-function. 1

CAPES / Vários estudos numéricos investigaram cavidades fechadas sob o efeito da convecção natural preenchidas com fluidos newtonianos generalizados (FNG) nos últimos anos pelas aplicações diretas em trocadores de calor compactos, no resfriamento de sistemas eletrônicos e na engenharia de polímeros. Neste trabalho é realizada a investigação numérica do processo de convecção natural de fluidos de lei de Potência e de Bingham em cavidades fechadas, aquecidas lateralmente e preenchidas com meios heterogêneos e bloco centrado. O meio heterogêneo é constituído de blocos sólidos, quadrados, desconectados e condutores de calor. Como parâmetros são utilizados a faixa de Rayleigh de 104 à 107, índice de potência n de 0, 6 à 1, 6, número de Bingham de 0, 5 até Bimax , sendo investigado da influência do número de Prandtl para cada modelo de fluido. Nas cavidades com meio heterogêneo são utilizadas as quantidades de blocos de 9, 16, 36 e 64, mantendo-se a razão entre a condutividade térmica do sólido e do fluido κ = 1. Para as cavidades com bloco centrado, são utilizados os tamanhos adimensionais de 0, 1 à 0, 9 com κ = 0, 1; 1 e 10. A modelagem matemática é realizada pelas equações de balanço de massa, de quantidade de movimento e de energia. As simulações são conduzidas no programa comercial ANSYS FLUENT R . Inicialmente são resolvidos problemas com fluidos newtonianos em cavidade limpa, seguida de cavidade preenchida com meio heterogêneo e posteriormente bloco centrado para validação da metodologia de solução. Na segunda etapa é realizada o estudo com os modelos de fluidos de lei de Potência e de Bingham seguindo a mesma sequência. Os resultados são apresentados na forma de linhas de corrente, isotermas e pelo número de Nusselt médio na parede quente. De maneira geral, a transferência de calor na cavidade é regida pelo número de Rayleigh, tamanho e condutividade térmica dos blocos, pelo índice de potência para o modelo de lei de Potência e do número de Bingham para o modelo de Bingham. O número de Prandtl tem grande influência nos dois modelos de fluidos. O meio heterogêneo reduz a transferência de calor na cavidade quando interfere na camada limite térmica para ambos os fluidos, sendo feita uma previsão analítica para o fluido de lei de Potência. Para bloco centrado, a interferência na camada limite com fluido de lei de Potência também foi prevista analiticamente. A transferência de calor aumentou com bloco de baixa condutividade térmica e pouca interferência e com bloco de alta condutividade térmica e grande interferência, para ambos os fluidos. / Many studies have been carried out in square enclosures with generalized Newtonian fluids with natural convection in past few years for directly applications in compact heat exchangers, cooling of electronics systems and polymeric engineering. The natural convection in square enclosures with differently heated sidewalls, filled with power-law and Bingham fluids in addition with heterogeneous medium and centered block are analyzed in this study. The heterogeneous medium are solid, square, disconnected and conducting blocks. The parameters used are the Rayleigh number in the range 104 - 107 , power index n range of 0, 6 - 1, 6, Bingham number range of 0, 5 - Bimax , being the influence of Prandtl number investigated for each fluid model. The number of blocks for heterogeneous medium are 9, 16, 36 and 64, keeping constant solid to fluid conductive ratio, κ = 1. For enclosures with centered block are used the nondimensional block size from 0, 1 to 0, 9, with solid to fluid conductive ratio in range κ = 0, 1; 1 and 10. Mathematical modeling is done by mass, momentum and energy balance equations. The solution of equations have been numerically solved in ANSYS FLUENT R software. Firstly, numerical solutions for validation with Newtonian fluids in clean enclosures are conducted, followed by enclosures with heterogeneous medium and centered block. Subsequently, numerical solutions of power-law and Bingham fluids with same enclosures configurations are conducted. The results are reported in the form of streamlines, isotherms and average Nusselt number at hot wall. In general, the heat transfer process in enclosure is governed by Rayleigh number, size and thermal conductivity of the blocks, power index n for power-law fluid and Bingham number for Bingham fluid. Both fluid models are very sensitive with Prandtl number changes. Heterogeneous medium decrease heat transfer in enclosure when affects thermal boundary layer for both fluid models. One analytical prediction was made for power-law fluid. An increase in heat transfer occurs with low thermal conductivity block and few interference and with high thermal conductivity block and great interference, for both fluids.

CAPES / Vários estudos numéricos investigaram cavidades fechadas sob o efeito da convecção natural preenchidas com fluidos newtonianos generalizados (FNG) nos últimos anos pelas aplicações diretas em trocadores de calor compactos, no resfriamento de sistemas eletrônicos e na engenharia de polímeros. Neste trabalho é realizada a investigação numérica do processo de convecção natural de fluidos de lei de Potência e de Bingham em cavidades fechadas, aquecidas lateralmente e preenchidas com meios heterogêneos e bloco centrado. O meio heterogêneo é constituído de blocos sólidos, quadrados, desconectados e condutores de calor. Como parâmetros são utilizados a faixa de Rayleigh de 104 à 107, índice de potência n de 0, 6 à 1, 6, número de Bingham de 0, 5 até Bimax , sendo investigado da influência do número de Prandtl para cada modelo de fluido. Nas cavidades com meio heterogêneo são utilizadas as quantidades de blocos de 9, 16, 36 e 64, mantendo-se a razão entre a condutividade térmica do sólido e do fluido κ = 1. Para as cavidades com bloco centrado, são utilizados os tamanhos adimensionais de 0, 1 à 0, 9 com κ = 0, 1; 1 e 10. A modelagem matemática é realizada pelas equações de balanço de massa, de quantidade de movimento e de energia. As simulações são conduzidas no programa comercial ANSYS FLUENT R . Inicialmente são resolvidos problemas com fluidos newtonianos em cavidade limpa, seguida de cavidade preenchida com meio heterogêneo e posteriormente bloco centrado para validação da metodologia de solução. Na segunda etapa é realizada o estudo com os modelos de fluidos de lei de Potência e de Bingham seguindo a mesma sequência. Os resultados são apresentados na forma de linhas de corrente, isotermas e pelo número de Nusselt médio na parede quente. De maneira geral, a transferência de calor na cavidade é regida pelo número de Rayleigh, tamanho e condutividade térmica dos blocos, pelo índice de potência para o modelo de lei de Potência e do número de Bingham para o modelo de Bingham. O número de Prandtl tem grande influência nos dois modelos de fluidos. O meio heterogêneo reduz a transferência de calor na cavidade quando interfere na camada limite térmica para ambos os fluidos, sendo feita uma previsão analítica para o fluido de lei de Potência. Para bloco centrado, a interferência na camada limite com fluido de lei de Potência também foi prevista analiticamente. A transferência de calor aumentou com bloco de baixa condutividade térmica e pouca interferência e com bloco de alta condutividade térmica e grande interferência, para ambos os fluidos. / Many studies have been carried out in square enclosures with generalized Newtonian fluids with natural convection in past few years for directly applications in compact heat exchangers, cooling of electronics systems and polymeric engineering. The natural convection in square enclosures with differently heated sidewalls, filled with power-law and Bingham fluids in addition with heterogeneous medium and centered block are analyzed in this study. The heterogeneous medium are solid, square, disconnected and conducting blocks. The parameters used are the Rayleigh number in the range 104 - 107 , power index n range of 0, 6 - 1, 6, Bingham number range of 0, 5 - Bimax , being the influence of Prandtl number investigated for each fluid model. The number of blocks for heterogeneous medium are 9, 16, 36 and 64, keeping constant solid to fluid conductive ratio, κ = 1. For enclosures with centered block are used the nondimensional block size from 0, 1 to 0, 9, with solid to fluid conductive ratio in range κ = 0, 1; 1 and 10. Mathematical modeling is done by mass, momentum and energy balance equations. The solution of equations have been numerically solved in ANSYS FLUENT R software. Firstly, numerical solutions for validation with Newtonian fluids in clean enclosures are conducted, followed by enclosures with heterogeneous medium and centered block. Subsequently, numerical solutions of power-law and Bingham fluids with same enclosures configurations are conducted. The results are reported in the form of streamlines, isotherms and average Nusselt number at hot wall. In general, the heat transfer process in enclosure is governed by Rayleigh number, size and thermal conductivity of the blocks, power index n for power-law fluid and Bingham number for Bingham fluid. Both fluid models are very sensitive with Prandtl number changes. Heterogeneous medium decrease heat transfer in enclosure when affects thermal boundary layer for both fluid models. One analytical prediction was made for power-law fluid. An increase in heat transfer occurs with low thermal conductivity block and few interference and with high thermal conductivity block and great interference, for both fluids.

Le chapitre 1 introduit les modèles et donne les principaux résultats obtenus. Dans le chapitre 2, on présente des simulations numériques d'un modèle macroscopique en deux dimensions. La méthode de discrétisation par éléments finis utilisée est décrite. Pour le cas test de l'écoulement autour d'un cylindre, les phénomènes en jeu dans les fluides vieillissants sont observés. Le chapitre 3 concerne l'étude mathématique de la version unidimensionnelle du système d'équations aux dérivées partielles utilisé pour les simulations. On montre que le problème est bien posé et on examine le comportement en temps long de la solution. Dans le dernier chapitre, des équations macroscopiques sont dérivées à partir d'une équation mésoscopique. L'analyse mathématique de cette équation mésoscopique est également menée / This thesis is devoted to the modelling, the mathematical analysis and the simulation of non-Newtonian fluids. Some fluids in an intermediate liquid-solid phase are particularly considered: aging fluids. Modelling scales are macroscopic and mesoscopic. In Chapter 1, we introduce the models and give the main results obtained. In Chapter 2, we present numerical simulations of a macroscopic two-dimensional model. The finite element method used for discretization is described. For the flow past a cylinder test-case, phenomena at play in aging fluids are observed. The Chapter 3 contains a mathematical analysis of the one-dimensional version of the system of partial differential equations used for the simulations. We show well-posedness and investigate the longtime behaviour of the solution. In the last chapter, macroscopic equations are derived from a mesoscopic equation. The mathematical analysis of this mesoscopic equation is also carried out

L’objectif de cette thèse est de développer un modèle numérique 3D afin d’étudier le phénomène de laves torrentielles ; écoulements visqueux fortement chargés en matière solide, surgissant en montagne lors d’orages violents. Aujourd’hui, la prévision des zones de vulnérabilité s’appuie sur des outils de calcul 0D, 1D ou 2D. Or ces outils ne peuvent représenter pleinement le comportement à surface libre des écoulements du fait de nombreuses approximations et hypothèses. Ainsi cette thèse met en œuvre un code numérique 3D pour étudier ces écoulements. Ce travail aboutit à la construction d’un modèle 3D à partir de données réelles de terrain. Plusieurs scénarios ont été étudiés et comparés à des résultats issus d’un modèle 2D. Les résultats mettent en évidence les apports non négligeables de la modélisation 3D : zones d’étalement et de dépôt, phénomènes d’encombrement, modélisation fine des écoulements dans les zones chenalisées. Préalablement, le modèle 3D a été validé en comparant les résultats numériques à des données expérimentales issues de la littérature, pour des typologies d’écoulement représentatives de celles observées sur des sites grandeur nature. / The objective of this thesis is to develop a 3D numerical model to assess debris flow. These viscous flows, heavily loaded with solid matter, form when heavy rain occurs in mountains. Today, forecasts of potentially impacted areas are based on 0D, 1D and 2D numerical tools. However, these tools cannot fully represent the free surface behaviour of debris flows due to the approximations and assumptions on which they are based. Thus, this work utilises a 3D numerical code to study this phenomenon. A specific model is built with real field data. Several flow scenarios are studied and compared with a 2D numerical model. The results highlight the significant benefits of a 3D approach by providing information on the fine representation of flow dynamics over the catchment area. The model also predicts the impact of debris flow (overflowing on a road bridge) and the zones of deposition and spreading. It highlights possible congestion phenomena and reproduces flows in the channels by fully accounting for parietal friction, capabilities not provided by 2D models. Prior to this application, the 3D model was evaluated with five sets of experimental data to validate its ability to represent viscoplastic flows. Different types of flows are studied and are representative of those observed on real sites when debris flow occur.