The surface oscillations in a deformable wall are known to induce an instability in the adjacent flow even in the absence of inertia. This instability, if understood properly, can be exploited to generate a well-mixed flow pattern with improved transport coefficients in microfluidic systems, wherein the benefits of inertial instabilities can not be realised. In order to utilise the wall deformability in micro-devices as well as other biotechnological applications, the quantitative knowledge of the critical parameter for the on-set of instability and the nature of bifurcation in the region of transition point are essential. With this objective, a major portion of this thesis deals with the stability analysis of flow past a flexible surface. For Newtonian flow over a deformable solid medium, the analyses of hydrodynamic stability in two flow regimes are presented: the viscous mode instability in the limit of zero Reynolds number, and the wall mode instability in the limit of high Reynolds number. The flexible solid in both analyses is described as a neo-Hookean solid continuum of finite thickness. The previous work on viscous instability using the same solid model ignored the viscous dissipation in the solid. In the present study, a purely elastic neo-Hookean model is augmented to incorporate the viscous stresses accounting for the dissipative mechanism in an aqueous gel-like solid medium. The linear stability analysis for this neo-Hookean viscoelastic solid shows a dramatic influence of solid viscosity on the stability behaviour. The important parameter here is where ηr is the solid viscosity relative to the fluid viscosity and H is the solid-to-fluid thickness ratio. While the effect solid viscosity is stabilizing for a further increase in viscosity in the regime reduces the critical shear rate for transition, indicating a destabilizing influence of solid viscosity. The weakly nonlinear analysis indicates that the bifurcation is subcritical for most values of H when ηr =0. However, for non-zero solid viscosity, the analysis reveals a range of ηr for which the nature of bifurcation is supercritical. The results are in contrast to the behaviour for the Hookean (linear) elastic solid, for which the effect of solid viscosity is always stabilising and the bifurcation is subcritical for all values of H and ηr. For the wall mode instability, critical parameters for the linear and the neo-Hookean elastic solid are found to be very close. The weakly nonlinear analysis of the wall mode instability shows that the instability is driven to a supercritically stable branch, indicating the possibility of a stable complex flow pattern which is ) correction to the base flow. The amplitude of the supercritically bifurcated equilibrium state, A1e, is derived in the vicinity of the critical point, and its scaling with the flow Reynolds number is obtained. The nonlinear analysis is also carried out using the asymptotic analysis in small parameter Re−1/3. The asymptotic results are found to be in good agreement with the numerical solutions for For a polymeric flow over a deformable solid medium, the viscous instability is analysed by extending the viscous mode for the Newtonian fluid to the fluid with finite elasticity. The viscoelastic fluid is described by an Oldroyd-B model which introduces two additional parameters: the Weissenberg number, W , and β, the ratio of solvent-to-solution viscosity. The polymer viscosity parameter β is an indirect measure of polymer concentration with the extreme cases of β =1 representing the Newtonian fluid and β =0the upper convected Maxwell fluid. The analysis considers both the linearly elastic and the neo-Hookean models to describe the deformable solid. The analysis reveals the presence of two classes of modes: the finite wavelength modes and the shortwave modes. The behaviour of the finite wavelength modes is similar for both the models of solid medium. The effect of increasing fluid Weissenberg number and also increasing polymer concentration (achieved by reducing β below 1) on the finite wavelength instability is stabilising. The viscous instability ceases to exist for W larger than a certain maximum value Wmax. The behaviour of the shortwave mode is remarkably different for both the models of solid. Using the shortwave asymptotic, the differences are elucidated and it is shown that the shortwave instabilities in both the models are qualitatively different modes. For a linear elastic solid model, the shortwave mode is attributed to the normal-stresses in polymeric fluid with high Weissenberg number. This mode does not exist for the Newtonian flow and is a downstream travelling disturbance wave. On the other hand, the shortwave mode for the neo-Hookean model is attributed to the normal-stress difference in the elastic solid. Hence, this mode does exist for the Newtonian fluid and is an upstream travelling disturbance wave. The role of polymer concentration in the criticality of finite wavelength and shortwave modes is examined for a wide range of Weissenberg number. The results are condensed in a map showing the stability boundaries in parametric space covering β, W and H. The weakly nonlinear analysis reveals that the bifurcation of linear instability is subcritical when there is no dissipation in the solid. The nature of bifurcation, however, changes to supercritical when the viscous effects in the solid are taken into account. The final problem of this thesis deals with the flow past a rigid surface. Here, the stability of base profile in a plane Couette flow of dilute polymeric fluid is studied at moderate Reynolds number. Three variants of Oldroyd-B model have been analysed, viz. the classical Oldroyd-B model, the diffusive Oldroyd-B model, and the non-homogeneous Oldroyd-B model. The Newtonian wall modes are modified marginally for the polymeric fluid described by the classical Oldroyd-B model. The Oldroyd-B model with artificial diffusivity introduces the additional ‘diffusive modes’ which scale with P´eclet number. The diffusive modes become the slowest decaying modes, in comparison to the wall modes, for large wavenumber disturbances. For these two models, the polymeric flow is linearly stable. Using the equilibrium flow method, wherein the nonlinear flow is assumed to be at the transition point, the finite amplitude disturbances are analysed, and the threshold energy necessary for subcritical transition is estimated. The third variant of Oldroyd-B model accounts for non-homogeneous polymer concentration coupled with the stress field. This model exhibits an instability in the linear analysis. The ‘concentration mode’ becomes unstable when the fluid Weissenberg number exceeds a certain transition value. This instability is driven by the stress-induced fluctuations in polymer number density.

Chemical modification is an important way to improve the properties of existing polymers, and one of the important examples is the hydrogenation of nitrile butadiene rubber (NBR) in organic solvent by homogeneous catalysis in order to extend its application. This process has been industrialized for many years to provide high performance elastomers (HNBR) for the automotive industry, especially those used to produce components in engine compartments. In the current commercial process, a batch reactor is employed for the hydrogenation step, which is labor intensive and not suitable for large volume of production. Thus, novel hydrogenation devices such as a continuous process are being developed in our research group to overcome these drawbacks. In order to make the process more practical for industrial application, high concentration polymer solutions should be targeted for the continuous hydrogenation. However, many problems are encountered due to the viscosity of the high concentration polymer solution, which increases tremendously as the reaction goes on, resulting in severe mass transfer and heat transfer problems. So, hydrogenation kinetics in high concentration NBR solution, as well as the rheological properties of this viscous solution are very essential and fundamental for the design of novel hydrogenation processes and reactor scale up. In the present work, hydrogenation of NBR in high concentration solution was carried out in a batch reactor. A commercial rhodium catalyst, Wilkinson’s catalyst, was used with triphenylphosphine as the co-catalyst and chlorobenzene as the solvent. The reactor was modified and a PID controller was tuned to fit this strong exothermic reaction. It was observed that when NBR solution is in a high concentration the kinetic behavior was greatly affected by mass transfer processes, especially the gas-liquid mass transfer. Reactor internals were designed and various agitators were investigated to improve the mechanical mixing. Experimental results show that the turbine-anchor combined agitator could provide superior mixing for this viscous reaction system. The kinetic behavior of NBR hydrogenation under low catalyst concentration was also studied. It was observed that the hydrogenation degree of the polymer could not reach 95% if less than 0.1%wt catalyst (based on polymer mass) was used, deviating from the behavior under a normal catalyst concentration. The viscosity of the NBR-MCB solutions was measured in a rotational rheometer that has a cylinder sensor under both room conditions and reaction conditions. Parameters that might affect the viscosity of the solutions were studied, especially the hydrogenation degree of polymer. Rheological properties of NBR-MEK solutions, as well as NBR melts were also studied for relevant information. It is concluded that the hydrogenation kinetics deviates from that reported by Parent et al. [6] when polymer is in high concentration and/or catalyst is in low concentration; and that the reaction solution (HNBR/NBR-MCB solution) deviates from Newtonian behavior when polymer concentration and hydrogenation degree are high.

Single ventricle congenital heart defects afflict 2 per every 1000 births. They are characterized by cyanotic mixing between the de-oxygenated blood coming back from the systemic circulation and the oxygenated blood from the pulmonary circulation. Prior to introduction of the Fontan procedure in 1971, surgical options for single ventricle patients were limited. The Fontan operation involves a series of three palliative procedures aimed at the separation of systemic and pulmonary circulations and reducing the long term effects of chronic hypoxia and ventricular volume overload. The total cavopulmonary connection (TCPC) is completed in the final stage of the surgery with the anastomosis of the inferior vena cava (IVC) and superior vena cava to the pulmonary arteries. Improved quantification and visualization of flow structures within the TCPC has the potential to aid in the planning and design of the Fontan operation. Despite significant development of phase contrast magnetic resonance imaging (PC MRI) for in vivo flow measurements, it is not routinely applied in children with single ventricle congenital heart disease. Limited technologies available for post-processing of PC MRI data has prevented clinicians and scientists from conducting the detailed hemodynamic analyses necessary to better understand the physiology of the single ventricle circulation. This thesis attempts to bridge the gap between PC MRI and fluid dynamics, by developing the necessary post-processing technologies for PC MRI, and then applying these techniques for characterizing single ventricle hemodynamics.

Chemical modification is an important way to improve the properties of existing polymers, and one of the important examples is the hydrogenation of nitrile butadiene rubber (NBR) in organic solvent by homogeneous catalysis in order to extend its application. This process has been industrialized for many years to provide high performance elastomers (HNBR) for the automotive industry, especially those used to produce components in engine compartments. In the current commercial process, a batch reactor is employed for the hydrogenation step, which is labor intensive and not suitable for large volume of production. Thus, novel hydrogenation devices such as a continuous process are being developed in our research group to overcome these drawbacks. In order to make the process more practical for industrial application, high concentration polymer solutions should be targeted for the continuous hydrogenation. However, many problems are encountered due to the viscosity of the high concentration polymer solution, which increases tremendously as the reaction goes on, resulting in severe mass transfer and heat transfer problems. So, hydrogenation kinetics in high concentration NBR solution, as well as the rheological properties of this viscous solution are very essential and fundamental for the design of novel hydrogenation processes and reactor scale up. In the present work, hydrogenation of NBR in high concentration solution was carried out in a batch reactor. A commercial rhodium catalyst, Wilkinson’s catalyst, was used with triphenylphosphine as the co-catalyst and chlorobenzene as the solvent. The reactor was modified and a PID controller was tuned to fit this strong exothermic reaction. It was observed that when NBR solution is in a high concentration the kinetic behavior was greatly affected by mass transfer processes, especially the gas-liquid mass transfer. Reactor internals were designed and various agitators were investigated to improve the mechanical mixing. Experimental results show that the turbine-anchor combined agitator could provide superior mixing for this viscous reaction system. The kinetic behavior of NBR hydrogenation under low catalyst concentration was also studied. It was observed that the hydrogenation degree of the polymer could not reach 95% if less than 0.1%wt catalyst (based on polymer mass) was used, deviating from the behavior under a normal catalyst concentration. The viscosity of the NBR-MCB solutions was measured in a rotational rheometer that has a cylinder sensor under both room conditions and reaction conditions. Parameters that might affect the viscosity of the solutions were studied, especially the hydrogenation degree of polymer. Rheological properties of NBR-MEK solutions, as well as NBR melts were also studied for relevant information. It is concluded that the hydrogenation kinetics deviates from that reported by Parent et al. [6] when polymer is in high concentration and/or catalyst is in low concentration; and that the reaction solution (HNBR/NBR-MCB solution) deviates from Newtonian behavior when polymer concentration and hydrogenation degree are high.

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