In this dissertation, we study the behavior of microscopic organisms utilizing lateral and spiral bending waves to swim in a fluid. More specifically, spermatozoa encounter different fluid environments filled with mucus, cells, hormones, and other large proteins. These networks of proteins and cells are assumed to be stationary and of low volume fraction. They act as friction, possibly preventing or enhancing forward progression of the swimmers. The flow in the medium is described as a viscous fluid with a resistance term known as a Brinkman fluid. It depends on the Darcy permeability parameter affecting the swimming patterns of the flagella. To further understand these effects we study the asymptotic swimming speeds of an infinite-length swimmer propagating planar or spiral bending waves in a Brinkman fluid. We find that, up to the second order expansion, the swimming speeds are enhanced as the resistance increases. The work to maintain the planar bending and the torque exerted on the fluid are also examined. The Stokes limits of the swimming speeds, the work and the torque are recovered as resistance goes to zero. The analytical solutions are compared with numerical results of finite-length swimmers obtained from the method of Regularized Brinkmanlets (MRB). The study gives insight on the effects of the permeability, the length and the radius of the cylinder on the performance of the swimmers. In addition, we develop a grid-free numerical method to study the bend and twist of an elastic rod immersed in a Brinkman fluid. The rod is discretized using a Kirchhoff Rod (KR) model. The linear and angular velocity of the rod are derived using the MRB. The method is validated through a couple of benchmark examples including the dynamics of an elastic rod, and the planar bending of a flagellum in a Brinkman fluid. The studies show how the permeability and stiffness coefficients affect the waveforms, the energy, and the swimming speeds of the swimmers. Also, the beating pattern of the spermatozoa flagellum depends on the intracellular concentrations of calcium ([Ca2+]). An increase of [Ca2+] is linked to hyperactivated motility. This is characterized by highly asymmetrical beating, which allows spermatozoa to reach the oocyte (egg) or navigate along the female reproductive tract. Here, we couple the [Ca2+] to the bending model of a swimmer in a Brinkman fluid. This computational framework is used to understand how internal flagellar [Ca2+] and fluid resistance in a Brinkman fluid alter swimming trajectories and flagellar bending.

archives@tulane.edu / Discrete fracture networks (DFNs) can be modeled with polygonal representations that are useful for geophysical modeling of nuclear waste containment and hydrofrac- turing. Flow and transport calculations are possible, but computationally expensive, limiting the feasibility for model uncertainty quantification. Graphs are used to re- duce model complexity and computation time. We present the formulation of using a graph as a reduced model for DFNs and pose the inversion problem central to this research. We present a novel alternative to Darcy’s law on graphs using the well known Brinkman formulation on the continuum. We apply the Levenberg-Marquardt algorithm to optimize graphs, calibrating them to observed data through the inversion problem. We present the deficiencies in physically motivated graphs, and show how optimized graphs produce better results overall. Our solution finds lumped parameters representing the fracture properties, and is used to reduce the computational time required for particle transport calculations. Breakthrough curves are produced on our obtained solutions, which closely match the high fidelity model. We present examples of creating these reduced models for DFNs with 500 fractures to illustrate the methodology and optimization scheme used to obtain an improved result over a previous formulation. / 1 / Jaime Lopez-Merizalde

Fluid-structure interactions play a central role in an overwhelming number of physical phenomena. All fluid dynamic students are familiar with the common assumption of a "smooth boundary". While this assumption often is enough to provide a high level understanding of the dynamics and physics at hand in practice this is not true.  Much of the detail and the unique phenomena can be traced back to surface properties that deviate from this elementary assumption.  In this work we investigate three problems all motivated by the existence of non-smooth surfaces.   The first paper considers how inhomogeneous surfaces can generate a lift force for lubricated contacts. This work showcases how subtle changes to surface texture or chemistry modeled by a slip length can invoke non-trivial forces. These forces result in striking particle trajectories not possible in the presence of a smooth no-slip wall.    The next work focuses on porous surfaces. Often the geometry of surfaces in nature and industry are complex covering a wide range of length scales. Resolving all the scales of motion arising from fluid interaction with such surfaces are computationally expensive. Effective equations are often applied to reduce the cost of such simulations. The Brinkman equation is one common model choice for free-fluid and porous surface interface. Despite the common application of the Brinkman equation, fundamental questions about what the effective viscosity should be remain open. We compare pore-scale Stokes flow solutions to the Brinkman model for several porous surfaces. This study provides a scaling for the effective viscosity as well as error quantification of the Brinkman model.    Lastly, we investigate how porous surfaces modify a turbulent boundary layer. Streamwise preferential porous surfaces have recently been suggested as a surface modification that has the potential to reduce drag. We compare particle image velocimetry measurements with direct numerical simulations focusing on the near wall features that are modified from the canonical smooth wall case. We present some preliminary turbulent statistics and flow visualizations in the current report.

Les procédés d’élaboration de pièces composites par infusion de résine, malgré leurs nombreux avantages, peinent à s’imposer dans les phases de production industrielle en raison de difficultés pour les maitriser. Ainsi, en partenariat avec ESI Group, un modèle complet pour la simulation de ces procédés est développé à l’ENSM-SE depuis les travaux précurseurs de P. Celle.Nos travaux portent sur la généralisation de ce modèle afin de traiter des cas, ainsi que sur son extension à la simulation des écoulements « post-infusion ». L'approche repose sur un découpage du domaine en trois zones (drainant, préformes imprégnées, préformes sèches) consistant ainsi à coupler un écoulement de Stokes dans le drainant à un écoulement de Darcy dans les préformes. De plus, l'influence mutuelle de la résine sur le comportement des préformes et de la déformation des préformes sur la perméabilité est considérée, à travers la loi de Terzaghi et des lois exprimant la perméabilité en fonction de la fraction de fibres, paramètre accessible uniquement dans une approche 3D mécanique couplée. Enfin, le procédé est découpé en trois phases : compression initiale des préformes sèches, remplissage et « post-infusion ». Les méthodes numériques, développées dans ces travaux, s'appliquent à des cas réels d'infusion souvent mis de côté dans les publications récentes car inaccessibles, impliquant des perméabilités très faibles (~10-15 m²), un drainant fin (~1 mm) et des géométries complexes.Cette approche innovante a été implémentée dans un code de calcul industriel (ProFlotTM), validée analytiquement sur des cas tests et expérimentalement sur des cas industriel dans le cadre du projet européen INFUCOMP. / Composite manufacturing processes by resin infusion, despite their many benefits, struggle to establish themselves in the industrial production phases due to difficulties to control them. So, in partnership with ESI Group, a comprehensive model for the simulation of these processes is developed at the ENSM-SE since the pioneering work of P. Celle.Our work focuses on the generalization of this model to handle complex industrial cases in three dimensions, as well as its extension to “post-infusion” flow simulation. The approach is based on three domains decomposition of the field (Distribution medium, impregnated preforms, dry preforms) consisting in coupling a Stokes flow in the distribution medium with a Darcy flow in the preforms. In addition, the mutual influence of the resin on the preforms and of the preforms deformation on the permeability is considered, through Terzaghi’s law and models expressing the permeability as a function of the fibre fraction, data only accessible with a 3D coupled mechanical approach. Finally, the process is divided into three phases: initial compression of dry preforms, filling and “post-infusion”. The numerical methods developed in this work, apply to real infusion cases often discarded in recent publications, involving very low permeability (~10-15 m²), thin distribution medium (~ 1 mm) and complex geometries (3D curved).This innovative approach has been implemented in an industrial simulation code (ProFlotTM), validated analytically over test cases and experimentally over industrial cases in the European project INFUCOMP.

[pt] Rochas carbonáticas possuem atributos únicos que as distinguem das siliciclasticas e necessitam de diferentes métodos de estudo para caracterizar sua textura. Rochas carbonáticas são resultado de interações entre processos deposicionais químicos e biológicos. Os processos diagenéticos que formam e alteram estas rochas contribuem para o acúmulo de heterogeneidades. Devido ao alto teor de heterogeneidade, as propriedades medidas (e.g. porosidade e permeabilidade) variam com a escala de investigação e estudos com o compromisso a significância dos detalhes e representação espacial. Ampliar uma escala menor para maior, exige procedimentos que preservem a essência dos processos físicos de um nível mais detalhada para um mais grosseiro. Métodos simplificados para dimensionar propriedades não aditivas em outras escalas, tal como permeabilidade, geralmente não honram a heterogeneidade presente em sistemas complexos. Com isso a dinâmica dos fluidos em rochas complexas exige abordagens e métodos mais sofisticados. Este estudo é focado em desenvolver uma metodologia de avaliação da permeabilidade como parâmetro de escala para meios porosos heterogêneos. A permeabilidade é retro calculada ao emular o experimento de Darcy e resolver o fluxo no meio poroso utilizando uma formulação de elementos finitos para equação de Brinkman. O estudo iniciou com foco em sistemas paramétricos de células periódicas e posteriormente a duas microtomogra fia de rochas carbonáticas, do qual uma foi selecionada para um estudo de representatividade espacial. As células periódicas foram utilizadas para avaliar os efeitos da formação de canais e espaços vazios, em analogia a fraturas e vugos, assim como os efeitos da permeabilidade da matriz porosa. As amostras carbonáticas consistem de casos nos quais é possível observar a presença de uma combinação dos fenômenos estudadas. Por fim um estudo de representatividade foi conduzido segmentando uma amostra de micro-tomografia em suficientes sub-amostras que capazes de reproduzir a heterogeneidade espacial da amostra original. / [en] Carbonate rocks have unique attributes that distinguish them from siliciclastics and that require diferent methods of study to characterize their texture. Carbonates rocks are formed as a result of close interactions between biological and chemical depositional processes. The underlying diagenetic processes that form and alter these rocks contribute to a build-up of heterogeneities. Because of the high heterogeneity content measured properties (e.g. porosity and permeability) change with the scale of investigation and studies have struggle with a trade-off between significance of details and space representativeness. Extending a smaller scale to a larger requires scaling up procedures that preserves the essence of physical processes at one level to be summarized at the coarser level. Simplistic methods for scaling-up non-additive properties such as permeability generally do not honour the original heterogeneity present in complex systems. Therefore the dynamics of fluid flow in complex rocks demand more sophisticate methods and approaches. This study was focused in developing a methodology to evaluate the permeability as a scaling-up parameter for heterogeneous porous media. The permeability is back-calculated by emulating Darcy s experiment and solving the pore-scale ow using a Finite element formulation of Brinkman flow equation. The study was initially focused on parametric systems of periodic cells and later extended to two micro-tomography carbonate samples in which one has been selected for a spatial representativeness study. The parametric cells were used to evaluate the shape effects of channels and void spaces in an analogy to geological fractures and vugs as well as the permeability of the porous matrix. The micro-tomography carbonate samples consisted of a real case scenario in which, to a certain degree, could be observed a combination of the previously studied periodic cells. Finally a representativeness study was conducted segmenting the micro-tomography sample into suficiently sub-samples that would be capable of reproducing the spatial heterogeneity of the sample.

[pt] O presente trabalho foca em uma investigação numérica do escoamento através de meios porosos com macroporos, como os carbonatos. A presença de vugs e fraturas tem um forte efeito na característica do escoamento. O fluxo através da matriz porosa é geralmente descrito pela equação de Darcy e o fluxo através dos macroporos pela equação de Stokes. O acoplamento entre essas duas abordagens distintas traz grande complexidade para a modelagem de tais fluxos. Neste trabalho, utilizamos a formulação de Brinkman que é capaz de descrever ambos os fluxos através da matriz porosa e macroporos com uma única equação diferencial. Resolvemos o sistema de equações diferenciais usando o modelo de elementos finitos e implementamos o código com auxílio da plataforma FEniCS. Primeiro, resolvemos o fluxo 1-D através de placas paralelas com uma das paredes sendo um material poroso. O objetivo foi comparar as previsões obtidas pela formulação de Brinkman com aquela obtida usando a condição de contorno de Beavers-Joseph. Então, resolvemos um fluxo 2-D através de um meio poroso com macroporos. A geometria da estrutura porosa foi obtida a partir de cortes 2D de imagens tomográficas de carbonatos. O objetivo desta análise foi avaliar uma permeabilidade equivalente em função da área e estrutura dos macroporos. / [en] The present work focuses on a numerical investigation of the flow through porous media with macropores, such as carbonates. The presence of vugs and fractures have a strong effect on the flow characteristics. The flow through the porous matrix is usually described by the Darcy equation and the flow through the macropores by Stokes equation. The coupling between these two distinct approaches brings great complexity to the modeling of such flows. In this work, we use Brinkman formulation that is able to describe the flow, both in the porous matrix and macropores with a single differential equation. We solved the set of differential equations using the finite element model and implemented the code in the FEniCS platform. We first solved the 1-D flow through parallel plates with one of the walls being a porous material. The goal was to compare the predictions obtained with the Brinkman formulation to that obtained by using the Beavers-Joseph boundary condition. Then, we solved a 2-D flow through a porous medium with macropores. The geometry of the pore structure was obtained from 2D slices of tomographic images of carbonates. The goal of this analysis was to evaluate an equivalent permeability as a function of macropores area and structures.

Les aquifères karstiques sont caractérisés par un écoulement rapide dans les conduits karstiques et un écoulement lent dans la roche encaissante (matrice). En raison de cet écoulement rapide, ces aquifères sont fortement vulnérables à la pollution. L'objectif principal de cette thèse est d'évaluer l'importance des échanges d'eau à l'interface entre les conduits et la matrice et par conséquent l'impact de ces échanges sur le transport de soluté dans le système karstique. Pour investiguer l'apport de l'équation de Brinkman à l'évaluation des écoulements d'eau, un modèle dont la géométrie du conduit est décrite par la fonction sinus a été établi. Deux scénarii sont appliqués. Le scénario A emploie la loi de Darcy dans le conduit et la matrice. Le scénario B emploie l'équation de Brinkman et la loi de Darcy dans le conduit et la matrice, respectivement. Le terme supplémentaire de l'équation de Brinkman décrivant les forces de cisaillement joue un rôle prédominant sur l'échange d'eau entre le conduit et la matrice lorsque les conduits sont sinueux. Le modèle hydrodynamique - transport a été appliqué au système karstique du Val d'Orléans. La calibration a démontré que le meilleur diamètre moyen du conduit est de 5 m, la perméabilité dans la roche encaissante est de 5,7×10-10 m2 et celle du conduit de 1,65×10-5 m2 à 5,5×10-5 m2. Les prédictions du modèle hydrodynamique - transport ont été comparées aux crues de la Loire observées dans les années 1992 et 1993 durant lesquelles les concentrations en chlorure (traceur naturel conservatif) ont été dosées dans la Loire et à la source du Bouillon. Avec un coefficient d'emmagasinement dans la matrice de 0,9 m-1, la distance pénétrée par l'eau de la crue dans la matrice est d'environ 530 m en deux mois pour une crue de 2 m (exemple de 1993).

Les procédés d'élaboration de pièces composites par infusion de résine, malgré leurs nombreux avantages, peinent à s'imposer dans les phases de production industrielle en raison de difficultés pour les maitriser. Ainsi, en partenariat avec ESI Group, un modèle complet pour la simulation de ces procédés est développé à l'ENSM-SE depuis les travaux précurseurs de P. Celle.Nos travaux portent sur la généralisation de ce modèle afin de traiter des cas, ainsi que sur son extension à la simulation des écoulements " post-infusion ". L'approche repose sur un découpage du domaine en trois zones (drainant, préformes imprégnées, préformes sèches) consistant ainsi à coupler un écoulement de Stokes dans le drainant à un écoulement de Darcy dans les préformes. De plus, l'influence mutuelle de la résine sur le comportement des préformes et de la déformation des préformes sur la perméabilité est considérée, à travers la loi de Terzaghi et des lois exprimant la perméabilité en fonction de la fraction de fibres, paramètre accessible uniquement dans une approche 3D mécanique couplée. Enfin, le procédé est découpé en trois phases : compression initiale des préformes sèches, remplissage et " post-infusion ". Les méthodes numériques, développées dans ces travaux, s'appliquent à des cas réels d'infusion souvent mis de côté dans les publications récentes car inaccessibles, impliquant des perméabilités très faibles (~10-15 m²), un drainant fin (~1 mm) et des géométries complexes.Cette approche innovante a été implémentée dans un code de calcul industriel (ProFlotTM), validée analytiquement sur des cas tests et expérimentalement sur des cas industriel dans le cadre du projet européen INFUCOMP.

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第21151号 / 農博第2277号 / 新制||農||1059(附属図書館) / 学位論文||H30||N5125(農学部図書室) / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 村上 章, 教授 川島 茂人, 教授 藤原 正幸 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Different generalized Newtonian fluids (where the normal stresses were neglected) were considered in this study. Analytical expressions were derived for time independent, fully developed velocity profiles of Herschel-Bulkley fluids (including the simplifications thereof: Newtonian, power law and Bingham plastic fluids) and Casson fluids through open channel sections. Both flow through cylindrical pipes (Hagen-Poiseuille flow) and parallel plates (plane Poiseuille flow) were brought under consideration. Equations were derived for the wall shear stresses in terms of the average channel velocities. These expressions for plane Poiseuille flow were then utilized in the modelling of flow through homogeneous, isotropic porous media. Flow through parallel plates was extended and a possibility of a moving lower wall (plane Couette-Poiseuille flow) was included for Herschel-Bulkley fluids (and the simplifications thereof). The velocity of the wall was assumed to be opposite to the pressure gradient (thus in the streamwise direction) yielding three different possible flow scenarios. These equations were again revisited in the study on flow over porous structures. Averaging of the microscopic momentum transport equation was carried out by means of volume averaging over an REV (Representative Elementary Volume). Flow through parallel plates enclosing a homogeneous porous medium (assumed homogeneous up to the external boundary) was studied at the hand of Brinkman’s equation. It was as- sumed (also for non-Newtonian fluids) that the term dominating outside the external boundary layer area is directly proportional to the superficial velocity that is, since only the viscous flow regime was considered, referred to as the ‘Darcy’ velocity if the diffusive Brinkman term is completely neglected. For a shear thinning or shear thickening fluid, the excess superficial velocity term was included in the proportionality coefficient that is constant for a particular fluid traversing a particular porous medium subjected to a specific pressure gradient. For such fluids only the inverse functions could be solved. If the ‘Darcy’ velocity is not reached within the considered domain, Gauss’s hypergeo- metric function had to be utilized. For Newtonian and Bingham plastic fluids, direct solutions were obtained. The effect of the constant yield stress was embedded in the proportionality coefficient. For linear flow, the proportionality coefficient consists of both a Darcy and a Forch- heimer term applicable to the viscous and inertial flow regimes respectively. Secondary averaging for different types of porous media was accomplished by using an RUC (Representative Unit Cell) to estimate average interstitial properties. Only homoge- neous, isotropic media were considered. Expressions for the apparent permeability as well as the passability in the Forchheimer regime (also sometimes referred to as the non-Darcian permeability) were derived for the various fluid types. Finally fluid flow in a domain consisting of an open channel adjacent to an infinite porous domain is considered. The analytically derived velocity profiles for both plane Couette- Poiseuille flow and the Brinkman equation were matched by assuming continuity in the shear stress at the porosity jump between the two domains. An in-house code was developed to simulate such a composite domain numerically. The difference between the analytically assumed constant apparent permeability in a macro- scopic boundary layer region as opposed to a dependency of the varying superficial velocity was discussed. This code included the possibility to alter the construction of the domain and to simulate axisymmetrical flow in a cylinder. / AFRIKAANSE OPSOMMING: Verskeie veralgemeende Newtoniese vloeistowwe (waarvan die normaalspannings ignoreer- baar is) word in hierdie studie beskou. Analitiese uitdrukkings vir tyd-onafhanklike, ten volle ontwikkelde snelheidsprofiele vir Herschel-Bulkley vloeistowwe (wat die vereen- voudigde weergawes daarvan insluit: Newtoniese, magswet- en Bingham-plastiek vloei- stowwe), sowel as Casson vloeistowwe, is afgelei vir vloei deur ‘n oop kanaal. Beide vloei deur silindriese pype (Hagen-Poiseuille vloei) en parallelle plate (vlak-Poiseuille vloei) is oorweeg. Vergelykings vir die skuifspannings op ‘n wand in terme van die gemiddelde snelhede is afgelei. Hierdie uitdrukking wat vir vlak-Poiseuille vloei verkry is, is in die modellering van vloei deur homogene, isotropiese poreuse media ook gebruik. Vloei deur parallelle plate is uitgebrei en die moontlikheid van ‘n bewegende onderste wand (vlak-Couette-Poiseuille vloei) is ondersoek vir Herschel-Bulkley vloeistowwe (en die vereenvoudigings daarvan). Dit word aangeneem dat die snelheid van die wand in die teenoorgestelde rigting as die drukgradiënt georiënteer is (dus in die stroomgewyse rigting) wat dan tot drie verskillende moontlike vloeigevalle lei. Hierdie vergelykings is weer in die studie van vloei oor poreuse strukture gebruik. Die gemiddelde van die mikroskopiese momentum transportvergelyking is bereken oor die volume van ‘n REV (“Representative Elementary Volume”). Vloei deur parallelle plate wat ‘n homogene poreuse medium omsluit (waar die medium homogeen aanvaar word tot by die eksterne grens) is bestudeer aan die hand van Brinkman se vergelyking. Daar is aanvaar (ook vir nie-Newtoniese vloeistowwe) dat die dominante term buite die eksterne grenslaaggebied direk eweredig is aan die oppervlaksnelheid en, aangesien slegs vloei in die viskeuse gebied oorweeg word, daarna verwys word as die “Darcy”- snelheid, indien die diffusiewe Brinkman-term heeltemal weglaatbaar is. Vir ‘n span-ningsverdunnende of -verdikkende vloeistof, word die oortollige oppervlaksnelheidsterm ingesluit by die proporsionaliteitskoëffisiënt wat konstant is vir ‘n spesifieke vloeistof wat deur ‘n sekere poreuse medium, onderhewig aan ‘n spesifieke drukgradiënt, vloei. Vir sulke vloeistowwe kon slegs die inverse funksies opgelos word. As die “Darcy”- snelheid nie binne die betrokke gebied bereik word nie, is daar van Gauss se hipergeometriese funksie gebruik gemaak. Vir Newtoniese en Bingham-plastiek vloeistowwe is egter direkte oplossings verkry. Die effek van die konstante toegeespanning is ingebed in die proporsionaliteitskoëffisiënt. Vir lineêre vloei bestaan die proporsionaliteitskoëffisiënt uit beide ‘n Darcy- en ‘n Forch- heimer-term wat van toepassing is in die viskeuse- en traagheidsvloeigebiede onder- skeidelik. Sekondˆere gemiddeldes vir verskillende tipes poreuse media is verkry; deur gebruik te maak van ‘n RUC (“Representative Unit Cell”) kan interstisiële gemiddelde eienskappe geskat word. Slegs homogene, isotrope media is in oorweging gebring. Uit- drukkings vir die o¨enskynlike deurlaatbaarheid sowel as die deurdringbaarheid in die Forchheimer-gebied (ook soms na verwys as die nie-Darcy deurlaatbaarheid) is afgelei vir die verskillende vloeistoftipes. Ten slotte is vloeistofvloei in ‘n gebied wat bestaan uit ‘n oop kanaal aangrensend aan ‘n oneindige poreuse domein ondersoek. Die analities-afgeleide snelheidsprofiele vir beide vlak-Couette-Poiseuille vloei en die Brinkman-vergelyking is gekoppel deur ‘n kontinu¨ıteit in die skuifspanning by die poreuse-sprong tussen die twee gebiede te aanvaar. ‘n Interne numeriese kode is ontwikkel om so ‘n saamgestelde domein numeries te simuleer. Die verskil tussen die analities konstant-aanvaarde deurlaatbaarheid in ‘n makroskopiese grenslaagstreek, eerder as ‘n afhanklikheid met die veranderende opper- vlaksnelheid, is bespreek. Hierdie kode sluit ook die moontlikheid in om die domein te herkonstrueer, asook om die simulasie van aksiaal-simmetriese vloei in ‘n silinder te ondersoek.