We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.

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For the latter half of the last century renormalization methods have played an important part in tackling problems in fundamental physics and in providing a deeper understanding of systems with many interacting scales or degrees of freedom with strong coupling. The study of turbulence is no exception, and this thesis presents an investigation of renormalization techniques available in the study of the statistical theory of homogeneous and isotropic turbulence. The thesis consists of two parts which assess the two main renormalization approaches available in modeling turbulence. In particular we will be focusing on the renormalization procedures developed by McComb and others. The first part of this thesis will discuss Renormalization Group (RG) approaches to turbulence, with a focus on applications to reduce the degrees of freedom in a large-eddy simulation. The RG methods as applied to classical dynamical systems will be reviewed in the context of the Navier-Stokes equations describing fluid flow. This will be followed by introducing a functional based formalism of a conditional average first introduced by McComb, Roberts and Watt [Phys. Rev A 45, 3507 (1992)] as a tool for averaging out degrees of freedom needed in an RG calculation. This conditional average is then used in a formal RG calculation applied to the Navier-Stokes equations, originally done by McComb and Watt [Phys. Rev. A 46, 4797 (1992)], and later revised by Mc- Comb and Johnston [Physica A 292, 346 (2001)]. A correction to the summing of the time-integral detailed in the latter work is shown to introduce an extra viscous life-time term to the denominator of the increment to the renormalized viscosity and is shown to have a negligible effect in the numerical calculations. We follow this study by outlining some problems with the previous approach. In particular it is shown that a cross-term representing the interaction between high and low wavenumber modes which was neglected in the previous studies on the grounds that it does not contribute to energy dissipation, does in fact contribute significantly. A heuristic method is then put forward to include the effects of this term in the RG calculation. This leads to results which agree qualitatively with numerical calculations of eddy-viscosities. We finish this part of the thesis with an application of the RG method to the modeling of a passive scalar advected by a turbulent velocity field. The second part of this thesis will begin by reviewing Eulerian renormalized perturbation theory attempts in closing the infinite moment hierarchy introduced by averaging the Navier-Stokes equations. This is followed by presenting a new formulation of the local energy transfer theory (LET) of McComb et. al. [J. Fluid Mech. 245, 279 (1992)] which resolves some problems of previous derivations. In particular we show by the introduction of time-ordering that some previous problems with the exponential representation of the correlator can be overcome. Furthermore, we show that the singularity in the LET propagator equation cancels by way of a counter-term. We end this study by introducing a single-time Markovian closure based on LET which, unlike other Markovian closures, does not rely on any arbitrary parameters being introduced in the theory.

A principle concern for engineered CO2 storage is long-term security. Surface leakage (‘seepage’) of injected CO2 to the surface is economically and environmentally undesirable. Italy is a region of intense natural CO2 degassing; 308 CO2 seeps are catalogued which exhibit different surface characteristics, and a number CO2 rich reservoirs were discovered when drilling for hydrocarbons. These seeps and reservoirs provide excellent natural analogues for seeps that might arise from breached carbon stores. This thesis explores the geological controls on the crustal plumbing of CO2 fluids to model the processes governing CO2 seep locations and distribution, and characteristics; and their consequences on human health risk. Risk of human death from accidental CO2 poisoning at all seep types is low (10-8 yr-1) and several factors influence risk of human mortality. Seeps distribute on two spatial scales; on a local scale (<5 km) seeps are clustered and aligned with subsidiary geologic structures, while on large scales seep clusters are discrete, and align with regional structures. Within clusters, seep locations are influenced by fault maturity, the presence of lithological boundaries and seep manifestation, which is determined by the flow properties of the outcropping lithology and local topography. Sealing and seeping CO2 reservoirs are identified, and their geological characteristics compared. Italian reservoirs successfully retain large CO2 columns at a range of reservoir conditions. Reservoirs which have hydrostatic pressure conditions in the overburden, determined from well logs, are located close to surface CO2 seeps and recent extensional faults. Where there is significant overpressure above hydrostatic in the overburden, there are no seeps present above the reservoir structure. Overpressure of reservoir fluids may enhance fluid flow rates but is not a necessary condition for CO₂ leakage. Geothermal conditions influences the style of leakage at depths and towards the surface. Total CO2 degassing from dry Italian CO2 seeps is 3.5 ± 0.5 Mt(CO2)yr-1. It would take thousands of years for the effectiveness of a commercial scale store to be significantly reduced if it leaks to form a single seep with the mean flux rates modeled in Italy. If a seep cluster develops, the storage effectiveness will reduce more rapidly, and could negate engineered CO₂ storage as a climate mitigation strategy. The research presented in this thesis contributes to a body of knowledge which directly informs site selection procedure for carbon storage and maximise the long term storage potential for CCS. Thorough scientific understanding of the geological processes governing fluid escape is crucial to assure the scientific, political and public communities that safe, long-term carbon storage can be realised as an effective climate mitigation technology.

A teoria da percolação tem se revelado muito útil no tratamento de inúmeros fenômenos da natureza. Devido a sua grande versatilidade, esta teoria é objeto de intensa pesquisa. Aqui, propomos novas formas de percolação e as estudamos através de simulações numéricas. Na primeira parte de nosso trabalho, investigamos a estrutura dos aglomerados gerados pelo modelo de percolação por invasão múltipla. Estimamos os valores das dimensões fractais do esqueleto, do esqueleto elástico, dos pontos de estrangulamento e dos menores caminhos, como função dos parâmetros do modelo. Por ter uma estrutura geométrica bastante estabilizada, o modelo otimizado pode vir a ser muito útil no tratamento de problemas com diluição da mecânica estatística. O modelo de percolação atenuada foi concebido para permitir que, durante o processo de invasão, os poros grandes possam também ser ocupados. Esta ocupação ocorre com uma probabilidade que diminui quando o tamanho do poro aumenta.Estimamos cuidadosamente os limiares de percolação e construímos os diagramas de fase correspondentes. Verificamos que os limiares de percolação de nosso modelo não satisfazem a conjectura de Galam e Mauger. Estudamos o efeito da inércia em fluidos escoando através de meios porosos incorporando uma caminhada de N passos ao modelo de percolação por invasão. A magnitude da inércia é proporcional ao parâmetro N, que representa o número de poros seqüencialmente invadidos após a ruptura do perímetro, em cada etapa do processo. Investigamos este modelo em duas e três dimensões. Verificamos que no caso bidimensional, as caminhadas de N passos são facilmente bloqueadas o que leva ao surgimento de um limite superior para o número de passos efetivamente realizados. Nossas estimativas das dimensões fractais dos aglomerados (como função do parâmetro N), indicam que este modelo pertence a uma classe de universalidade diferente daquela da percolação por invasão ordinária. Propomos um modelo de percolação para tratar um processo de solidificação de dois fluidos imiscíveis na presença de impurezas móveis. O movimento das impurezas ocorre devido a uma interação repulsiva de curto alcance observada experimentalmente por Ulhmann, Chalmers e Jackson (UCJ). Dependendo das concentrações de fluidos e impurezas, pode haver a formação de uma fase sólida que percola todo o sistema. Construímos o diagrama de fases deste modelo no espaço das concentrações e calculamos seus expoentes críticos. Nossos resultados indicam que o modelo pertence à mesma classe de universalidade que a percolação ordinária. Finalmente, estudamos um processo de percolação por invasão na presença de impurezas que se movem segundo o mecanismo UCJ. Encontramos um valor crítico para a concentração de impurezas, acima do qual não mais existe percolação. O perfil de aceitação aproxima-se de uma função de Heavyside, com o ponto de descontinuidade dependendo da concentração de impurezas. / Percolation theory provides a quantitative and conceptual model for the understanding of many natural phenomena. Here, we present new kinds of percolation and study them using Monte Carlo simulation techniques. We start studying the cluster, the backbone and the elastic backbone structures of the multiple invasion percolation for both the perimeter and the optimized versions. The behavior of the mass, the number of cutting sites and loops are investigated and their corresponding scaling exponents are estimated. By construction, the mass of the optimized model scales exactly with the gyration radius of the cluster - we verify that this also happens with the backbone. Our simulation shows that the red sites almost disappear, indicating that the cluster has achieved a high degree of connectivity. We propose a new kind of invasion percolation, which permits that, besides small pores, large pores being also occupied. In our model, the occupation probability of a pore diminishes with the pore\'s size. We estimate their corresponding percolation thresholds and show that they do not satisfy the Galam and Mauger conjecture. In order to take into account the inertia of the invader fluid, a new kind of invasion percolation is introduced. In this model, which we named N-steps invasion percolation, the inertia forces are controlled by the number N of pores (or steps) invaded after the perimeter rupture. The new model belongs to a different class of universality and has its fractal dimension depending on N. A blocking phenomenon takes place in two dimensions. It imposes an upper bound value on N. For pores sizes larger than the critical threshold, the acceptance profile exhibits a permanent tail. We also introduce a model for the solidification process of two immiscible fluids interacting repulsively with mobile impurities on a two dimensional square lattice. In the space of the fluids and impurities concentrations the phase diagram exhibits a critical curve separating a percolating from a non-percolating phase. Estimated values for the fractal dimension and the exponent ? of the order parameter, reveal that the critical exponents do not vary along this curve, i.e., they are independent of the impurities concentration. The universality class we find is that of the ordinary percolation. Finally, based on the main ideas of the dynamic epidemic and invasion percolation models, we propose a model to describe the cleaning process of a dirty porous medium by fluid injection. An analysis of the acceptance profiles strongly indicates that this model is a kind of self-organized system.

Generalizamos o modelo de percolação por invasão de maneira que vários sítios possam ser simultaneamente invadidos. Propomos dois tipos de generalização: na primeira, o fluxo de fluido invasor e controlado através do perímetro do aglomerado, enquanto que na segunda modificação, o crescimento e governado pela relação de escala entre a massa e o raio de giração dos aglomerados. Estudamos cuidadosamente tanto o perfil de aceitação quanto as dimensões fractais (\'D IND.F\') dos aglomerados assim crescidos. No modelo baseado nas relações de escala, \'D IND.F\' pode ser tratado como um mero parâmetro real que pode assumir qualquer valor no intervalo (0, ?). Nos intervalos (0, \'91 SOB.48\') e (2, ?), o sistema e frustrado. Para \'D IND.F\' > 2, o modelo exibe um fenômeno interessante: em algumas etapas ocorrem explosões no crescimento da massa dos aglomerados (bursts). Na região [\'91 SOB.48\',2], os aglomerados obedecem exatamente e em qualquer escala a relação M ~ RgDF entre a massa m e o raio de giração \'RG\'. Acreditamos que estes fractais cuja estrutura completamente e estabilizada possam ser muito úteis no tratamento de problemas de diluição da mecânica estatística. / We generalize the standard site invasion percolation model to permit simultaneous invasion of several sites. We propose two kinds of generalizations: one in which the invasion flux is controlled by the perimeter size and another where the scaling properties command the growth process. The acceptance profile as well as the fractal dimension \'D IND.F\' are carefully studied. In the model based on scaling relation, \'D IND.F\' can be treated as a mere real parameter in the range (0, ?). In the intervals (0, \'91 SOB.48\') and (2, ?) the system is frustrated. For \'D IND.F\' > 2 the model exhibits also an interesting burst phenomenon that is explained in the text. In the region [\'91 SOB.48\',2], the clusters obey exactly and in any scale the relation M ~ RgDF between the mass M and gyration radius Rg. These stable random fractals may be very useful in the study of dilute systems.

An integrated field , microstructure, fracture statistic , geochemistry , and laboratory permeability study of the East Fork and White Rock fault zones , of similar age and tectonic regime but different structural level and hydrogeologic history , provides detailed information about the internal deformation and fluid flow processes in fault zones . The primary conclusions of this research are: 1) Fault zones can be separated into subzones of protolith, damaged zone , and gouge /cataclasite , based on physical morphology and permeability structure . At deep structural levels, gouge/cataclasite zones are more evolved (thicker with increased grain size reduction) due to strain localization , higher pressure and temperature, and fluid/rock interaction ; 2) Deformation mechanisms evolved from primarily brittle fracturing and faulting in the damaged zone to extreme, fluid-enhanced chemical breakdown and cataclasis which localized strain in the fault core. Deformation in the deep-level-fault core may be a combination of frictional and quasiplastic mechanisms, and is largely controlled by extremely fine-grained clays, zeolites , and other phyllosilicates that may have acted as a thermally pressurized, fluid-saturated lubricant; 3) Permeability in fault zones was temporally heterogeneous and anisotropic (permeability of damaged zone>protolith>gouge /cataclasite, permeability along fault> permeability across fault); 4) Volume loss was concentrated in the fault cores and was negligible at intermediate structural levels and high at deep structural levels in the semi-brittle to brittle regime ; 5) Fluid flow and solute transport were concentrated upwards and subparallel to the fault in the damaged zone ; 6) Faults at both the local and regional scale acted as fluid flow conduit/barrier systems depending upon the evolutionary stage and interval in the seismic cycle ; 7) Fluid/rock volume ratios , fluid flux , and fluid/rock volume ratios over time ranged from ⋍ 103 to 104, 10-6 ms-1 to 10-9 ms-1, and 0.05 L/m3 rock•yr to 0.50 L/m3 rock•yr, respectively, suggesting that enormous quantities of fluids passed through the fault zones; 8) Box counting fractal analyses of fault zone fractures showed that fracture spatial and density distribution is scale-invariant at the separate scales of outcrop , hand-sample , and thin section, but self-affine from outcrop to thin-section scale; 9) Linear fractal analysis depicts clustering and density distribution as a function of orientation, and may be a quick, robust method of estimating two-dimensional fracture permeability; and 10) Fractal analysis of fractures is not a comprehensive statistical method, but can be used as another supplemental statistical parameter.

In order to understand water quality in the coastal ocean and its effects on human health, the necessity arises to locate the sources of contaminants and track their transport throughout the ocean. Dynamical systems methods are applied to the study of transport of enterococci as an indicator of microbial concentration in the vicinity of Hobie Beach, an urban, subtropical beach in Miami, FL that is used for recreation and bathing on a daily basis. Previous studies on water quality have shown that Hobie Beach has high microbial levels despite having no known point source. To investigate the cause of these high microbial levels, a combination of measured surface drifter trajectories and numerically simulated flows in the vicinity of Hobie Beach is used. The numerically simulated flows are used to identify Lagrangian Coherent Structures (LCSs), which provide a template for transport in the study area. Surface drifter trajectories are shown to be consistent with the simulated flows and the LCS structure. LCSs are then used to explain the persistent water contamination and unusually high concentrations of microbes in the water off of this beach as compared with its neighboring beaches. From the drifter simulations, as well as field experiments, one can see that passive tracers are trapped in the area along the coastline by LCS. The Lagrangian circulation of Hobie Beach, influenced primarily by tide and land geometry causes a high retention rate of water near the shore, and can be used to explain the elevated levels of enterococci in the water.

Osteocytes are believed to be the mechanosensory cells that detect and respond to mechanical loading. Physiological loading by oscillatory fluid flow (OFF) activates osteocytes to increase intracellular calcium concentration and release prostaglandin E2 (PGE2). Osteocytes are also sensitive to chemical stimulations such as serotonin, which can also increase PGE2 release. However, it is unclear whether mechanical stimulation can influence osteocyte sensitivity towards serotonin. In this thesis, MLO-Y4 osteocyte-like cells were subjected to serotonin with or without precondition by OFF, the responses of intracellular calcium and PGE2 release were measured. Serotonin increased intracellular calcium and PGE2 release in osteocytes. The effects were significantly reduced by OFF precondition, suggesting mechanical precondition by OFF can reduce osteocyte sensitivity towards serotonin. In terms of mechanisms, OFF- and serotonin-induced calcium responses depended on intra- and extracellular calcium stores. ATP was found to partially mediate OFF modulation of serotonin-induced PGE2 release but not calcium.

Osteocytes are believed to be the mechanosensory cells that detect and respond to mechanical loading. Physiological loading by oscillatory fluid flow (OFF) activates osteocytes to increase intracellular calcium concentration and release prostaglandin E2 (PGE2). Osteocytes are also sensitive to chemical stimulations such as serotonin, which can also increase PGE2 release. However, it is unclear whether mechanical stimulation can influence osteocyte sensitivity towards serotonin. In this thesis, MLO-Y4 osteocyte-like cells were subjected to serotonin with or without precondition by OFF, the responses of intracellular calcium and PGE2 release were measured. Serotonin increased intracellular calcium and PGE2 release in osteocytes. The effects were significantly reduced by OFF precondition, suggesting mechanical precondition by OFF can reduce osteocyte sensitivity towards serotonin. In terms of mechanisms, OFF- and serotonin-induced calcium responses depended on intra- and extracellular calcium stores. ATP was found to partially mediate OFF modulation of serotonin-induced PGE2 release but not calcium.