<p> The purpose of this thesis is to prove the existence of a unique solution to a system of partial differential equations which models the flow of a compressible barotropic fluid under periodic boundary conditions. The equations come from modifying the compressible Navier-Stokes equations. The proof utilizes the method of successive approximations. We will define an iteration scheme based on solving a linearized version of the equations. Then convergence of the sequence of approximate solutions to a unique solution of the nonlinear system will be proven. The main new result of this thesis is that the density data is at a given point in the spatial domain over a time interval instead of an initial density over the entire spatial domain. Further applications of the mathematical model are fluid flow problems where the data such as concentration of a solute or temperature of the fluid is known at a given point. Future research could use boundary conditions which are not periodic.</p><p>

<p> This work presents a novel polytopal finite-element framework that addresses the collective issues of discretization sensitivity and mesh generation for computational solid mechanics problems. The use of arbitrary polygonal and polyhedral shapes in place of canonical isoparametric elements seeks to remediate issues pertaining to meshing and mesh quality (particularly for irregularly shaped elements), while maintaining many of the desirable features of a traditional finite element method. </p><p> A general class of <i>partitioned element methods</i> (PEM) is proposed and analyzed, constituting a family of approaches for constructing piecewise polynomial approximations to harmonic shape functions on arbitrary polytopes. Such methods require a geometric partition of each element, and under certain conditions will directly yield integration consistency. Two partitioned element methods are explored in detail, including a novel approach herein referred to as the <i>discontinuous Galerkin partitioned-element method</i> (DG-PEM). An implementational framework for the DG-PEM is presented, along with a discussion of its associated numerical challenges. </p><p> The numerical precision of the PEM is explored via classical patch tests and single element tests for a representative sampling of polygonal element shapes. Solution sensitivity with respect to element shape is examined for a handful of problems, including a mesh convergence study in the nearly incompressible regime. Finally, the efficacy of the DG-PEM is assessed for a number of benchmark problems involving large deformations and nonlinear material behavior.</p><p>

<p> Suspensions and dispersions are an important class of complex fluids frequently encountered in a variety of industrial processes and are prominent in many consumer products such as beauty creams and food dressing. The extensive use of suspensions can be partly attributed to their unique rheological properties such as shear-induced normal stresses, yield stress, time-dependent viscosity and shear thinning. These rheological properties are a direct result of the interplay between the suspension microstructure and flow and have consequences for material processing. The quantitative understanding of suspension rheology so far has been dominated by empirical models. However, such models are either very specialized to particular flows, involve numerous/unphysical parameters, or are inadequate to describe rheological phenomena such as normal stresses. Alternatively, microscopic approaches have primarily been successful in addressing idealized cases and/or small length/time scales. Therefore, the goal of this thesis is to develop new and improved classes of continuum models that clearly connect the suspension microstructure under flow to the observed macroscopic rheology. </p><p> In this thesis, new, generally multiscale methods are applied towards developing robust constitutive models for suspension rheology. Two primary modeling approaches are employed to advance the modeling of suspension rheology in this thesis. First is a bottom-up approach that starts from a microscopic description of the suspension microstructure (e.g., the evolving aggregate size distribution) that is then coupled to an empirical/phenomenological equation to allow for the evaluation of the shear stress. The shortcoming of using a phenomenological stress expression is counterbalanced by the accurate microstructure picture provided by a microscopic framework. The second technique is a top-down approach that starts from a macroscopic description of the system through the use of state variables whose dynamic equations are developed within the Hamiltonian-enhanced Non-Equilibrium Thermodynamics framework. The key benefit of this latter approach is that the expressions for the stress tensor and microstructure, with the latter represented by a second rank tensor, emerge self-consistently from the framework. Moreover, the generated equations are applicable to general flows. The multiscale nature of suspension microstructure implies that depending on the phenomena of interest, one or the other or a combination of the two approaches may be favored. Regardless of the approach taken, a recurrent theme in this work is the clear association of the observed macroscopic rheological behavior with an underlying microscopic picture. Finally, for all the suspensions emphasized in this thesis i.e., thixotropic, polydisperse, noncolloidal and emulsions, the corresponding rheological models developed are validated against experimental/simulation data revealing their predictive capability. </p><p> A number of important specific accomplishments are achieved in this thesis. To begin with, a population balance-based constitutive model for thixotropic suspensions is developed. Unlike alternative phenomenological models currently in use, a population balance-based model incorporates parameters with clear physical meaning. As a result, the resultant model holds promise for inverse design of thixotropic materials such as pastes that are used in many industrial processes. Next, the use of a conformation tensor as an internal variable to represent changes in suspension microstructure during material deformation is also demonstrated. For the first time, a comprehensive conformation tensor-based framework for suspensions, with a rigor approaching that performed previously for polymeric system, is realized. When applied to dilute emulsions, the conformation tensor-based rheological model that results is in exact agreement with existing asymptotic microscopic theory. In the same emulsion system, effects of microinertia and Ostwald ripening have also been included within a conformation tensor-based model for the first time. In concentrated suspensions, the conformation based theory has been shown to be capable of describing emerging secondary structure in the particle configuration leading to first and second normal stress differences that are both negative. Additional advances have also been made to develop self-consistent approximations for polydisperse suspension viscosity and testing them against prototype experiments. On a broader level, this work provides a number of methodologies for systematic constitutive model development in complex fluids. From an engineering perspective, the results of this thesis can be used to improve upon existing numerical tools, e.g., computational fluid dynamics, to allow for accurate simulation of industrial processes such as extrusion and screen printing of thixotropic pastes, suspensions and emulsions. </p><p>

<p> Oceanic motion outside thin boundary layers is primarily a mixture of quasi-geostrophic flow and internal waves with either near-inertial frequencies or the frequency of the semidiurnal lunar tide. This dissertation seeks a deeper understanding of waves and flow through reduced models that isolate their nonlinear and coupled evolution from the Boussinesq equations. Three physical-space models are developed: an equation that describes quasi-geostrophic evolution in an arbitrary and prescribed field of hydrostatic internal waves; a three-component model that couples quasi-geostrophic flow to both near-inertial waves and the near-inertial second harmonic; and a model for the slow evolution of hydrostatic internal tides in quasi-geostrophic flow of near-arbitrary scale. This slow internal tide equation opens the path to a coupled model for the energetic interaction of quasi-geostrophic flow and oceanic internal tides. </p><p> Four results emerge. First, the wave-averaged quasi-geostrophic equation reveals that finite-amplitude waves give rise to a mean flow that advects quasi-geostrophic potential vorticity. Second is the definition of a new material invariant: Available Potential Vorticity, or APV. APV isolates the part of Ertel potential vorticity available for balanced-flow evolution in Eulerian frames and proves necessary in the separating waves and quasi-geostrophic flow. The third result, hashed out for near-inertial waves and quasi-geostrophic flow, is that wave-flow interaction leads to energy exchange even under conditions of weak nonlinearity. For storm-forced oceanic near-inertial waves the interaction often energizes waves at the expense of flow. We call this extraction of balanced quasi-geostrophic energy 'stimulated generation' since it requires externally-forced rather than spontaneously-generated waves. The fourth result is that quasi-geostrophic flow can encourage or 'catalyze' a nonlinear interaction between a near-inertial wave field and its second harmonic that transfers energy to the small near-inertial vertical scales of wave breaking and mixing. </p>

This thesis describes the development of a numerical modelling strategy for simulating the flow around a shrouded spiral bevel gear. The strategy is then applied to a series of parametric variations of key shroud parameters. The shroud and gear in question are generic, although based upon those employed in the internal gear box of a Rolls-Royce aeroengine. The need to shroud the gear comes from the fact that a spiral bevel gear, when rotated, acts like a fan. Work is done by the gear to move the surrounding fluid, usually air with oil particles suspended in it, which creates a parasitic loss, referred to as the windage power loss. The work within this thesis is part of a larger project which has investigated how windage power loss can be affected by geometric features of gears and shrouds. This is important as for large diameter (>200mm) bevel gears running at high speeds (>10,000 RPM) the windage power loss forms a substantial part of the total power loss. The modelling strategy has been developed in this work by studying 4 different fluid flow settings: Taylor-Couette flow, Conical Taylor-Couette flow, an unshrouded spiral bevel gear, and a shrouded spiral bevel gear. Work on Taylor-Couette flow provided a basic setting in which to trial various numerical techniques and gain familiarity with the commercial CFD program which would be used throughout this thesis (FLUENT), along with the meshing program GAMBIT. It gave an understanding of the flow, which was then used to simulate the flow in a modification of Taylor-Couette flow where the cylinders are replaced with cones, called Conical Taylor-Couette flow. Comparisons were made between 4 popular turbulence models, allowing a decision to be made on the `best' turbulence model to use in the modelling of a shrouded gear, and to start to develop the strategy. This strategy was then applied to the more complex geometry of an unshrouded gear, simulating experimental data which had been created on an in-house rig. To confirm the applicability of the strategy to modelling shrouded spiral bevel gears, it was applied to two shrouds for which experimental data was available. It showed that numerical modelling can capture the relative performance of the shrouds well. The work then continued by considering a series of parametric variations, whereby 3 key shroud parameters are each varied in 3 manners, producing 27 variations. Each of these parameters can affect the windage power loss: an assessment of how much each parameter affects windage power loss has been given. A description of the flow field in `good' and `bad' cases has been given, and through approximating the flow by using the compressible form of Bernoulli's equation, reasons for a `bad' shroud being `bad' have been presented.

The implementation of the δ₀₊ᵣ model in a finite element program is discussed. The idea of considering damage as a structural performance helps to avoid singularity. Strategies in drift correction is considered. The generalized time finite element method (GTFEM) is also discussed and implemented. It shows improved accuracy and stability with highly non-linear material properties.

Mathematical models for rotary lip seals is developed to study the basic stability and dynamic response of the lip seals. In this thesis, a linear stability analysis is performed to find the stability characteristics of such a mechanical system. Then, a finite element algorithm for the numerical integration of the system of coupled differential equations governing the dynamic behavior of the lip seals is presented.

The low areal weight requirements of telescopes in aerospace applications has driven the study on composite mirrors for several years. For example, the primary parabolic mirror in a balloon-borne, Cassegrain telescope required an optical quality better than 30 microns in figure RMS error. A parametric study on composite sandwich mirrors was conducted by using finite element analysis as well as optical analysis. The factors covered the cell sizes, core materials, core thicknesses, face layups, and support configurations. Based on theoretical calculations, many high quality spherical composite sandwich mirrors were generated by using a non-heat curing process. The CFRP faces and Nomex core were chosen as the baseline materials for mirror fabrication due to their high strength and low weight. The proposed replication method applied an interface layer between face and surface coat to eliminate print-through problems. Many important goals have been realized in those mirror samples with optical laser interferometer testing. These include the figure RMS error less than 2 microns and the surface RMS error less than 0.05 micron. The areal weights of the mirror samples are less than 7 kg/m². The thermal stability of these mirrors was observed from the optical results with thermal cycling tests. The proposed 2-meter parabolic composite sandwich mirror, with an areal weight of less than 10 kg/m², would consist of either [0/90/45/-45](s) layup faces with an optimal 3'' core or (QC) layup faces with a total core thickness of 5 inches. Both a ring support around the equator and the 18-point Hindle-type support would lead to the best optical quality under both self weight and thermal loading.

The study of a thin, incompressible Newtonian fluid layer trapped between two almost parallel, sliding surfaces has been actively pursued in the last decades. This subject includes lubrication applications such as slider bearings or the sealing of non-pressurized fluids with rotary lip seals. When a viscous lubricant flows between an elastic body and a rigid surface, the contact geometry may undergo substantial deformation affecting the flow field of the lubricant. Therefore, a coupled model between an elastic ring and the fluid film underneath it is proposed. Initially, a linear stability analysis is performed. Then, non-linear calculations are presented showing that the system deformations are able to induce mixing of lubricant throughout the sealed region. In the second part of this work, the flow of lubricant fluid through the micro-gap of rotary lip seals is analyzed theoretically and numerically from a different perspective. The study is carried out assuming that a 'small-gap' parameter δ attains an extreme value in the Navier-Stokes equations. The precise meaning of small-gap is achieved by the limit δ = 0, and the numerical solution of the resulting set of equations predicts transport of lubricant through the contact region due to centrifugal instabilities. Numerical results obtained with the finite element method are presented. In particular, the influence of inflow and outflow boundary conditions, and their importance in the simulated flow are discussed. To this aim, the penalty method for incompressible flows in presence of variable body forces is re-examined with the help of well-known examples, yielding a corrected formulation that is more accurate and faster than standard finite element methods found in the literature.

The inhomogeneities situated in materials render the diversification of composite families, and can provide synergistic effects for tailoring materials to a specified and often hostile environment. The work presented here focuses on the fracture and stress analysis of systems with various inhomogeneities. In Chapter 1, interactions among cracks and rigid-line inclusions are investigated. Rigid-line inclusions are represented by a distribution of forces while cracks are modeled by the standard dislocation approach. Chapter 2 presents an analysis of composite systems with interacting cracks and a dilute distribution of inclusions. A damage analysis procedure is developed to evaluate the effective properties of such composites. Chapter 3 examines multiple void-crack interactions. The formulation is based on a mixture of dislocations and tractions. Chapter 4 presents an approach to modeling bridged crack systems. A fully regular integral equation formulation is developed and the approach is ideally suited for the analysis of systems with large number of closely spaced inhomogeneities. The integral equations of different forms, developed throughout the dissertation can also be utilized to evaluate and verify various micromechanical models. The possible future extensions and the major limitations of the present work are briefly discussed in Chapter 5.