In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

Dynamic response of materials to high-speed and high-intensity loading conditions is important in several applications including high-speed flows with droplets, bubbles and particles, and hyper-velocity impact and penetration processes. In such high-pressure physics problems, simulations encounter challenges associated with the treatment of material interfaces, particularly when strong nonlinear waves like shock and detonation waves impinge upon them. To simulate such complicated interfacial dynamics problems, a fixed Cartesian grid approach in conjunction with levelset interface tracking is attractive. In this regard, a sharp interface Cartesian grid-based, Ghost Fluid Method (GFM) is developed for resolving embedded fluid, elasto-plastic solid and rigid (solid) objects in hyper-velocity impact and high-intensity shock loaded environment. The embedded boundaries are tracked and represented by virtue of the level set interface tracking technique. The evolving multi-material interface and the flow are coupled by meticulously enforcing the boundary conditions and jump relations at the interface. In addition, a tree-based Local Mesh Refinement scheme is employed to efficiently resolve the desired physics. The framework developed is generic and is applicable to interfaces separating a wide range of materials and for a broad spectrum of speeds of interaction (O(km/s)). The wide repertoire of problems solved in this work demonstrates the flexibility, stability and robustness of the method in accurately capturing the dynamics of the embedded interface. Shocks interacting with large ensembles of particles are also computed.

In this thesis, the seminal work of Stokes concerning the ow of a Navier-Stokesuid due to a suddenly accelerated or oscillating plate and the ow due to torsionaloscillations of an innitely long rod in a Navier-Stokes uid is extended to a uid withpressure dependent viscosity. The viscosity of many uids varies signicantly withpressure, a fact recognized by Stokes; and Barus, in fact, conducted experiments thatshowed that the variation of the viscosity with pressure was exponential. Given sucha tremendous variation, we study how this change in viscosity aects the nature of thesolution to these problems. We nd that the velocity eld, and hence the structureof the vorticity and the shear stress at the walls for uids with pressure dependentviscosity, are markedly dierent from those for the classical Navier-Stokes uid.

A prerequisite for aeroelastic stability prediction inturbomachines is the understanding of the fluctuatingaerodynamic forces acting on the blades. Unsteady transonicflows are complex because of mutual interactions betweentravelling pressure waves, outlet disturbances, shock motion,and fluctuating turbulent boundary layers. Complex phenomenaappear in the shock/boundary layer region and produce phaselags and high time harmonics, which can give a significantcontribution to the overall unsteady lift and torque, andtherefore affect flutter boundaries, cause large localstresses, or even severely damage the turbomachine. The present research work is concerned with theunderstanding of phenomena associated with travelling waves innon-uniform transonic flows and how they affect the unsteadypressure distribution on the surface as well as the far fieldradiated sound. In similitude with turbomachines potentialinteraction, the emphasis was put on the unsteady interactionof upstream propagating acoustic waves with an oscillatingshock in 2D and 3D nozzle flows. Both numerical andexperimental studies are carried out and compared with eachother. Results shows that the unsteady pressure distribution, bothon the bump surface and within the channel, results from thesuperposition of upstream and downstream propagating waves. Itis believed that outlet pressure perturbations propagateupstream in the nozzle, interact in the high subsonic flowregion according to the acoustic blockage theory, and arepartly reflected or absorbed by the oscillating shock,depending on the frequency of the perturbations and theintensity of the SBLI. Furthermore the shock motion amplitudeis found to be related to the mean flow gradients and localwave length of the perturbations rather than to the shockboundary layer interaction. The phase angle between incomingpressure perturbations and the shock motion increases with theperturbation frequency but also depends on the intensity of theSBLI. Additionally the phase angle "shift" observed underneaththe shock location linearly increases with the perturbationfrequency and the shock strength. Such phase shift is criticalregarding aeroelastic stability and might have a significantimpact on the phase angle of the overall aerodynamic forceacting on the blade and shift the aerodynamic damping fromstable to exciting. <b>Keywords:</b>Shock Boundary Layer Interaction, ShockMotion, Unsteady Flows, Nozzle Flows, Potential Interaction,Back Pressure Perturbations.

Aerothermodynamics is an integral component in the design and implementation of hypersonic transport systems. Accurate estimates of the aerodynamic forces and heat transfer rates are critical in trajectory analysis and for payload weight considerations. The present work seeks to investigate the ability of an unstructured Cartesian grid framework in modeling hypersonic viscous flows. The effectiveness of modeling viscous phenomena in hypersonic flows using the immersed boundary ghost cell methodology of this solver is analyzed. The capacity of this framework to predict the surface physics in a hypersonic non-reacting environment is investigated. High velocity argon gas flows past a 2-D cylinder are simulated for a set of freestream conditions (Reynolds numbers), and impact of the grid cell sizes on the quality of the solution is evaluated. Additionally, the formulation is verified over a series of hypersonic Mach numbers for the flow past a hemisphere, and compared to experimental results and empirical estimates. Next, a test case that involves flow separation and the interaction between a hypersonic shock wave and a boundary layer, and a separation bubble is investigated using various adaptive mesh refinement strategies. The immersed boundary ghost cell approach is tested with two temperature clipping strategies, and their impact on the overall solution accuracy and smoothness of the surface property predictions are compared. Finally, species diffusion terms in the conservation equations, and collision cross-section based transport coefficients are installed, and hypersonic flows in thermochemical nonequilibrium environments are studied, and comparisons of the off-surface flow properties and the surface physics predictions are evaluated. First, a 2-D cylinder in a hypersonic reacting air flow is tested with an adiabatic wall boundary condition. Next, the same geometry is tested to evaluate the viscous chemistry prediction capability of the solver with an isothermal wall boundary condition, and to identify the strengths and weaknesses of the immersed boundary ghost cell methodology in computing convective heating rates in such an environment.

La descripción del flujo de fluidos involucra la solución de las ecuaciones de Navier-Stokes compresible, un problema muy complejo cuya estructura matemática no es del todo comprendida. Por lo tanto, mediante análisis asintótico, se pueden derivar modelos simplificados bajo ciertas hipótesis sobre el problema hechas en términos de parámetros adimensionales que miden la importancia relativa de los diferentes procesos físicos. Los flujos a baja velocidad se pueden describir por diferentes modelos que incluyen las ecuaciones de Navier Stokes incompresible cuya matemática es mucho mas conocida. Sin embargo, algunos flujos importantes no se pueden considerar incompresibles debido a la presencia de efectos térmicos. En esta clase de problemas se pueden derivar otra clase de ecuaciones simplificadas: las ecuaciones de Boussinesq y las ecuaciones de bajo numero de Mach.La complejidad de estos problemas matemáticos hace que su solución numérica sea muy difícil. En estos problemas el método de los elementos finitos es inestable, lo que en la práctica implica soluciones numéricas que presentan oscilaciones nodo a nodo de naturaleza no física. En las ecuaciones de Navier Stokes incompresible, dos fuentes bien conocidas de inestabilidad son la condición de incompresibilidad y la presencia del término convectivo. Muchas técnicas de estabilización utilizadas hoy en día se basan en la separación de escalas, descomponiendo la incógnita en una parte gruesa inducida por la discretización del domino y una parte fina de subescala. Modelar la subescala y su influencia conduce a un problema modificado para la escala gruesa que resulta estable.Aunque las técnicas de estabilización son ampliamente utilizadas hoy en día, importantes problemas permanecen abiertos. Contribuyendo a su comprensión, en este trabajo se analizan varios aspectos del modelado de las subescalas. Para problemas escalares de segundo orden, se encuentra la dependencia de la subescala con el tamaño de la malla en el caso general de mallas anisótropas. Estas ideas son extendidas a sistemas de ecuaciones para considerar el problema de Oseen. También se analiza el modelado de las subescalas en problemas transitorios, obteniendo un mejor esquema de integración temporal para el problema de escala gruesa. Para considerar flujos a baja velocidad, se presenta la extensión de estas técnicas a problemas no lineales acoplados, lo que esta íntimamente relacionado con el problema del modelado de la turbulencia, que es un tema en si mismo.Los flujos acoplados térmicamente, aparte del interés intrínseco que merecen, son importantes desde un punto de vista ingenieril. Una solución precisa del problema de flujo es necesaria para definir las cargas térmicas sobre las estructuras, que en muchos casos responden fuertemente, haciendo el problema acoplado. Esta clase de problemas, que motivaron este trabajo, incluyen la respuesta estructural en el caso de un incendio. / A general description of a fluid flow involves the solution of the compressible Navier-Stokes equations, a very complex problem whose mathematical structure is not well understood. Therefore, simplified models can be derived by asymptotic analysis under some assumptions on the problem, made in terms of dimensionless parameters that measure the relative importance of different physical processes. Low speed flows can be described by several models including the incompressible Navier Stokes equations whose mathematical structure is much better understood. However many important flows cannot be considered as incompressible, even at low speed, due to the presence of thermal effects. In such kind of problems another class of simplified equations can be derived: the Boussinesq equations and the Low Mach number equations.The complexity of these mathematical problems makes their numerical solution very difficult. For these problems the standard finite element method is unstable, what in practice means that node to node oscillations of non physical nature may appear in the numerical solution. In the incompressible Navier Stokes equations, two well known sources of numerical instabilities are the incompressibility constraint and the presence of the convective terms. Many stabilization techniques used nowadays are based on scale separation, splitting the unknown into a coarse part induced by the discretization of the domain and a fine subgrid part. The modelling of the subgrid scale and its influence leads to a modified coarse scale problem that now can be shown to be stable. Although stabilization techniques are nowadays widely used, important problems remain open. Contributing to their understanding, several aspects of the subgrid scale modelling are analyzed in this work. For second order scalar problems, the dependence of the subgrid scale on the mesh size, in the general anisotropic case, is clarified. These ideas are extended to systems of equations to consider the Oseen problem. The modelling of the subgrid scales in transient problems is also analyzed, leading to an improved time discretization scheme for the coarse scale problem. To consider low speed flow models, the extension of these techniques to nonlinear and coupled problems is presented, something that is intimately related to the problem of turbulence modelling, which a entire subject on its own right. Thermally coupled flow problems, despite the intrinsic interest they deserve, are important from an engineering point of view. An accurate solution of a flow problem is needed to define thermal loads on structures which, in many cases have a strong response, making the problem coupled. This kind of problems, that motivated this work, include the problem of a structural response in the case of fires.

In this research, least-squares based finite element formulations and their applications in fluid mechanics are presented. Least-squares formulations offer several computational and theoretical advantages for Newtonian as well as non-Newtonian fluid flows. Most notably, these formulations circumvent the inf-sup condition of Ladyzhenskaya-Babuska- Brezzi (LBB) such that the choice of approximating space is not subject to any compatibility condition. Also, the resulting coefficient matrix is symmetric and positive-definite. It has been observed that pressure and velocities are not strongly coupled in traditional leastsquares based finite element formulations. Penalty based least-squares formulations that fix the pressure-velocity coupling problem are proposed, implemented in a computational scheme, and evaluated in this study. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. These penalty based formulations produce accurate results for even low penalty parameters (in the range of 10-50 penalty parameter). A stress based least-squares formulation is also being proposed to couple pressure and velocities. Stress components are introduced as independent variables to make the system first order. The continuity equation is eliminated from the system with suitable modifications. Least-squares formulations are also developed for viscoelastic flows and moving boundary flows. All the formulations developed in this study are tested using several benchmark problems. All of the finite element models developed in this study performed well in all cases. A method to exploit orthogonality of modal bases to avoid numerical integration and have a fast computation is also developed during this study. The entries of the coefficient matrix are calculated analytically. The properties of Jacobi polynomials are used and most of the entries of the coefficient matrix are recast so that they can be evaluated analytically.

In this thesis, the seminal work of Stokes concerning the ow of a Navier-Stokesuid due to a suddenly accelerated or oscillating plate and the ow due to torsionaloscillations of an innitely long rod in a Navier-Stokes uid is extended to a uid withpressure dependent viscosity. The viscosity of many uids varies signicantly withpressure, a fact recognized by Stokes; and Barus, in fact, conducted experiments thatshowed that the variation of the viscosity with pressure was exponential. Given sucha tremendous variation, we study how this change in viscosity aects the nature of thesolution to these problems. We nd that the velocity eld, and hence the structureof the vorticity and the shear stress at the walls for uids with pressure dependentviscosity, are markedly dierent from those for the classical Navier-Stokes uid.

The world we live in is increasingly integrated. For the work of economists, increasing international integration bears a significant importance. The present thesis is essentially a work on International Economics. As such it is no exception in that it consists of different chapters, all of which address different issues in the field. The first two chapters are theoretical in nature, whereas the third is empirical. The last chapter provides a technical reference to mathematical problems encountered in the first chapter. The first chapter is concerned with one of the negative effect of international trade: terms-of-trade uncertainty. It asks (and answers) the question why economic agents in practice fail to (completely) hedge away foreign price uncertainty in the presence of well-developed forward markets, even though theoretically they should obtain a full-hedge. The second chapter explores some of the effects international capital flows bring to a country that opens up its capital account. The third chapter describes the evolution of international capital flows and trade flows over that last decade. The last chapter is concerned with the existence of explicit demand schedules under expected utility maximization when the random variable is nonlinearly transformed. / Die Welt in der wir leben ist durch zunehmende Integration in fast allen Bereichen des Lebens gekennzeichnet. In der ökonomischen Sphäre wird diese zunehmende Integration auch oft mit dem Begriff Globalisierung beschrieben. Zwei Hauptmerkmale der Globalisierung sind dabei zunehmende internationale Kapital- und Handelsströme. Die vorliegende Dissertation beschäftigt sich mit ausgesuchten Teilaspekten dieser Phänomene. &amp;quot;Warum sichern sich die Wirtschaftssubjekte in der Praxis nur unvollständig gegenüber Wechselkursrisiken ab, obwohl sie theoretisch eine vollständige Absicherung wählen sollten?&amp;quot;, &amp;quot;Welchen Einfluß hat die Herkunft von Kapital auf die Ökonomie?&amp;quot; sowie &amp;quot;Wie ist der empirische Befund zur Entwicklung von Handels- und Kapitalströmen in der letzten Dekade?&amp;quot; sind Fragen denen die vorliegende Arbeit nachgeht. Ein Kapitel mit Ergebnissen zur Existenz von expliziten Nachfragefunktionen unter Erwartungsnutzenmaximierung bei zugrundeliegender nichtlinearer Transformation der Zufallsvariablen (eine Fragestellung die im Rahmen der Bearbeitung von Kapitel 1 auftaucht) beschliesst die Arbeit.

<p>A prerequisite for aeroelastic stability prediction inturbomachines is the understanding of the fluctuatingaerodynamic forces acting on the blades. Unsteady transonicflows are complex because of mutual interactions betweentravelling pressure waves, outlet disturbances, shock motion,and fluctuating turbulent boundary layers. Complex phenomenaappear in the shock/boundary layer region and produce phaselags and high time harmonics, which can give a significantcontribution to the overall unsteady lift and torque, andtherefore affect flutter boundaries, cause large localstresses, or even severely damage the turbomachine.</p><p>The present research work is concerned with theunderstanding of phenomena associated with travelling waves innon-uniform transonic flows and how they affect the unsteadypressure distribution on the surface as well as the far fieldradiated sound. In similitude with turbomachines potentialinteraction, the emphasis was put on the unsteady interactionof upstream propagating acoustic waves with an oscillatingshock in 2D and 3D nozzle flows. Both numerical andexperimental studies are carried out and compared with eachother.</p><p>Results shows that the unsteady pressure distribution, bothon the bump surface and within the channel, results from thesuperposition of upstream and downstream propagating waves. Itis believed that outlet pressure perturbations propagateupstream in the nozzle, interact in the high subsonic flowregion according to the acoustic blockage theory, and arepartly reflected or absorbed by the oscillating shock,depending on the frequency of the perturbations and theintensity of the SBLI. Furthermore the shock motion amplitudeis found to be related to the mean flow gradients and localwave length of the perturbations rather than to the shockboundary layer interaction. The phase angle between incomingpressure perturbations and the shock motion increases with theperturbation frequency but also depends on the intensity of theSBLI. Additionally the phase angle "shift" observed underneaththe shock location linearly increases with the perturbationfrequency and the shock strength. Such phase shift is criticalregarding aeroelastic stability and might have a significantimpact on the phase angle of the overall aerodynamic forceacting on the blade and shift the aerodynamic damping fromstable to exciting.</p><p><b>Keywords:</b>Shock Boundary Layer Interaction, ShockMotion, Unsteady Flows, Nozzle Flows, Potential Interaction,Back Pressure Perturbations.</p>