High-quality silicon crystals provide the basis of many industrial technological advances, including computers and telecommunication devices. The increasing size and extremely high quality requirements of silicon wafers have made furnace design and crystal manufacturing a very challenging task. Numerical simulations have become an essential and powerful tool to overcome the difficulties of the experimental approach with a view to understanding the crystal growth process but also to finding an appropriate path to optimize the crystal pulling conditions in industry.</br></br> This thesis deals with the use of alternating and steady transverse magnetic fields in silicon growth from the melt. The use of magnetic fields represents a powerful tool to damp out turbulence and control the melt flow. This technique can also be used to heat the system. We focus on the numerical modeling of (i) induction heating in the Floating Zone process and of (ii) melt convection under the effect of transverse magnetic fields in the Czochralski process. For each of these topics, our work is subdivided in two parts : firstly mathematical modeling, based on asymptotic or Fourier expansions, and secondly numerical implementation and simulation of the considered processes. </br></br> First, a theoretical and numerical model of the alternating magnetic field distribution (as generated by induction heating) has been developed by means of an asymptotic expansion technique. Moreover, a new methodology has been developed to calculate the thermal and mechanical effects of alternating magnetic fields on the liquid conductor flow, leading to accurate expressions for the equivalent magnetic heat flux and surface stresses in the 2D and 3D cases. Second, investigation of the effect of a transverse magnetic field on the melt flow in semi-conductor crystal growth has been performed by the simplified FLET method (“Fourier Limited Expansion Technique”.)

We study the capture of a particle into resonance at a potential hole with dissipative perturbation and periodic outside force. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance.

In recent years, N-hetereocyclic carbenes (NHCs) have received considerable attention as organocatalysts due to their unusual ability to induce a reversal of reactivity (Umpolung) in aldehydes. Indeed, NHCs unique properties have been applied to the efficient and metal-free synthesis of organic compounds that have proven elusive using traditional approaches.<p> My Masters research program has been focused on the use of NHCs as organocatalysts in ring expansion reactions and their applications to cascade reactions.<p> During my Masters studies, an NHC-catalyzed efficient ring expansion of 4-, 5-, and 6-membered oxacycloalkane-2-carboxaldehydes to generate the corresponding lactone derivatives was developed. This reaction provides access to a variety of lactones using readily available NHCs under mild conditions.<p> Then, the ring-expansion lactonization has been successfully extended to an efficient lactamization using azacycloalkane-carboxaldehydes, which could provide functionalized lactams in moderate yields under mild conditions.<p> In addition, intrigued by the possibility of effecting the Umpolung of electron-poor dienes using NHC catalysts, the ring-expansion lactonization was applied to an attempted Diels-Alder-ring expansion cascade reaction. Though no cascade reactions were observed, some very interesting results were obtained, and those results will guide future investigations in this area.

In recent years, N-hetereocyclic carbenes (NHCs) have received considerable attention as organocatalysts due to their unusual ability to induce a reversal of reactivity (Umpolung) in aldehydes. Indeed, NHCs unique properties have been applied to the efficient and metal-free synthesis of organic compounds that have proven elusive using traditional approaches.<p> My Masters research program has been focused on the use of NHCs as organocatalysts in ring expansion reactions and their applications to cascade reactions.<p> During my Masters studies, an NHC-catalyzed efficient ring expansion of 4-, 5-, and 6-membered oxacycloalkane-2-carboxaldehydes to generate the corresponding lactone derivatives was developed. This reaction provides access to a variety of lactones using readily available NHCs under mild conditions.<p> Then, the ring-expansion lactonization has been successfully extended to an efficient lactamization using azacycloalkane-carboxaldehydes, which could provide functionalized lactams in moderate yields under mild conditions.<p> In addition, intrigued by the possibility of effecting the Umpolung of electron-poor dienes using NHC catalysts, the ring-expansion lactonization was applied to an attempted Diels-Alder-ring expansion cascade reaction. Though no cascade reactions were observed, some very interesting results were obtained, and those results will guide future investigations in this area.

One of the central difficulties in many models of glacier and ice sheet flow lies in the prescription of boundary conditions at the bed. Often, processes which occur there dominate the evolution of the ice mass as they control the speed at which the ice is able to slide over the bed. In part I of this thesis, we study two complications to classical models of glacier and ice sheet sliding. First, we focus on the effect of cavity formation on the sliding of a glacier over an undeformable, impermeable bed. Our results do not support the widely used sliding law $u_b = C\tau_b^pN^{-q}$, but indicate that $\tau_b/N$ actually decreases with $u_b/N$ at high values of the latter, as suggested previously for simple periodic beds by Fowler (1986). The second problem studied is that of an ice stream whose motion is controlled by bed obstacles with wavelengths comparable to the thickness of ice. By contrast with classical sliding theory for ice of constant viscosity,the bulk flow velocity does not depend linearly on the driving stress. Indeed, the bulk flow velocity may even be a multi-valued function of driving stress and ice thickness. In the second part of the thesis, attention is turned to the formation of drumlins. The viscous till model of Hindmarsh (1998) and Fowler (2000) is analysed in some detail. It is shown that the model does not predict the formation of three-dimensional drumlins, but only that of two-dimensional features, which may be interpreted as Rogen moraines. A non-linear model allows the simulation of the predicted bedforms at finite amplitude. Results obtained indicate that the growth of bedforms invariably leads to cavitation. A model for travelling waves in the presence of cavitation is also developed, which shows that such travelling waves can indeed exist. Their shape is, however, unlike that of real bedforms, with a steep downstream face and no internal stratification. These results indicate that Hindmarsh and Fowler's model is probably not successful at describing the processes which lead to the formation of streamlined subglacial bedforms.

This thesis examines certain aspects of diffraction and scattering of high frequency waves, utilising and extending upon the Geometrical Theory of Diffraction (GTD). The first problem considered is that of scattering of electromagnetic plane waves by a perfectly conducting thin body, of aspect ratio O(k^1/2), where k is the dimensionless wavenumber. The edges of such a body have a radius of curvature which is comparable to the wavelength of the incident field, which lies inbetween the sharp and blunt cases traditionally treated by the GTD. The local problem of scattering by such an edge is that of a parabolic cylinder with the appropriate radius of curvature at the edge. The far field of the integral solution to this problem is examined using the method of steepest descents, extending the recent work of Tew [44]; in particular the behaviour of the field in the vicinity of the shadow boundaries is determined. These are fatter than those in the sharp or blunt cases, with a novel transition function. The second problem considered is that of scattering by thin shells of dielectric material. Under the assumption that the refractive index of the dielectric is large, approximate transition conditions for a layer of half a wavelength in thickness are formulated which account for the effects of curvature of the layer. Using these transition conditions the directivity of the fields scattered by a tightly curved tip region is determined, provided certain conditions are met by the tip curvature. In addition, creeping ray and whispering gallery modes outside such a curved layer are examined in the context of the GTD, and their initiation at a point of tangential incidence upon the layer is studied. The final problem considered concerns the scattering matrix of a closed convex body. A straightforward and explicit discussion of scattering theory is presented. Then the approximations of the GTD are used to find the first two terms in the asymptotic behaviour of the scattering phase, and the connection between the external scattering problem and the internal eigenvalue problem is discussed.

The classical theory of channelized subglacial drainage,due orginally to Röthlisberger (1972) and Nye (1976), considers water flow in an ice channel overlying a rigid, impermeable bed. At steady flow, creep closure of the channel walls is counteracted by melt-back due to heat dissipation, and this leads to an equilibrium relation between channel water pressure and discharge. More generally, such a balance exhibits an instability that can be used to describe the mechanics of catastrophic flood events known as `jökulhlaups'. In this thesis, we substantiate these developments by exploring a detailed model where the channel is underlain by subglacial till and the flow supports a sediment load. Attention is given to the physics of bed processes and its effect on channel morphology. In particular, we propose a theory in which the channel need not be semi-circular, but has independently evolving depth and width determined by a local balance between melting and closure, and in which sediment erosion and deposition is taken into account. The corresponding equilibrium relation indicates a reverse dependence to that in the classical model, justifying the possibility of the subglacial canals envisaged by Walder and Fowler (1994). Theoretical predictions for sediment discharge are also derived. Regarding time-dependent flood drainage, we demonstrate how rapid channel widening caused by bank erosion can explain the abrupt recession observed in the flood hydrographs. This allows us to produce an improved simulation of the 1972 jökulhlaup from Grímsvötn, Iceland, and self-consistently, a plausible estimate for the total sediment yield. We also propose a mechanism for the observed flood initiation lake-level at Grímsvötn. These investigations expose the intimate interactions between drainage and sediment transport, which have profound implications on the hydrology, sedimentology and dynamics of ice masses, but which have received little attention.

This thesis is motivated by the need to calculate the electromagnetic fields produced by sources radiating in the presence of conductors. We begin by reviewing existing theory concerning sources in the presence of flat structures. Various extensions to the canonical Sommerfeld problem are considered. In particular we investigate the asymptotic solution for a finite source that focusses its energy at a point. In chapter 5 we review and extend the asymptotic results concerning illumination of a convex perfect conductor by an incident plane wave and outline the procedure for decoupling the electromagnetic surface field into two scalar modes. In chapter 6 we place a source on a perfect conductor and obtain a complete asymptotic solution for the fields. Special attention is paid to the asymptotic structure that smoothly matches between the leading order lit and shadow regions. We also investigate the degenerate case where one of the curvatures of the perfect conductor is zero. The case where the source is just off the surface is also investigated. In chapter 8 we use the Euler-Maclaurin summation formula to cheaply calculate the fields due to complicated arrays of point dipoles. The final chapter combines many earlier results to consider more general sources on the surface of a perfect conductor. In particular we must introduce new asymptotic regions for open sources. This then enables us to consider the focussing of the surface field due to a finite source. The nature of the surface and geometrical optics fields depends on the size of the source in comparison to the curvatures of the surface on which they lie. We discuss this in detail and conclude with the practical example of a spiral antenna.

Frost heave describes the phenomenon whereby soil freezing causes upwards surface motion due to the action of capillary suction imbibing water from the unfrozen region below. The expansion of water on freezing is a small part of the overall surface heave and it is the flow of water towards the freezing front which is largely responsible for the uplift. In this thesis, we analyse a model of frost heave due to Miller (1972, 1978) which is referred to as `secondary frost heave'. Secondary frost heave is characterised by the existence of a `partially frozen zone', underlying the frozen soil, in which ice and water coexist in the pore space. In the first part of the thesis we follow earlier work of Fowler, Krantz and Noon where we show that the Miller model for incompressible soils can be dramatically simplified. The second part of the thesis then uses this simplification procedure to develop simplified models for saline and compressible soils. In the latter case, the development of the theory leads to the consideration of non-equilibrium soil consolidation theory and the formation of segregated massive ice within permafrost. The final part of the thesis extends the simplified Miller model to the analysis of differential frost heave and the formation of patterned ground (e.g. earth hummocks and stone circles). We show that an instability mechanism exists which provides a plausible theory for the formation of these types of patterned ground.

Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains viii, 140 p.; also includes graphics. Includes bibliographical references (p. 139-140).