No description available.

The effect of structuring, in the form of magnetic or density inhomogeneities, on the magnetohydrodynamic (mhd) waves of an infinite plasma is investigated. The appropriate dispersion formulae, in both Cartesian and cylindrical polar coordinate geometries, are derived. The main properties of the allowable modes in structured plasmas are described, particularly those featuring in a slender inhomogeneity. The inclusion of non-adiabatic effects is examined, specifically for a thermally dissipative, unstratified, finite structure and for a slender inhomogeneity in a stratified medium. The dissipative time scales of slender structures are shown to have a dependence on the Peclet number. Growth factors appropriate to these time scales for the overstable motions of a thermally dissipative, Boussinesq fluid are derived. For the linear analysis of a slender structure it is shown that the dispersive nature of the waves is deducible from the simplified one-dimensional equations. The analysis is extended, for slender structures, to nonlinear motions and the governing equation representing an effective balance between nonlinear, dispersive and dissipative effects, the Benjamin-Ono-Burgers equation, is established. The solutions of this equation are considered and, for weakly-dissipative systems, are shown to be slowly decaying solitons. The importance, in the context of group velocity, of the dispersive nature of waves in ducted structures is discussed and analogies are made with other ducted waves, for example, the Love waves of seismology. It is suggested that the behaviour of such waves, following an impulse, may account for the range of oscillatory behaviour, the quasi-periodic and short time scales, observed in both the solar corona and Earth's magnetosphere. Density variations across a structure and the structure's curvature, with possible applications to coronal loops, are also considered. Further suggestions for possibly identifying some of the theoretical results with observed behaviour in sunspots, chromospheric fibrils and spicules are also made.

A numerical model has been developed which simulates the three-dimensional stability and transition of a periodically forced free shear layer in an incompressible fluid. Unlike previous simulations of temporally evolving shear layers, the current simulations examine spatial stability. The spatial model accommodates features of free shear flow, observed in experiments, which in the temporal model are precluded by the assumption of streamwise periodicity; e.g., divergence of the mean flow and wave dispersion. The Navier-Stokes equations in vorticity-velocity form are integrated using a combination of numerical methods tailored to the physical problem. A spectral method is adopted in the spanwise dimension in which the flow variables, assumed to be periodic, are approximated by finite Fourier series. In complex Fourier space, the governing equations are spatially two-dimensional. Standard central finite differences are exploited in the remaining two spatial dimensions. For computational efficiency, time evolution is accomplished by a combination of implicit and explicit methods. Linear diffusion terms are advanced by an Alternating Direction Implicit/Crank-Nicolson scheme whereas the Adams-Bashforth method is applied to convection terms. Nonlinear terms are evaluated at each new time level by the pseudospectral (collocation) method. Solutions to the velocity equations, which are elliptic, are obtained iteratively by approximate factorization. The spatial model requires that inflow-outflow boundary conditions be prescribed. Inflow conditions are derived from a similarity solution for the mean inflow profile onto which periodic forcing is superimposed. Forcing functions are derived from inviscid linear stability theory. A numerical test case is selected which closely parallels a well-known physical experiment. Many of the aspects of forced shear layer behavior observed in the physical experiment are captured by the spatial simulation. These include initial linear growth of the fundamental, vorticity roll-up, fundamental saturation, eventual domination of the subharmonic, vortex pairing, emergence of streamwise vorticity, and temporary stabilization of the secondary instability. Moreover, the spatial simulation predicts the experimentally observed superlinear growth of harmonics at rates 1.5 times that of the fundamental. Superlinear growth rates suggest nonlinear resonances between fundamental and harmonic modes which are not captured by temporal simulations.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Doctor of Philosophy, Johannesburg, 2013. / The (1+3) dimensional wave and Klein-Gordon equations are constructed using the covariant d'Alembertian operator on several spacetimes of interest. Equations on curved geometry inherit the nonlinearities of the geometry. These equations display interesting properties in a number of ways. In particular, the number of symmetries and therefore, the conservation laws reduce depending on how curved the manifold is. We study the symmetry properties and conservation laws of wave equations on Freidmann-Robertson-Walker, Milne, Bianchi, and de Sitter universes. Symmetry structures are used to reduce the number of unknown functions, and hence contribute to nding exact solutions of the equations. As expected, properties of reduction procedures using symmetries, variational structures and conservation laws are more involved than on the well known at (Minkowski) manifold.

This research deals with several novel aspects of finite element formulations and methodology in parallel adaptive simulation of flow problems. Composite macroelement schemes are developed for problems of thin fluid layers with deforming free surfaces or decomposing material phases; experiments are also run on divergence-free formulations that can be derived from the same element classes. The constrained composite nature and C¹ continuity requirements of these elements raises new issues, especially with respect to adaptive refinement patterns and the treatment of hanging node constraints, which are more complex than encountered with standard element types. This work combines such complex elements with these applications and with parallel adaptive mesh refinement and coarsening (AMR/C) techniques for the first time. The use of adaptive macroelement spaces also requires appropriate programming interfaces and data structures to enable easy and efficient implementation in parallel software. The algorithms developed for this work are implemented using object-oriented designs described herein. One application class of interest concerns heated viscous thin fluid layers that have a deformable free surface. These problems occur in both normal scale laboratory and industrial applications and in micro-fluidics. Modeling this flow via depth averaging gives a nonlinear boundary value problem describing the transient evolution of the film thickness. The model is dominated by surface tension effects which are described by a combination of nonlinear second and fourth-order operators. This research work also includes studies using the divergence-free forms constructed from these elements for certain classes of non-Newtonian fluids such as the Powell-Eyring and Williamson shear-thinning viscosity models. In addition to the target problems we conduct verification studies in support of the simulation development. In the final application investigated, C¹ elements are used in conforming finite element approximations of the Cahn-Hilliard phase field model for moving interface and phase separation problems. The nonlinear Cahn-Hilliard equation combines anti-diffusive configurational free energy based terms with a fourth-order interfacial free energy based term. Numerical studies include both manufactured and physically significant problems, including parametric studies of directed pattern self-assembly in phase decomposition of thin films. The main new contributions include construction of C¹ and div-free macroelement classes suitable for AMR/C with nonconforming hanging node meshes; a posteriori error estimation for fourth-order problems using these and other element classes; use of projection operators to automate the correct treatment of constraints at hanging nodes and through AMR/C steps; design of supporting data structures and algorithms for implementation in a parallel object oriented framework; variational formulations, methodology and numerical experiments with nonlinear fourth-order flow and transport models; and parametric and Monte Carlo studies of directed phase decomposition. / text

No description available.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of California, San Diego, 1997. / Vita. Includes bibliographical references (leaves 230-237).

Thesis (Ph. D.)--University of Hawaii, 1997. / Includes bibliographical references (leaves 184-190).

The nonlinear propagation of a visible, continuous wave laser beam was studied in three types of polymer systems that harness photochemical reactions: (i) a photopolymerization to create permanent self-written structures, (ii) a photo-oxidation hosted within a polymer matrix and (iii) a reversible photoisomerization that triggers the contraction of a photoresponsive hydrogel. The process of self-trapping was characterized by monitoring the spatial intensity profiles over time. The mechanism of each material was determined with a series of control experiments in order to confirm the nature of the nonlinear response, including their reversibility and intensity-dependence. These observations led to the study of interactions between self-trapped beams. Two beams under linear conditions will pass through one another, but two beams travelling in a nonlinear medium will interact and influence one another. The interactions of two beams introduced into the aforementioned photochemical systems were investigated and revealed a rich diversity of phenomena including: (i) the attraction between beams, (ii) merging of beams into a single waveguide, (iii) nonlocal attraction between beams, (iv) orbiting of beams, (v) switching of beam positions, and (vi) inhibition of the self-trapping of a neighbouring beam. Each observation is dependent on a detailed understanding of the underlying mechanism of refractive index change. Numerical simulations supplement some of these experiments and provide further evidence for the nonlinear mechanisms. The formation of permanent self-written structures with these nonlinear waves offers the opportunity to create seamless 3D printed materials with prismatic geometries. Several macroscopic objects were constructed using nonlinear waves from incoherent LEDs and amplitude masks. Decomposition of 3D objects into prismatic elements was carried out using an algorithm that breaks an object into individual pieces. Using a multi-step printing process, several prismatic elements can be combined to form a target object. The results of these experimental and theoretical studies improve upon the current understanding of the dynamics of nonlinear light propagation in photochemical systems. These insights may allow us to harness other nonlinear effects and develop new materials for applications such as optical communication, computing and 3D printing. / Thesis / Doctor of Science (PhD) / The nonlinear propagation of a visible, continuous wave laser beam was studied in three types of polymer systems that harness photochemical reactions: (i) a photopolymerization to create permanent self-written structures, (ii) a photo-oxidation hosted within a polymer matrix and (iii) a reversible photoisomerization that triggers the contraction of a photoresponsive hydrogel. Photochemical changes to the material lead to self-induced light-guiding structures that influence the behaviour of light. These self-trapped beams can interact with one another inside of a nonlinear medium, giving rise to a rich diversity of phenomena including: (i) the attraction between beams, (ii) merging of beams into a single waveguide, (iii) nonlocal attraction between beams, (iv) orbiting of beams, (v) switching of beam positions, and (vi) inhibition of the self-trapping of a neighbouring beam. Each observation is dependent on a detailed understanding of the underlying mechanism of refractive index change. Numerical simulations supplement some of these experiments and provide further evidence for the nonlinear mechanisms. The formation of permanent self-written structures with these nonlinear waves offers the opportunity to create seamless 3D printed materials with prismatic geometries. Several macroscopic objects were constructed using nonlinear waves from incoherent LEDs and amplitude masks. Decomposition of 3D objects into prismatic elements was carried out using an algorithm that breaks an object into individual pieces. Using a multi-step printing process, several prismatic elements can be combined to form a target object. The results of these experimental and theoretical studies improve upon the current understanding of the dynamics of nonlinear light propagation in photochemical systems. These insights may allow us to harness other nonlinear effects and develop new materials for applications such as optical communication, computing and 3D printing.