A dissertation submitted for the degree of Masters of Science, School of Computational and Applied Mathematics, University of Witwatersrand, Johannesburg, 2014. / The magma equation which models the migration of melt upwards through the Earth’s mantle is considered. The magma equation depends on the permeability and viscosity of the solid mantle which are assumed to be a function of the voidage . It is shown using Lie group analysis that the magma equation admits Lie point symmetries provided the permeability and viscosity satisfy either a power law, or an exponential law for the voidage or are constant. The conservation laws for the magma equation for both power law and exponential law permeability and viscosity are derived using the multiplier method. The conserved vectors are then associated with Lie point symmetries of the magma equation. A rarefactive solitary wave solution for the magma equation is derived in the form of a quadrature for exponential law permeability and viscosity. Finally small amplitude and large amplitude approximate solutions are derived for the magma equation when the permeability and viscosity satisfy exponential laws.

An efficient three-dimensional structured solver for the Euler and Navier-Stokes equations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multi-stage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of one-quarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M -+ 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and Engquist-Osher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi one-dimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. March 2016. / We investigate the two-dimensional turbulent wake and derive the governing equations for the mean velocity components using both the eddy viscosity and the Prandtl mixing length closure models to complete the system of equations. Prandtl’s mixing length model is a special case of the eddy viscosity closure model. We consider an eddy viscosity as a function of the distance along the wake, the perpendicular distance from the axis of the wake and the mean velocity gradient perpendicular to the axis of thewake. We calculate the conservation laws for the system of equations using both closure models. Three main types of wakes arise from this study: the classical wake, the wake of a self-propelled body and a new wake is discovered which we call the combination wake. For the classical wake, we first consider the case where the eddy viscosity depends solely on the distance along the wake. We then relax this condition to include the dependence of the eddy viscosity on the perpendicular distance from the axis of the wake. The Lie point symmetry associated with the elementary conserved vector is used to generate the invariant solution. The profiles of the mean velocity show that the role of the eddy viscosity is to increase the effective width of the wake and decrease the magnitude of the maximum mean velocity deficit. An infinite wake boundary is predicted fromthis model. We then consider the application of Prandtl’s mixing length closure model to the classical wake. Previous applications of Prandtl’s mixing length model to turbulent wake flows, which neglected the kinematic viscosity of the fluid, have underestimated the width of the boundary layer. In this model, a finite wake boundary is predicted. We propose a revised Prandtl mixing length model by including the kinematic viscosity of the fluid. We show that this model predicts a boundary that lies outside the one predicted by Prandtl. We also prove that the results for the two models converge for very large Reynolds number wake flows. We also investigate the turbulentwake of a self-propelled body. The eddy viscosity closure model is used to complete the system of equations. The Lie point symmetry associated with the conserved vector is derived in order to generate the invariant solution. We consider the cases where the eddy viscosity depends only on the distance along the wake in the formof a power law and when a modified version of Prandtl’s hypothesis is satisfied. We examine the effect of neglecting the kinematic viscosity. We then discuss the issues that arisewhenwe consider the eddy viscosity to also depend on the perpendicular distance from the axis of the wake. Mean velocity profiles reveal that the eddy viscosity increases the boundary layer thickness of the wake and decreases the magnitude of the maximum mean velocity. An infinite wake boundary is predicted for this model. Lastly, we revisit the discovery of the combination wake. We show that for an eddy viscosity depending on only the distance along the axis of the wake, a mathematical relationship exists between the classical wake, the wake of a self-propelled body and the combination wake. We explain how the solutions for the combination wake and the wake of a self-propelled body can be generated directly from the solution to the classical wake. / GR 2016

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves 88-95)

Abstract: We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether's theorem are given, in the sense which enlarges the scope of its application. The dissertation looks at extending the second Noether's theorem to optimal control problems which are invariant under symmetry depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. Furthermore, we look at the Conservation Laws, i.e. conserved quantities along Euler-Lagrange extremals, which are obtained on the basis of Noether's theorem. And finally we obtain a generalization of Noether's theorem for optimal control problems. The generalization involves a one-parameter family of smooth maps which may depend also on the control and a Lagrangian which is invariant up to an addition of an exact differential. / (M.Sc.) North-West University, Mafikeng Campus, 2005

The non-Newtonian incompressible power-law uid in jet ow models is investigated. An important feature of the model is the de nition of a suitable Reynolds number, and this is achieved using the standard de nition of a Reynolds number and ascertaining the magnitude of the e ective viscosity. The jets under examination are the two-dimensional free, liquid and wall jets. The two-dimensional free and wall jets satisfy a di erent partial di erential equation to the two-dimensional liquid jet. Further, the jets are reformulated in terms of a third order partial di erential equation for the stream function. The boundary conditions for each jet are unique, but more signi - cantly these boundary conditions are homogeneous. Due to this homogeneity the conserved quantities are critical in the solution process. The conserved quantities for the two-dimensional free and liquid jet are constructed by rst deriving the conservation laws using the multiplier approach. The conserved quantity for the two-dimensional free jet is also derived in terms of the stream function. For a Newtonian uid with n = 1 the twodimensional wall jet gives a conservation law. However, this is not the case for the two-dimensional wall jet for a non-Newtonian power-law uid. The various approaches that have been applied in an attempt to derive a conservation law for the two-dimensional wall jet for a power-law uid with n 6= 1 are discussed. In conjunction with the attempt at obtaining conservation laws for the two-dimensional wall jet we present tenable reasons for its failure, and a feasible way forward. Similarity solutions for the two-dimensional free jet have been derived for both the velocity components as well as for the stream function. The associated Lie point symmetry approach is also presented for the stream function. A parametric solution has been obtained for shear thinning uid free jets for 0 < n < 1 and shear thickening uid free jets for n > 1. It is observed that for values of n > 1 in the range 1=2 < n < 1, the velocity pro le extends over a nite range. For the two-dimensional liquid jet, along with a similarity solution the complete Lie point symmetries have been obtained. By associating the Lie point symmetry with the elementary conserved vector an invariant solution is found. A parametric solution for the two-dimensional liquid jet is derived for 1=2 < n < 1. The solution does not exist for n = 1=2 and the range 0 < n < 1=2 requires further investigation.

Doctor of Philosophy / Department of Mathematics / Marianne Korten / Charles N. Moore / We start with linear single variable conservation laws and examine the conditions under which a local extrapolation method (LEM) with upwinding underlying scheme is total variation diminishing TVD. The results are then extended to non-linear conservation laws. For this later case, we restrict ourselves to convex flux functions f, whose derivatives are positive, that is, fœ A0 and fœ A0. We next show that the Goodman-LeVeque flux satisfies the conditions for the LEM to be applied to it. We make heavy use of the CFL conditions, the geometric properties of convex functions apart from the martingle type properties of functions which are increasing, continuous, and differentiable.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science. March 28, 2013 / The search for solutions to the important differential equations arising in extended surface heat transfer continues unabated. Extended surfaces, in the form of longitudinal fins are considered. First we consider the steady state problem and then the transient heat transfer models. Here, thermal conductivity and heat transfer coefficient are assumed to be functions of temperature. Thermal conductivity is considered to be given by the power law in one case and by the linear function of temperature in the other; whereas heat transfer coefficient is only given by the power law. Explicit analytical expressions for the temperature profile, fin efficiency and heat flux for steady state problems are derived using the one-dimensional Differential Transform Method (1D DTM). The obtained results from 1D DTM are compared with the exact solutions to verify the accuracy of the proposed method. The results reveal that the 1D DTM can achieve suitable results in predicting the solutions of these problems. The effects of some physical parameters such as the thermo-geometric fin parameter and thermal conductivity gradient, on temperature distribution are illustrated and explained. Also, we apply the two-dimensional Differential Transform Method (2D DTM) to models describing transient heat transfer in longitudinal fins. Furthermore, conservation laws for transient heat conduction equations are derived using the direct method and the multiplier method, and finally we find Lie point symmetries associated with the conserved vectors.

One of the more recently established methods of analysis of di erentials involves the invariance properties of the equations and the relationship of this with the underlying conservation laws which may be physical. In a variational system, conservation laws are constructed using a well known formula via Noether's theorem. This has been extended to non variational systems too. This association between symmetries and conservation laws has initiated the double reduction of di erential equations, both ordinary and, more recently, partial. We apply these techniques to a number of well known equations like the damped driven Schr odinger equation and a transformed PT symmetric equation(with Schr odinger like properties), that arise in a number of physical phenomena with a special emphasis on Schr odinger type equations and equations that arise in Optics.

Research considered investigates the optimal control problem which is constrained by a hyperbolic conservation law (HCL). We carried out a comparative study of the solutions of the optimal control problem subject to each one of the two di erent types of hyperbolic relaxation systems [64, 92]. The objective was to employ the adjoint-based optimization to minimize the cost functional of a matching type between the optimal solution and the target solution. Numerical tests were then carried out and promising results obtained. Finally, an extension was made to the adjoint-based optimization approach to apply second-order schemes for the optimal control problem in which also good numerical results were observed.