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.

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.

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Faculty of Science School of Computational and Applied Mathematics. May 2015. / In this thesis we present numerical simulations of the gravitational collapse of isothermal clouds of one solar mass at a temperature of 10K. We will consider two types of initial conditions – initially uniform spheres and perturbed Bonnor-Ebert spheres. The aim of the performed numerical simulations is to investigate the core bounce described by Hayashi and Nakano [1]. They reported that if strong enough, the shock wave would be capable of ionizing the gas in the collapsing cloud. The simulations are performed using two numerical methods: the TVD MUSCL scheme of van Leer using a Roe flux on a uniform grid and the TVD Runge-Kutta time-stepping using a Marquina flux on a non-uniform grid. These two particular methods are used because of their differences in numerical structure. Which allows us to confidently make statements about the nature of the collapse, particularly with regards to the core bounce. The convergence properties of the two methods are investigated to validate the solutions obtained from the simulations. The numerical simulations have been performed only in the isothermal regime by using the Truelove criterion [2] to terminate the simulation before central densities become large enough to cause artificial fragmentation. In addition to the numerical simulations presented in this thesis, we also introduce new, analytical solutions for the steady-state accretion of an isothermal gas onto a spherical core as well as infinite cylinders and sheets. We present the solutions and their properties in terms of the Lambert function with two parameters, γ and m. In the case of spherical accretion we show that the solution for the velocity perfectly matched the solutions of Bondi [3]. We also show that the analytical solutions for the density – in the spherical case – match the numerical solutions obtained from the simulations. From the agreement of these solutions we propose that the analytical solution can provide information about the protostellar core (in the early stages of its formation) such as the mass.

Lee Wai Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 129-130). / Text in English; abstract also in Chinese. / Lee Wai Shing. / Contents --- p.i / List of Figures --- p.v / Abstract --- p.vii / Acknowledgement --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Background of this research project --- p.2 / Chapter 1.3 --- Conservative systems and open systems -normal modes (NM's) vs quasi-normal modes (QNM's) --- p.3 / Chapter 1.4 --- Appetizer ´ؤ What our problems are like --- p.6 / Chapter 1.5 --- A brief overview of the following chapters --- p.7 / Chapter Chapter 2. --- Inversion of conservative systems- perturbative inversion --- p.9 / Chapter 2.1 --- Overview --- p.9 / Chapter 2.2 --- Way to introduce the additional information --- p.9 / Chapter 2.3 --- General Formalism --- p.11 / Chapter 2.4 --- Example --- p.15 / Chapter 2.5 --- Further examples --- p.19 / Chapter 2.6 --- Effects of noise --- p.23 / Chapter 2.7 --- Conclusion --- p.25 / Chapter Chapter 3. --- Inversion of conservative systems - total inversion --- p.26 / Chapter 3.1 --- Overview --- p.26 / Chapter 3.2 --- Asymptotic behaviour of the eigenfrequencies --- p.26 / Chapter 3.3 --- General formalism --- p.28 / Chapter 3.3.1 --- Evaluation of V(0) --- p.28 / Chapter 3.3.2 --- Squeezing the interval - evaluation of the potential at other positions --- p.32 / Chapter 3.4 --- Remarks --- p.36 / Chapter 3.5 --- Conclusion --- p.37 / Chapter Chapter 4. --- Theory of Quasi-normal Modes (QNM's) --- p.38 / Chapter 4.1 --- Overview --- p.38 / Chapter 4.2 --- What is a Quasi-normal Mode (QNM) system? --- p.38 / Chapter 4.3 --- Properties of QNM's in expectation --- p.40 / Chapter 4.4 --- General Formalism --- p.41 / Chapter 4.4.1 --- Construction of Green's function and the spectral represen- tation of the delta function --- p.42 / Chapter 4.4.2 --- The generalized norm --- p.45 / Chapter 4.4.3 --- Completeness of QNM's and its justification --- p.46 / Chapter 4.4.4 --- Different senses of completeness --- p.48 / Chapter 4.4.5 --- Eigenfunction expansions with QNM's 一 the two-component formalism --- p.49 / Chapter 4.4.6 --- Properties of the linear space Γ --- p.51 / Chapter 4.4.7 --- Klein-Gordon equation - The delta-potential system --- p.54 / Chapter 4.5 --- Studies of other QNM systems --- p.54 / Chapter 4.5.1 --- Wave equation - dielectric rod --- p.55 / Chapter 4.5.2 --- Wave equation ´ؤ string-mass system --- p.57 / Chapter 4.6 --- Summary --- p.58 / Chapter Chapter 5. --- Inversion of open systems- perturbative inversion --- p.59 / Chapter 5.1 --- Overview --- p.59 / Chapter 5.2 --- General Formalism --- p.59 / Chapter 5.3 --- Example 1. Klein-Gordon equation ´ؤ delta-potential system --- p.66 / Chapter 5.3.1 --- Model perturbations --- p.66 / Chapter 5.4 --- Example 2. Wave equation ´ؤ dielectric rod --- p.72 / Chapter 5.5 --- Example 3. Wave equation ´ؤ string-mass system --- p.76 / Chapter 5.5.1 --- Instability of the matrix [d] = [c]-1 upon truncation --- p.79 / Chapter 5.6 --- Large leakage regime and effects of noise --- p.81 / Chapter 5.7 --- Conclusion . . . --- p.84 / Chapter Chapter 6. --- Transition from open systems to conservative counterparts --- p.85 / Chapter 6.1 --- Overview --- p.85 / Chapter 6.2 --- Anticipations of what is going to happen --- p.86 / Chapter 6.3 --- Some computational experiments --- p.86 / Chapter 6.4 --- Reason of breakdown - An intrinsic error of physical systems --- p.87 / Chapter 6.4.1 --- Mathematical derivation of the breakdown behaviour --- p.90 / Chapter 6.4.2 --- Two verifications --- p.93 / Chapter 6.5 --- Another source of errors - An intrinsic error of practical computations --- p.95 / Chapter 6.5.1 --- Vindications --- p.96 / Chapter 6.5.2 --- Mathematical derivation of the breakdown --- p.98 / Chapter 6.6 --- Further sources of errors --- p.99 / Chapter 6.7 --- Dielectric rod --- p.100 / Chapter 6.8 --- String-mass system --- p.103 / Chapter 6.9 --- Conclusion --- p.105 / Chapter Chapter 7. --- A first step to Total Inversion of QNM systems? --- p.106 / Chapter 7.1 --- Overview --- p.106 / Chapter 7.2 --- Derivation for F(0) --- p.106 / Chapter 7.3 --- Example 一 delta potential system --- p.108 / Chapter Chapter 8. --- Conclusion --- p.111 / Chapter 8.1 --- A summary on what have been achieved --- p.111 / Chapter 8.2 --- Further directions to go --- p.111 / Appendix A. A note on notation --- p.113 / Appendix B. Asymptotic series of NM eigenvalues --- p.114 / Appendix C. Evaluation of functions related to RHS(x) --- p.117 / Appendix D. Asymptotic behaviour of the Green's function --- p.119 / Appendix E. Expansion coefficient an --- p.121 / Appendix F. Asymptotic behaviour of QNM eigenvalues --- p.123 / Appendix G. Properties of the inverse matrix [d] = [c]-1 --- p.125 / Appendix H. Matrix inverse through the LU decomposition method --- p.127 / Bibliography --- p.129

Wong Tak Kwong. / Thesis submitted in: August 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 67-72). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Basic Notations --- p.7 / Chapter 1.2 --- Riemann Problems --- p.10 / Chapter 1.3 --- Elementary Waves --- p.10 / Chapter 1.3.1 --- Rarefaction Waves --- p.11 / Chapter 1.3.2 --- Shock Waves --- p.11 / Chapter 1.3.3 --- Composite Waves --- p.13 / Chapter 1.4 --- Remarks --- p.14 / Chapter 2 --- Non-strictly Hyperbolic Conservation Laws --- p.16 / Chapter 2.1 --- Systems with Isolated Umbilic Degeneracy --- p.16 / Chapter 2.1.1 --- Mathematical Motivations --- p.17 / Chapter 2.2 --- Complex Burgers' Equation --- p.21 / Chapter 2.2.1 --- Introduction --- p.21 / Chapter 2.2.2 --- Basic Properties --- p.22 / Chapter 2.2.3 --- Riemann Solutions --- p.24 / Chapter 2.2.4 --- Under-Compressive Shocks --- p.31 / Chapter 3 --- Relaxation Approximation --- p.34 / Chapter 3.1 --- Basic Ideas of the Relaxation Approximation --- p.34 / Chapter 3.1.1 --- General Settings --- p.35 / Chapter 3.1.2 --- Subcharacteristic Condition --- p.36 / Chapter 3.2 --- Relaxation of Scalar Conservation Laws --- p.39 / Chapter 3.2.1 --- Perturbation Problems --- p.39 / Chapter 3.3 --- Jin-Xin Relaxation Systems --- p.42 / Chapter 3.3.1 --- Basic Ideas of the Jin-Xin Systems --- p.42 / Chapter 3.4 --- Zero-Relaxation Limit --- p.45 / Chapter 3.4.1 --- 2x2 Hyperbolic Relaxation Systems --- p.45 / Chapter 3.4.2 --- Jin-Xin Relaxation Systems --- p.48 / Chapter 4 --- Jin-Xin Relaxation Limit for the Complex Burgers' Equations --- p.51 / Chapter 4.1 --- Jin-Xin Relaxation Limit for the UCUI Solutions --- p.52 / Chapter 4.1.1 --- Main Statements --- p.52 / Chapter 4.1.2 --- Analysis on UCUI Solution --- p.53 / Chapter 4.1.3 --- Shock Profiles --- p.56 / Chapter 4.1.4 --- Re-scaled Relaxation System --- p.60 / Chapter 4.1.5 --- Proof of Theorem 4.1.1.3 --- p.63 / Bibliography --- p.67