3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item The symmetry structures of curved manifolds and wave equations(2017) Bashingwa, Jean Juste HarrissonKilling vectors are widely used to study conservation laws admitted by spacetime metrics or to determine exact solutions of Einstein field equations (EFE) via Killing’s equation. Its solutions on a manifold are in one-to-one correspondence with continuous symmetries of the metric on that manifold. Two well known spherically symmetric static spacetime metrics in Relativity that admit maximal symmetry are given by Minkowski and de-Sitter metrics. Some other spherically symmetric metrics forming interesting solutions of the EFE are known as Schwarzschild, Kerr, Bertotti-Robinson and Einstein metrics. We study the symmetry properties and conservation laws of the geodesic equations following these metrics as well as the wave and Klein-Gordon (KG) type equations constructed using the covariant d’Alembertian operator on these manifolds. As expected, properties of reduction procedures using symmetries are more involved than on the well known flat (Minkowski) manifold.Item Turbulent wake flows: lie group analysis and conservation laws(2016) Hutchinson, Ashleigh JaneWe 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.Item Symmetries and conservation laws of difference and iterative equations(2016-01-22) Folly-Gbetoula, Mensah KekeliWe construct, using rst principles, a number of non-trivial conservation laws of some partial di erence equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation. Symmetries and the more recent ideas and notions of characteristics (multipliers) for di erence equations are also discussed. We then determine the symmetry generators of some ordinary di erence equations and proceed to nd the rst integral and reduce the order of the di erence equations. We show that, in some cases, the symmetry generator and rst integral are associated via the `invariance condition'. That is, the rst integral may be invariant under the symmetry of the original di erence equation. We proceed to carry out double reduction of the di erence equation in these cases. We then consider discrete versions of the Painlev e equations. We assume that the characteristics depend on n and un only and we obtain a number of symmetries. These symmetries are used to construct exact solutions in some cases. Finally, we discuss symmetries of linear iterative equations and their transformation properties. We characterize coe cients of linear iterative equations for order less than or equal to ten, although our approach of characterization is valid for any order. Furthermore, a list of coe cients of linear iterative equations of order up to 10, in normal reduced form is given.Item Invariances, conservation laws and conserved quantities of the two-dimensional nonlinear Schrodinger-type equation(2014) Lepule, SeipatiSymmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.Item Numerical simulations of isothermal collapse and the relation to steady-state accretion(2015-05) Herbst, Rhameez SheldonIn 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.Item Nonclassical symmetry reductions and conservation laws for reaction-diffusion equations with application to population dynamics(2015-05-29) Louw, KirstenThis dissertation analyses the reaction-di usion equations, in particular the modi ed Huxley model, arising in population dynamics. The focus is on determining the classical Lie point symmetries, and the construction of the conservation laws and group-invariant solutions for reaction-di usion equations. The invariance criterion for determination of classical Lie point symmetries results in a system of linear determining equations which can be solved analytically. Furthermore, the Lie point symmetries associated with the conservation laws are determined. Reductions by associated Lie point symmetries are carried out. Nonclassical symmetry techniques are also employed. Here the invariance criterion for symmetry determination results in a system of nonlinear determining equations which may be solved albeit di cult. Nonclassical symmetries results in exact solutions which may not be constructed by classical Lie point symmetries. The highlight in construction of exact solution using nonclassical symmetries is the introduction of the modi ed Hopf-Cole transformation. In this dissertation, the di usion term and the coe cient of the source term are given as quadratic functions of space variable in one case, and the coe cient as the generalised power law in the other. These equations admit a number of classical Lie point symmetries. The genuine nonclassical symmetries are admitted when the source term of the reaction-di usion equation is a cubic.Item Group invariant solutions and conservation laws for jet flow models of non-Newtownian power-law fluids(2014-07-18) Magan, Avnish BhowanThe 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.Item Symmetries, conservation laws and reductions of Schrodinger systems of equations(2014-06-12) Masemola, PhetogoOne 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.Item An analysis of the symmetries and conservation laws of some classes of nonlinear wave equations in curved spacetime geometry(2013-08-08) Jamal, SThe (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.Item Analytical solutions and conservation laws of models describing heat transfer through extended surfaces(2013-07-29) Ndlovu, Partner LuyandaThe 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.