School of Computer Science and Applied Mathematics (ETDs)
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Item Double-diffusive convection in rotating fluids under gravity modulation(University of the Witwatersrand, Johannesburg, 2024-09) Mathunyane, Alfred Ntobeng; Duba, C. Thama; Mason, D.P.This study employs the method of normal modes and linear stability analysis to investigate double-diffusive convection in a horizontally layered, rotating fluid, specifically focusing on its application to oceanic dynamics. Double diffusive convection arises when opposing gradients of salinity and temperature interact within a fluid, a phenomenon known as thermohaline convection, and it is crucial for the understanding of ocean circulation and its role in climate change. With the increasing mass of water due to glaciers melting, fluid pressure variations occur, leading to slight fluctuations in gravity. We conduct both stationary and oscillatory stability analyses to determine the onset of double-diffusive convection under gravity modulation. Our analysis reveals that time-dependent periodic modulation of gravitational fields can stabilize or destabilize thermohaline convection for both stationary and oscillatory convection, with amplitude stabilizing and frequency destabilizing. The wavenumber in the y- direction also affects convection in the equatorial regions. This wavenumber exhibits destabilizing effects for large values and stabilizing effects for small values for both stationary and oscillatory convection. Rotation along with gravity modulation tends to destabilize the system for both stationary and oscillatory convection. The key difference between stationary and oscillatory convection is that oscillatory convection exhibits large values of the Rayleigh number, thus susceptible to overstability while stationary convection tends to have relatively smaller Rayleigh numbers and thus more stable. This research provides insights into the complex interplay between gravity modulation and thermohaline convection, contributing to our understanding of ocean dynamics and their implications for climate change.Item Symmetry reductions and approximate solutions for heat transfer in slabs and extended surfaces(University of the Witwatersrand, Johannesburg, 2023-06) Nkwanazana, Daniel Mpho; Moitsheki, Raseelo JoelIn this study we analyse heat transfer models prescribed by reaction-diffusion equations. The focus and interest throughout the work is on models for heat transfer in solid slabs (hot bodies) and extended surface. Different phenomena of interest are heat transfer in slabs and through fins of different shapes and profiles. Furthermore, thermal conductivity and heat transfer coefficients are temperature dependent. As a result, the energy balance equations that are produced are nonlinear. Using the theory of Lie symmetry analysis of differential equations, we endeavor to construct exact solutions for these nonlinear models. We will employ a number of symmetry techniques such as the classical Lie point symmetry methods, the nonclassical symmetry, nonlocal and nonclassical potential symmetry approach to construct the group-invariant solutions. In order to identify the forms of the heat source term that appear in the considered equation for which the principal Lie algebra (PLA) is extended by one element, we first perform preliminary group classification of the transient state problem. Also, we consider the direct group classification method. Invariant solutions are constructed after some reductions have been performed. One-dimensional Differential Transform Method (1D DTM) will be used when it is impossible to determine an exact solution. The 1D DTM has been benchmarked using some exact solutions. To solve the transient/unsteady problem, we use the two-dimensional Differential Transform Method (2D DTM). Effects of parameters appearing in the equations on the temperature distribution will be studied.