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<p> One of the most difficult and interesting aspects of the physics of collapse and outflow formation, as well as the evolution of the protostellar disk, is the role of hydromagnetic forces. However, magnetic fields are only coupled to the charged species present in poorly ionized molecular clouds. Ambipolar diffusion-the process by which magnetic fields "slip" in poorly ionized gas-strongly affects the initial cloud as well as the final observable structure through collisional heating. Also, as the gas becomes opaque to cosmic rays, the ionized structure of the accreting gas may become more complex, leading to a neutral 'dead zone' in a layered accretion disk (vital in determining planet masses in planet formation theories (Matsumura & Pudritz, 2005)). We omit possible effects of ionizing radiation in these early stages of formation. </p> <p> In this thesis, we perform fully 3D simulations (using the FLASH AMR code) and have implemented ambipolar diffusion in the MHD module of the code in addition to a broad treatment of cooling (Banerjee et al., 2006). This has allowed us to track the ionized gas and magnetic fields properly from the beginning of collapse down to the onset of outflows. We find that high accretion rates persist on the order 1 of 10-3 M0 yr-(where the core mass has reached about 0.1 M0 ) due to efficient extraction of angular momentum through magnetic processes. Magnetic braking is reduced by about 3/4 in the initial collapse relative to an ideal collapse of same initial conditions. This, with a reduction in magnetic pressure in the disk, leads to an increased rate of fragmentation. One of the major new results of this work is the discovery that outflows from disks still occur even in the presence of ambipolar diffusion. Surprisingly, they are initiated even earlier than outflows from idealized, completely ionized disks. They are generated by a magnetic tower mechanism at central densities of 1012 cm-3, as effective ram pressure on the wound up toroidal field is reduced, allowing it to push away from the disk earlier. </p> <p> We have also shown that the formation of a dead zone in these early stages is dependent on shielding of cosmic rays, in the absence of which a decoupled zone in the disk midplane forms. This region, where the accreting gas is effectively decoupled from the magnetic field, extends 10 AU in radius and (2-3) AU in height from the midplane. The global magnetic field threading such a complex accretion disk shows a dragged out structure, as coupled surface layers of the disk pull in the field. The disk is puffy due to drift heating and the initial stages of the outflow pushing out into the ambient medium. However, overall magnetic field build-up is still efficient, as values of the magnetic field in the disk are only reduced by half. </p> / Thesis / Master of Science (MSc)

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Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2020 / The flow of electrically conducting fluids in the presence of a magnetic field has wide applications in science, engineering and technology. Examples of the applications include industrial processes such as the cooling of reactors, extrusion of plastics, purification of crude oil, medical applications, aerodynamics and many more. The induced magnetic field usually act as a flow control mechanism, especially under intense heat. In this study a couple stress fluid in a channel will be used as the working fluid. Channel flow and heat transfer characteristics of couple stress fluids find applications in processes such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plates in a bath, tribology of thrust bearings and lubrication of engine rod bearings. One major characteristic that distinguishes the couple stress fluid from other non-Newtonian fluids is the inclusion of size-dependent microstructure that is of mechanical significance. As such, the couple stress constitutive model is capable of describing the couple stresses, the effect of body couples and the nonsymmetric tensors manifested in several real fluids of technological importance. A fully developed laminar magnetohydrodynamic (MHD) flow of an incompressible couple stress fluid through a vertical channel due to a steady-periodic temperature on the channel plates is investigated. Specifically, the effects of couple stresses and internal heat generation on MHD natural convection flow with steady-periodic heat input, the impact of magnetic field induction on the buoyancy-induced oscillatory flow of couple stress fluid with varying heating and a mixed convective two dimensional flow of unsteady MHD couple stress fluid through a channel field with porous medium are studied. Analytical methods and the semi-analytic Adomian decomposition method will be used to solve the resulting non-linear differential equations governing the flow systems. Useful results for velocity, temperature, skin friction and Nusselt number are obtained and discussed quantitatively. The effects of the various flow governing parameters on the flow field are investigated.

Thesis Ph.D. ((Applied Mathematics)) -- University of Limpopo, 2022 / The world is under threat from the devastating effects of the continued depletion of the Ozone layer. Increased global warming is causing catastrophic ecological damage and imbalance due to accelerated melting of glaciers, rampant runaway veld res, widespread floods and other extreme events. The delegates to the Cop26 Climate Change Summit were reminded that the continued burning of fossil fuels is releasing carbon into the atmosphere at an unprecedented pace and scale and that the world is already in trouble. Complete substitution of fossil fuels with clean energy sources is the only solution through which the world can be saved from the deleterious effects of global warming. However, total dependence on renewable energy sources can only be possible through novel technology that enables efficient energy utilization and conservation. For instance, the evolution of advanced techniques in manufacturing processes has led to the reduction in the size of various industrial and engineering designs that consume reduced amounts of energy. Efficient energy utilization in thermo-fluid flow systems can be achieved through entropy generation minimization. Entropy is a thermodynamic quantity that represents the unavailability of a system's thermal energy for conversion into mechanical work. In this study, thermodynamic analysis of reactive variable properties third-grade fluid flow in channels with varied geometries and subjected to different physical effects was investigated with the second law of thermodynamics as the area of focus. Entropy generation and inherent irreversibility analysis were the main focus of the study where the sensitivities of these quantities to the embedded parameters were numerically and graphically described and analysed. The semi-analytic Adomian decomposition method, the semi-implicit fi nite difference scheme and the spectral quasilinearisation method were employed to solve the nonlinear differential equations modelling the flow systems. The results reveal that the effects of the parameters on flow velocity, fluid temperature, entropy generation and inherent irreversibility cannot be neglected. In particular, conditions for entropy generation minimization were successfully established and documented. / University of Limpopo