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A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Doctor of Philosophy, 2013. / The aim of the thesis is to derive group invariant, exact, approximate analytical and numerical solutions for a two-dimensional laminar, non-Newtonian pre-existing hydraulic fracture propagating in impermeable and permeable elastic media. The fracture is driven by the injection of an incompressible, viscous non-Newtonian fluid of power law rheology in which the fluid viscosity depends on the magnitude of the shear rate and on the power law index n > 0. By the application of lubrication theory, a nonlinear diffusion equation relating the half-width of the fracture to the fluid pressure is obtained. When the interface is permeable the nonlinear diffusion equation has a leak-off velocity sink term. The half-width of the fracture and the net fluid pressure are linearly related through the PKN approximation. A condition, in the form of a first order partial differential equation for the leak-off velocity, is obtained for the nonlinear diffusion equation to have Lie point symmetries. The general form of the leak-off velocity is derived. Using the Lie point symmetries the problem is reduced to a boundary value problem for a second order ordinary differential equation. The leak-off velocity is further specified by assuming that it is proportional to the fracture half-width. Only fluid injection at the fracture entry is considered. This is the case of practical importance in industry. Two exact analytical solutions are derived. In the first solution there is no fluid injection at the fracture entry while in the second solution the fluid velocity averaged over the width of the fracture is constant along the length of the fracture. For other working conditions at the fracture entry the problem is solved numerically by transforming the boundary value problem to a pair of initial value problems. The numerical solution is matched to the asymptotic solution at the fracture tip. Since the fracture is thin the fluid velocity averaged over the width of the fracture is considered. For the two analytical solutions the ratio of the averaged fluid velocity to the velocity of the fracture tip varies linearly along the fracture. For other working conditions the variation is approximately linear. Using this observation approximate analytical solutions are derived for the fracture half-width. The approximate analytical solutions are compared with the numerical solutions and found to be accurate over a wide range of values of the power-law index n and leak-off parameter β. The conservation laws for the nonlinear diffusion equation are investigated. When there is fluid leak-off conservation laws of two kinds are found which depend in which component of the conserved vector the leak-off term is included. For a Newtonian fluid two conservation laws of each kind are found. For a non-Newtonian fluid the second conservation law does not exist. The behaviour of the solutions for shear thinning, Newtonian and shear thickening fluids are qualitatively similar. The characteristic time depends on the properties of the fluid which gives quantitative differences in the solution for shear thinning, Newtonian and shear thickening fluids.

A dissertation submitted to the Faculty of Science,University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. December 2016. / This research is concerned with the analysis of a two-dimensional Newtonian fluid-driven fracture in a permeable rock. The fluid flow in the fracture is laminar and the fracture is driven by the injection of a Newtonian fluid into it. Most of the problems in litera- ture involving fluid flow in permeable rock formation have been modeled with the use of Darcy's law. It is however known that Darcy's model breaks down for flows involv- ing high fluid velocity, such as the flow in a porous rock formation during hydraulic fracturing. The Forchheimer flow model is used to describe the non-Darcy fluid flow in the porous medium. The objective of this study is to investigate the problem of a fluid-driven fracture in a porous medium such that the flow in the porous medium is non-Darcy. Lubrication theory is applied to the system of partial di erential equations since the fracture that is considered is thin and its width slowly varies along its length. For this same reason, Perkins-Kern-Nordgren approximation is adopted. The theory of Lie group analysis of differential equations is used to solve the nonlinear coupled sys- tem of partial differential equations to obtain group invariant solutions for the fracture half-width, leak-o depth and length of the fracture. The strength of fluid leak-off at the fracture wall is classi ed into three forms, namely, weak, strong and moderate. A group invariant solution of the traveling wave form is obtained and an exact solution for the case in which there is weak fluid leak-off at the interface is found. A dimensionless parameter, F0, termed the Forchheimer number was obtained and investigated. Nu- merical results are obtained for both the case of Darcy and non-Darcy flow. Computer generated graphs are used to illustrate the analytical and numerical results. / MT2017

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Experiments investigating the flow over a body in a steady turn require the use of a rotating arm. This apparatus has several limitations, including the restriction of only making one revolution before the body passes through its own wake, which in turn places restrictions on the available time to record measurements. Kinematic transformation using appropriately curved bodies in rectilinear flow overcomes these limitations, but introduces new problems. In this study, Computational Fluid Dynamics (CFD) was used to compare the flow over a body of revolution in a steady turn to the flow generated by the equivalent curved body. To ensure that the angle of attack between the straight body and the curved flow streamlines are preserved for the corresponding curved body placed in straight streamlines, the method of Gurzhienko (1934) was used. This is the first time that modern CFD techniques have been used to analyse the method of curved bodies for three-dimensional applications.

Petrov-Galerkin finite element methods based on time-space elements are developed for the time-dependent multi-dimensional linear convection-diffusion equation. The methods introduce two parameters in conjunction with perturbed weighting functions. These parameters are determined locally using truncation error analysis techniques. In the one-dimensional case, the new algorithms are thoroughly analyzed for convergence and stability properties. Numerical schemes that are second order in time, third order in space and stable when the Courant number is less than or equal to one are produced. Extensions of the algorithm to nonlinear Navier-Stokes equations are investigated. In this case, it is found more efficient to use a Petrov-Galerkin method based on a one parameter perturbation and a semi-discrete Petrov-Galerkin formulation with a generalized Newmark algorithm in time. The algorithm is applied to the two-dimensional simulation of natural convection in a horizontal circular cylinder when the Boussinesq approximation is valid. New results are obtained for this problem which show the development of three flow regimes as the Rayleigh number increases. Detailed calculations for the fluid flow and heat transfer in the cylinder for the different regimes as the Rayleigh number increases are presented.

For aerospace and naval applications where low radiated noise levels are a requirement, rotor noise generated by inflow turbulence is of great interest. Inflow turbulence is stretched and distorted as it is ingested into a thrusting rotor which can have a significant impact on the noise source levels. This thesis studies the distortion of subsonic, high Reynolds number turbulent flow, with viscous effects ignored, that occur when a rotor is embedded in a turbulent boundary layer. The analysis is based on Rapid Distortion Theory (RDT), which describes the linear evolution of turbulent eddies as they are stretched by a mean flow distortion. Providing that the gust does not distort the mean flow streamlines the solution for a mean flow with shear is found to be the same as the solution for a mean potential flow with the addition of a potential flow gust. By investigating the inflow distortion of small-scale turbulence for various simple flows and rotor inflows with weak shear, it is shown that RDT can be applied to incompressible shear flows to determine the flow distortion. It is also shown that RDT can be applied to more complex flows modeled by the Reynolds Averaged Navier Stokes (RANS) equations. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2013.

Boundary layers are regions where turbulence develops easily. In the case where the flow occurs on a surface showing a certain degree of roughness, turbulence eddies will interact with the roughness elements and will produce an acoustic field. This thesis aims at predicting this type of noise with the help of the Computational Fluid Dynamics (CFD) simulation of a wall jet using the Reynolds Average Navier-Stokes (RANS) equations. A frequency spectrum is reconstructed using a representation of the turbulence with uncorrelated sheets of vorticity. Both aerodynamic and acoustic results are compared to experimental measurements of the flow. The CFD simulation of the flow returns consistent results but would benefit from a refinement of the grid. The surface pressure spectrum presents a slope in the high frequencies close to the experimental spectrum. The far field noise spectrum has a 5dB difference to the experiments. / by Jean-Baptiste Blanc. / Thesis (M.S.C.S.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.