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Singular behaviour of Non-Newtonian fluids
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Since 1996, a team at the Centre for Research in Applied Technology (CRATECH) at Peninsula Technikon, under NRF sponsorship and with industrial co-operation, has been involved in the simulation of Non-Newtonian flow behaviour in industrial processes, in particular, injection moulding of polymers. This study is an attempt to deal with some current issues of Non-Newtonian flow, in small areas, from the viewpoint of computational mechanics. It is concerned with the numerical simulation of Non-Newtonian fluid flows in mould cavities with re-entrant corners. The major complication that exists in this numerical simulation is the singularity of the stresses at the entry of the corner, which is responsible for nonintegrable stresses and the propagation of solution errors. First, the study focuses on the derivation of the equations of motion of the flow which leads to Navier- Stokes equations. Thereafter, the occurrence of singularities in the numerical solution of these equations is investigated. Singularities require special attention no matter what numerical method is used. In finite element analysis, local refinement around the singular point is often employed in order to improve the accuracy. However, the accuracy and the rate of convergence are not, in general, satisfactory. Incorporating the nature of singularity, obtained by an asymptotic analysis in the numerical solution, has proven to be a very effective way to improve the accuracy in the neighborhood of the singularity and, to speed up the rate of convergence. This idea has been successfully adopted in solving mainly fracture mechanics problems by a variety of methods: finite difference, finite elements, boundary and global elements, and spectral methods. In this thesis, the singular finite elements method (SFEM), similar in principle to the crack tip element used in fracture mechanics, is proposed to improve the solution accuracy in the vicinity of the singular point and to speed up the rate of convergence. This method requires minor modifications to standard finite element schemes. Unfortunately, this method could not be implemented in this study due to the difficulty in generating the mesh for the singular element. Only the standard finite element method with mesh refinement has been used. The results obtained are in accordance with what was expected.