In this thesis, a theoretical investigation is undertaken into fluid and mixing flows generated by various geometries for Newtonian and non-Newtonian fluids, on both sequential and parallel computer systems. The thesis begins by giving the necessary background to the mixing process and a summary of the fundamental characteristics of parallel architecture machines. This is followed by a literature review which covers accomplished work in mixing flows, numerical methods employed to simulate fluid mechanics problems and also a review of relevant parallel algorithms. Next, an overview is given of the numerical methods that have been reviewed, discussing the advantages and disadvantages of the different methods. In the first section of the work the implementation of the primitive variable finite element method to solve a simple two dimensional fluid flow problem is studied. For the same geometry colour band mixing is also investigated. Further investigational work is undertaken into the flows generated by various rotors for both Newtonian and non-Newtonian fluids. An extended version of the primitive variable formulation is employed, colour band mixing is also carried out on two of these geometries. The latter work is carried out on a parallel architecture machine. The design specifications of a parallel algorithm for a MIMD system are discussed, with particular emphasis placed on frontal and multifrontal methods. This is followed by an explanation of the implementation of the proposed parallel algorithm, applied to the same fluid flow problems as considered earlier and a discussion of the efficiency of the system is given. Finally, a discussion of the conclusions of the entire accomplished work is presented. A number of suggestions for future work are also given. Three published papers relating to the work carried out on the transputer networks are included in the appendices.

The 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.

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A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015. / This thesis is primarily concerned with the analysis of some nonlinear problems arising in the study of non-Newtonian fluid mechanics by employing group theoretic and compatibility approaches. It is well known now that many manufacturing processes in industry involve non-Newtonian fluids. Examples of such fluids include polymer solutions and melts, paints, blood, ketchup, pharmaceuticals and many others. The mathematical and physical behaviour of non-Newtonian fluids is intermediate between that of purely viscous fluid and that of a perfectly elastic solid. These fluids cannot be described by the classical Navier–Stokes theory. Striking manifestations of non-Newtonian fluids have been observed experimentally such as the Weissenberg or rod-climbing effect, extrudate swell or vortex growth in a contraction flow. Due to diverse physical structure of non-Newtonian fluids, many constitutive equations have been developed mainly under the classification of differential type, rate type and integral type. Amongst the many non-Newtonian fluid models, the fluids of differential type have received much attention in order to explain features such as normal stress effects, rod climbing, shear thinning and shear thickening. Most physical phenomena dealing with the study of non-Newtonian fluids are modelled in the form of nonlinear partial differential equations (PDEs). It is easier to solve a linear problem due to its extensive study as well due to

The e ects of non-Newtonian uids are investigated by means of two appropri- ate models studying a third and fourth grade uid respectively. The geometry of both these models is described by the unsteady unidirectional ow of an in-compressible uid over an in nite at rigid plate within a porous medium. The uid is electrically conducting in the presence of a uniform applied magnetic eld that occurs in the normal direction to the ow. The classical Lie symmetry approach is undertaken in order to construct group invariant solutions to the governing higher-order non-linear partial dif-ferential equations. A three-dimensional Lie algebra is acquired for both uid ow problems. In each case, the invariant solution corresponding to the non-travelling wave type is considered to be the most signi cant solution for the uid ow model under investigation since it directly incorporates the magnetic eld term. A numerical solution to the governing partial di erential equation is produced and a comparison is made with the results obtained from the analytical ap-proach. Finally, a graphical analysis is carried out with the purpose of observing the e ects of the emerging physical parameters. In particular, a study is carried out to examine the in uences of the magnetic eld parameter and the non-Newtonian fluid parameters.

The aim of this thesis was to understand how a non-Newtonian fluid like chocolate fluidises under vibration. The problem was approached both experimentally and computationally, employing the finite element method as implemented in oomph-lib. In order to model a non-Newtonian fluid its constitutive behaviour has to be known. For this purpose, rheological measurements on tempered chocolate were performed. The chocolate was modelled using generalised Newtonian models for the viscosity and it was found that the Sisko model provided the best fit to the data. The generalised Newtonian Navier-Stokes equations were implemented within oomph-lib and their validation against analytical solutions yielded excellent agreement. Both the experiments and the computations for a sessile drop of chocolate, which is vibrated vertically, spreading on a layer of the same fluid yielded qualitatively similar results. The parameter which controls the spreading is the acceleration arising from the vibration. The drop rests on the layer and does not spread under gravity and small accelerations. However, as the acceleration increases and exceeds a threshold the drop starts to spread out. For a constant amplitude vibration, the spreading rate increases with increasing frequency and for large frequencies approaches a theoretical spreading law which corresponds to viscous dominated spreading of a Newtonian drop.

Thesis (Ph.D.)--University of Delaware, 2006. / Principal faculty advisor: Romain Valéry Roy, Dept. of Mechanical Engineering. Includes bibliographical references.

As a logical extension of previous work conducted into viscoelastic atomization, initially motivated by the need to improve spray coating transfer efficiencies, an experimental investigation into the spray-surface interaction for a number of Newtonian and non-Newtonian substitute test liquids is presented. Three model elastic liquids of varying polymer molecular weight and three inelastic liquids of varying shear viscosity were sprayed upon a moving surface to isolate the effect of elasticity and shear viscosity, respectively, on spray impaction behavior. In addition, two liquids exhibiting shear thinning behavior and an industrial top of rail liquid friction modifier, KELTRACK, for use in the railroad industry, were included in the spray tests. High-speed photography was used to examine the impingement of these liquids on the surface. Ligaments, formed as a consequence of a liquid’s viscoelasticity, were observed impacting the surface for 300K PEO, 1000K PEO, and KELTRACK. These ligaments were broadly classified into four groups, based on their structure. Splashing of elastic liquid ligaments and droplets led to filamentary structures being expelled from the droplet periphery, which were then carried away by the atomizing air jet, leading to reductions in transfer efficiency. The effect of increasing elasticity amongst the three varying molecular weight elastic solutions was shown to increase the splash threshold; a similar effect was noted with increasing shear viscosity. Attempts were made at quantifying a critical splash-deposition limit for all test liquids however due to imaging system limitations, no quantitative conclusions could be made. For KELTRACK, both droplets and ligaments spread and deposited on the rail surface upon impact, with no observed splash or rebound. Splash was only noted when droplets impinged directly on a previously deposited liquid film and even then, splashing was well contained. Thus, KELTRACK’s current rheological formulation proved to be very effective in ensuring high coating transfer efficiencies.

As a logical extension of previous work conducted into viscoelastic atomization, initially motivated by the need to improve spray coating transfer efficiencies, an experimental investigation into the spray-surface interaction for a number of Newtonian and non-Newtonian substitute test liquids is presented. Three model elastic liquids of varying polymer molecular weight and three inelastic liquids of varying shear viscosity were sprayed upon a moving surface to isolate the effect of elasticity and shear viscosity, respectively, on spray impaction behavior. In addition, two liquids exhibiting shear thinning behavior and an industrial top of rail liquid friction modifier, KELTRACK, for use in the railroad industry, were included in the spray tests. High-speed photography was used to examine the impingement of these liquids on the surface. Ligaments, formed as a consequence of a liquids viscoelasticity, were observed impacting the surface for 300K PEO, 1000K PEO, and KELTRACK. These ligaments were broadly classified into four groups, based on their structure. Splashing of elastic liquid ligaments and droplets led to filamentary structures being expelled from the droplet periphery, which were then carried away by the atomizing air jet, leading to reductions in transfer efficiency. The effect of increasing elasticity amongst the three varying molecular weight elastic solutions was shown to increase the splash threshold; a similar effect was noted with increasing shear viscosity. Attempts were made at quantifying a critical splash-deposition limit for all test liquids however due to imaging system limitations, no quantitative conclusions could be made. For KELTRACK, both droplets and ligaments spread and deposited on the rail surface upon impact, with no observed splash or rebound. Splash was only noted when droplets impinged directly on a previously deposited liquid film and even then, splashing was well contained. Thus, KELTRACKs current rheological formulation proved to be very effective in ensuring high coating transfer efficiencies.