Abstract
The work reported in this thesis considers the steady flow of Newtonian and shear thinning fluids past unbounded and bounded spherical particles. In the bounded case, the sphere is located either (i) along the axis of a uniform tube or (ii) along the axis of a tapered tube. The objective of this study is to provide a better estimation of the hydrodynamic drag force for these cases. The effects of inertia (particle Reynolds number ranged from 0.01 to 100), shear thinning characteristics and the proximity of the walls (for the bounded cases) were investigated. The contributions of the individual pressure and viscous forces, as well as plots for the local static pressure and wall shear stress were also presented. The governing equations (continuity and momentum) have been solved numerically using the computational fluid dynamics software, FLUENT 5. The literature on the unbounded case is extensive, particularly at creeping regime. However, there exists a significant disagreement concerning the influence of shear thinning parameters on the drag force, see for example Laero et al. (1997). This study resolves the literature disagreement regarding the dependence of the drag correction factor on power law index by using Spriggs truncated model, Birds (1987), to describe the fluid inelastic shear thinning behaviour. At a given modified particle Reynolds number and power law index, the drag correction factor could have different values and trends depending on the value of the dimensionless shear rate H. The dimensionless shear rate here is defined as the ratio of the minimum shear rate, above which the power law model begins to describe the rheological behaviour, to the average effective shear created by the sphere. Not considering H could lead to inappropriate drag predictions at creeping and intermediate Reynolds numbers regimes. Expressions like those of Graham and Jones (1994) seem to be applicable only when power law conditions hold (H ~ 0). Computations were carried out for a bounded sphere in a uniform tube. Both the settling sphere in stationary fluids and stationary sphere in moving fluids were considered. For the former case, it was found that H has significant influence on the trends for size ratios alR<0.1. The effect of H gradually diminishes as the size ratio increases and was found to die out completely after size ratio of alR ~ 0.6. The results also suggested that the wall effects become less significant as the degree of shear thinning (pseudo-plasticity behaviour) increases, which is in-line with the general conclusions reached in the past by some experimentalists. On the other hand, the results showed that the wall correction factor on a stationary sphere becomes less than the settling sphere counterparts as the size ratio decreases and/or the power law index increases. Finally, the numerical results for the hydrodynamic drag force acting on a spherical particle positioned along the axis of a tapered tube were presented. These were obtained for tubes of standard contraction diameter ratio of 10:1 and three half-angle values, 10°, 20° and 30°. In all cases, the ratio of the sphere radius to the downstream tube radius is alR2=0.2. The drag results were expressed in terms of the wall correction factor, K2, which is defined as the ratio of the drag force on the sphere at a given separation distance to that on the sphere in an unbounded fluid (Stokes force). The separation distance is defined as the distance between the centre of the sphere and the inlet of the conical section of the tube. Results showed that the hydrodynamic drag force acting on a spherical particle at various separation distances increases substantially especially towards the exit. By way of an example, at creeping flow regime, the drag force on a sphere at the exit of a tube is two orders of magnitude larger than that on a sphere at the inlet. Wall correction factor values were also calculated for different power law indices. The wall correction factor increases with increasing shear-thinning behaviour (decreasing n). However, as the sphere moves downstream the influence of n gradually diminishes. The inertia effects seem to play a role only when the sphere is at a distance of 75% of the tube's axal length. The accuracy of the drag calculation procedure (mesh generation and boundary conditions) was established by carrying out comparisons with previously available experimental, analytical and/or numerical results for Newtonian and power law fluids whenever possible.