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# Department of Applied Mathematics

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### Browsing Department of Applied Mathematics by browse.metadata.advisor "Du Plessis, J. P."

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- ItemHydrodynamic permeability of staggered and non-staggered regular arrays of squares(Stellenbosch : Stellenbosch University, 2003-12) Lloyd, Cindy; Du Plessis, J. P.; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences.
Show more ENGLISH ABSTRACT: This work entails an analysis of two-dimensional Newtonian flow through a prismatic array of squares. Both in-line and staggered configurations are investigated, as well as the very low velocity Darcy regime, where Stokes' flow predominates, and the Forchheimer regime, where interstitial inertial effects such as recirculation are present. As point of departure two recently developed pore-scale models are discussed and their results compared to Stokes' flow computational analysis for flow through regular arrays of rectangles. The commercial CFX code is also used to analyse the problem and to determine the accuracy of the assumptions used for the development of the pore-scale models. Finally an improvement is suggested to the RRUC model towards more accurate prediction of permeabilities, especially for porosities below 75%, and whereby its quantitative predictive capability is thus enhanced considerably.Show more - ItemPore-scale modelling for fluid transport in 2D porous media(Stellenbosch : University of Stellenbosch, 2006-12) Cloete, Maret; Du Plessis, J. P.; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics.
Show more In the present study, a model to predict the hydrodynamic permeability of viscous flow through an array of solid phase rectangles of any aspect ratio is derived. This also involves different channel widths in the streamwise and the transverse flow directions which may be chosen irrespectively to the rectangular shape itself. It is shown how, with the necessary care taken during description of the interstitial geometry, a volume averaged approach can be used to obtain results identical to a direct method. Insight into the physical situation is gained during the modelling of the two-dimensional interstitial flow processes and resulting pressure distributions and this may prove valuable when the volume averaging method is applied to more complex three-dimensional cases. The analytical results show close correspondence to numerical calculations, except in the higher porosity range for which a more realistic model is needed. Tortuosity is studied together with its inverse. Correspondences and differences regarding the definitions for the average straightness of pathlines, expressed in literature, are examined. A new definition, allowing different channel widths in the streamwise and the transverse flow directions, for the tortuosity is derived from first principles. A general relation between newly derived permeability and tortuosity expressions was obtained. This equation incorporates many possible geometrical features for a two-dimensional unit cell for granules. Three possible staggering configurations of the solid phase along the streamwise direction are also included in this relation.Show more - ItemPore-scale modelling of transport phenomena in homogeneous porous media(Stellenbosch : Stellenbosch University, 1999-12) Diedericks, Gerhardus Petrus Jacobus; Du Plessis, J. P.; Stellenbosch University. Faculty of Science. Dept. of Mathematics.
Show more ENGLISH ABSTRACT: The main purpose of this study is to develop deterministic, process-based models of incompressible Newtonian flow and electrical c01iduction in homogeneous, anisotropic porous media. The foundation of the models is provided by the volume averaging theory which is used to obtain the macroscopic balance equations for momentum transport and electrical conduction. These volume averaged equations contain, amongst others, integral terms over the fluid-solid surface area where the integrands are related to the microscopic fluxes of the transport quantities. The closure modelling is conducted by employing a pore-scale model which requires explicit assumptions regarding the mean geometric properties of the porous medium microstructure and accounts for the configuration of the fluid-solid surface area. The pore-scale model also provides an estimate of the microscopic flow paths. The average geometry of different anisotropic materials, namely two types of foamlike materials, granular porous media and fibre beds, is captured in representative unit cells which form the core of the physical pore-scale model. This particular type of closure modelling further requires a direct transformation of microscopic fluxes to the macroscopic level. It is indicated, in context of the volume averaging theory, that microscopic fluxes may be estimated by the respective macroscopic channel average fluxes. The transformation of the microscopic flux to the channel average flux is accomplished through the flux related tortuosity tensor. New definitions for the tortuosity and lineality as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. It is shown that the definitions produce results which conform with several other published results and are applicable to anisotropic media. Application of the modelling technique to Newtonian flow results in momentum transport equations valid for both the Darcy and Forchheimer flow regimes. The coefficients appearing in these equations are expressed in terms of fluid properties and measurable geometric features of the porous medium. The predictions of the anisotropic foamlike materials are validated against experimental pressure gradient measurements for flow through a high porosity, anisotropic knitted wire mesh rolled up to form a cylindrical plug. The predictions compare reasonably well with the experimental results. The modelling approach is also applied to electrical conduction in anisotropic porous media saturated with an electrically conductive fluid. A macroscopic form of Ohm's law is derived as well as deterministic expressions for the formation factor. The formation factor predictions for isotropic porous media are compared to several experimental measurements as well as to semi-empirical expressions. The predictions compare favourably to the measurements.Show more