Browsing Doctoral Degrees (Applied Mathematics) by Subject "Newtonian fluids -- Mathematical models"
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- 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.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.