Abstracts Mathematics

Modelling of non-Newtonian fluid flow through and over porous media with the inclusion of boundary effects

by Maret Cloete

ENGLISH ABSTRACT: Different generalized Newtonian fluids (where the normal stresses were neglected) were considered in this study. Analytical expressions were derived for time independent, fully developed velocity profiles of Herschel-Bulkley fluids (including the simplifications thereof: Newtonian, power law and Bingham plastic fluids) and Casson fluids through open channel sections. Both flow through cylindrical pipes (Hagen-Poiseuille flow) and parallel plates (plane Poiseuille flow) were brought under consideration. Equations were derived for the wall shear stresses in terms of the average channel velocities. These expressions for plane Poiseuille flow were then utilized in the modelling of flow through homogeneous, isotropic porous media. Flow through parallel plates was extended and a possibility of a moving lower wall (plane Couette-Poiseuille flow) was included for Herschel-Bulkley fluids (and the simplifications thereof). The velocity of the wall was assumed to be opposite to the pressure gradient (thus in the streamwise direction) yielding three different possible flow scenarios. These equations were again revisited in the study on flow over porous structures. Averaging of the microscopic momentum transport equation was carried out by means of volume averaging over an REV (Representative Elementary Volume). Flow through parallel plates enclosing a homogeneous porous medium (assumed homogeneous up to the external boundary) was studied at the hand of Brinkman’s equation. It was as- sumed (also for non-Newtonian fluids) that the term dominating outside the external boundary layer area is directly proportional to the superficial velocity that is, since only the viscous flow regime was considered, referred to as the ‘Darcy’ velocity if the diffusive Brinkman term is completely neglected. For a shear thinning or shear thickening fluid, the excess superficial velocity term was included in the proportionality coefficient that is constant for a particular fluid traversing a particular porous medium subjected to a specific pressure gradient. For such fluids only the inverse functions could be solved. If the ‘Darcy’ velocity is not reached within the considered domain, Gauss’s hypergeo- metric function had to be utilized. For Newtonian and Bingham plastic fluids, direct solutions were obtained. The effect of the constant yield stress was embedded in the proportionality coefficient. For linear flow, the proportionality coefficient consists of both a Darcy and a Forch- heimer term applicable to the viscous and inertial flow regimes respectively. Secondary averaging for different types of porous media was accomplished by using an RUC (Representative Unit Cell) to estimate average interstitial properties. Only homoge- neous, isotropic media were considered. Expressions for the apparent permeability as well as the passability in the Forchheimer regime (also sometimes referred to as the non-Darcian permeability) were derived for the various fluid types. Finally fluid flow in a domain…