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

[摘要] ENGLISH ABSTRACT: Different generalized Newtonian fluids (where the normal stresses were neglected) wereconsidered in this study. Analytical expressions were derived for time independent,fully developed velocity profiles of Herschel-Bulkley fluids (including the simplificationsthereof: Newtonian, power law and Bingham plastic fluids) and Casson fluids throughopen channel sections. Both flow through cylindrical pipes (Hagen-Poiseuille flow) andparallel plates (plane Poiseuille flow) were brought under consideration. Equations werederived for the wall shear stresses in terms of the average channel velocities. Theseexpressions for plane Poiseuille flow were then utilized in the modelling of flow throughhomogeneous, isotropic porous media.Flow through parallel plates was extended and a possibility of a moving lower wall (planeCouette-Poiseuille flow) was included for Herschel-Bulkley fluids (and the simplificationsthereof). 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. Theseequations were again revisited in the study on flow over porous structures.Averaging of the microscopic momentum transport equation was carried out by meansof volume averaging over an REV (Representative Elementary Volume). Flow throughparallel plates enclosing a homogeneous porous medium (assumed homogeneous up tothe 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 externalboundary layer area is directly proportional to the superficial velocity that is, since onlythe viscous flow regime was considered, referred to as the 'Darcy' velocity if the diffusiveBrinkman term is completely neglected. For a shear thinning or shear thickening fluid,the excess superficial velocity term was included in the proportionality coefficient thatis constant for a particular fluid traversing a particular porous medium subjected to aspecific 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, directsolutions were obtained. The effect of the constant yield stress was embedded in theproportionality 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. Secondaryaveraging 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 aswell as the passability in the Forchheimer regime (also sometimes referred to as thenon-Darcian permeability) were derived for the various fluid types.Finally fluid flow in a domain consisting of an open channel adjacent to an infinite porousdomain is considered. The analytically derived velocity profiles for both plane Couette-Poiseuille flow and the Brinkman equation were matched by assuming continuity in theshear stress at the porosity jump between the two domains.An in-house code was developed to simulate such a composite domain numerically. Thedifference between the analytically assumed constant apparent permeability in a macro-scopic boundary layer region as opposed to a dependency of the varying superficialvelocity was discussed. This code included the possibility to alter the construction ofthe domain and to simulate axisymmetrical flow in a cylinder.

[发布机构] Stellenbosch University