On non-Newtonian fluid flow in ducts and porous media

Abstract

In this communication a three-shape-factor approach is developed to characterize both the flow of non-Newtonian fluids, in particular generalized Newtonian fluids, in an arbitrarily shaped duct and the flow over an isolated sphere. The flow of Herschel–Bulkley fluid and Meter fluid is studied. While the detailed solution for a Herschel–Bulkley fluid can be used to deduce the values of the shape factors, the simplified solution for a Meter fluid is also provided to enable for easy application. The interaction of macromolecules with fine capillary duct walls, i.e. the discontinuity in flow, is modeled using the Meter fluid model. At the low shear limit (or very low flow rate), the solutions of Chauveteau (1982, J. Rheol. 26, 111–142) and Kozicki et al. (1987, Chem. Engng Commun. 59, 137–160) are recovered. For a high flow rate, the present study predicts an increased pressure drop for surface-adhesive capillary and a pressure drop reduction for a wall macromolecule depleted/aligned system. The success in modeling flow in arbitrary shaped ducts and flow over isolated spheres using the same approach suggests that the present approach is also applicable to flow in porous media. When a volume averaging technique is employed to arrive at the continuum governing equations for flow of a generalized Newtonian fluid in porous media, the macroscopic viscosity is obtained. The macroscopic viscosity is the single quantity in the governing equations that must be determined for a non-Newtonian fluid flow in addition to the parameters already known for a Newtonian fluid flow. In the Newtonian limit, the macroscopic viscosity becomes identical to the Newtonian viscosity. Otherwise, the macroscopic viscosity is the apparent viscosity of the fluid under the average microscopic shear conditions in porous media. The expressions for the macroscopic viscosity for various fluids: namely, Herschel–Bulkley fluid, Meter fluid and Cross fluid, are derived in this communication. Porous medium matrix—macromolecule interactions due to flow discontinuity are studied using two phenomenological models: macromolecule retention and macromolecule alignment/depletion. In particular, the Meter fluid model is used to derive the effects of the porous medium matrix—macromolecule interactions. Predictions based on this study agree well with experimental data for flow of non-Newtonian fluids in packed beds and consolidated sandstones.