Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 1. Creeping flows

Abstract

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders, for cases in which fluid inertia can be neglected. The truncated power-law fluid model is used to define the relationship between the viscous stress and the rate-of-strain tensor. The flow is shown to be dominated by shear effects, not extension. Numerical simulation results are presented for the drag coefficient of the cylinders and the velocity variance components, and are compared against asymptotically valid analytical results. Square and hexagonal arrays are considered, both for crossflow in the plane perpendicular to the alignment vector of the cylinders (flows along the axes of the array as well as off-axis flows), and for flow along the cylinders. It is shown that the observed strong dependence of the drag coefficient on the power-law index (through which the stress tensor is related to the rate-of-strain tensor) can be described at all solid area fractions by scaling the drag on a cylinder with appropriate velocity and length scales. The velocity variance components show only a weak dependence on the power-law index. The results for steady-state drag and velocity variances are used in an approximate theory for flows accelerated from rest. The numerical simulation data for unsteady flows agree very well with the approximate theory.

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

“…In a companion paper [1] we have presented numerical simulation results for the creeping flow of inelastic non-Newtonian fluids through arrays of cylinders. The effects of fluid inertia in these flows is the subject of this second part.…”

“…As is described in [1], the main issue is the determination of the drag coefficient for a cylinder in the array, and most early studies have been restricted to creeping flows of Newtonian fluids. More recently, flows with small-but-finite and intermediate Reynolds numbers Re = ρaU/µ have been studied, where ρ is the fluid viscosity, a is the cylinder radius, U is the averaged velocity in the array and µ is the fluid viscosity.…”

“…In [1] it is shown that creeping flows through arrays of cylinders are dominated by shear, not extension. As in [1], we use the truncated power-law model for the relation between the viscous part of the local stress tensor τ ij and the rate-ofstrain tensor E ij in the fluid [6,Chapter 4]:…”

“…The case of n = 1 corresponds to a Newtonian fluid; n < 1 provides a model of a pseudoplastic or shear-thinning fluid and n > 1 corresponds to a shear-thickening fluid. As in [1], we shall consider here the range 0·5 ≤ n ≤ 1·5, and chooseγ 0 sufficiently small such that its value does not affect the results (corresponding to a power-law fluid model).…”

“…The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays.…”