Alan L. Graham

A constitutive equation for concentrated suspensions that accounts for shear‐induced particle migration

A constitutive equation for computing particle concentration and velocity fields in concentrated monomodal suspensions is proposed that consists of two parts: a Newtonian constitutive equation in which the viscosity depends on the local particle volume fraction and a diffusion equation that accounts for shear‐induced particle migration. Particle flux expressions used to obtain the diffusion equation are derived by simple scaling arguments. Predictions are made for the particle volume fraction and velocity fields for steady Couette and Poiseuille flow, and for transient start‐up of steady shear flow in a Couette apparatus. Particle concentrations for a monomodal suspension of polymethyl methacrylate spheres in a Newtonian solvent are measured by nuclear magnetic resonance (NMR) imaging in the Couette geometry for two particle sizes and volume fractions. The predictions agree remarkably well with the measurements for both transient and steady‐state experiments as well as for different particle sizes.

Migration of particles undergoing pressure-driven flow in a circular conduit

This study focuses on the demixing of neutrally buoyant suspensions of spheres during slow, pressure driven flows in circular conduits. Distributions of the solid fraction of particles, φ, and the suspension velocity, ν, are measured at different lengths from a static in-line mixer. Experiments were conducted over a range of volume average solids fractions, φbulk (0.10⩽φ⩽0.50), and at two different ratios of the particle radius, a, to the radius of the circular conduit, R (a/R=0.0256 and a/R=0.0625). At φbulk⩾0.20, the particles rapidly migrate to the low-shear-rate region in the center of the conduit. This migration results in a blunting of the ν profile, relative to the parabolic profile observed in homogeneous Newtonian fluids. For the flow geometry with the smaller ratio of a/R, the φ profile builds to a sharp maximum or cusp in the center. Particle structures are observed in the experiments with the higher a/R. The entrance lengths for the development of the φ and ν fields, Lφ and Lν, respectively, a...

Particle migration in a Couette apparatus: Experiment and modeling

Suspensions comprised of neutrally buoyant spheres in Newtonian fluids undergoing creeping flow in the annular region between two rotating, coaxial cylinders (a wide-gap Couette) display a bulk migration of particles towards regions of lower shear rate. A series of experiments are performed to characterize this particle migration, including the influence of particle size, surface roughness, and volume fraction. Little, if any, effect of particle surface roughness is observed. An existing continuum diffusive-flux model [Phillips et al. (1992)] for predicting particle concentration profiles in monomodal suspensions is evaluated using the current series of experimental data. This model predicts a dependence of the migration rate on the square of the suspended particles’ radius, a2; whereas the present experiments indicate that systems with average particle volume fractions of 50% display a rate that scales with a3. Previous use of the diffusive-flux model has assumed constant values for diffusion coefficient...

Flow-aligned tensor models for suspension flows

Abstract Models to describe the transport of particles in suspension flows have progressed considerably during the last decade. In one class of models, designated as suspension balance models, the stress in the particle phase is described by a constitutive equation, and particle transport is driven by gradients in this stress. In another class of models, designated as diffusive flux models, the motion of particles within the suspension is described through a diffusion equation based on shear rate and effective viscosity gradients. Original implementations of both classes of models lacked a complete description of the anisotropy of the particle interactions. Because of this, the prediction of particle concentration in torsional flows in parallel plate and cone-and-plate geometries did not match experimental data for either class of models. In this work, the normal stress differences for the suspension balance formulation are modeled using a frame-invariant flow-aligned tensor. By analogy, the diffusive flux model is reformulated using the same flow-aligned tensor, which allows separate scaling arguments for the magnitude of the diffusive flux to be implemented in the three principal directions of flow. Using these flow-aligned tensor formulations, the main shortcomings of the original models are eliminated in a unified manner. Steady-state and transient simulations are performed on various one-dimensional and two-dimensional flows for which experimental data are available, using finite-difference and finite-element schemes, respectively. The results obtained are in good agreement with experimental data for consistent sets of empirical constants, without the need for ad hoc additional terms.

Modelling of concentrated suspensions using a continuum constitutive equation

We simulate the behaviour of suspensions of large-particle, non-Brownian, neutrally-buoyant spheres in a Newtonian liquid with a Galerkin, finite element, Navier–Stokes solver into which is incorporated a continuum constitutive relationship described by Phillips et al . (1992). This constitutive description couples a Newtonian stress/shear-rate relationship (where the local viscosity of the suspension is dependent on the local volume fraction of solids) with a shear-induced migration model of the suspended particles. The two-dimensional and three-dimensional (axisymmetric) model is benchmarked with a variety of single-phase and two-phase analytic solutions and experimental results. We describe new experimental results using nuclear magnetic resonance imaging to determine non-invasively the evolution of the solids-concentration profiles of initially well-mixed suspensions as they separate when subjected to slow flow between counter-rotating eccentric cylinders and to piston-driven flow in a pipe. We show good qualitative and quantitative agreement of the numerical predictions and the experimental measurements. These flows result in complex final distributions of the solids, causing rheological behaviour that cannot be accurately described with typical single-phase constitutive equations.

A new constitutive model for fibre suspensions: flow past a sphere

A new phenomenological constitutive equation for homogeneous suspensions of macrosized fibres is proposed. In the model, the local averaged orientation of the fibres is represented by a director field, which evolves in time in a manner similar to the rotation of a prolate spheroid. The stress is linear in the strain rate, but the viscosity is a fourth-order tensor that is directly related to the director field. In the limit of low-volume fractions of fibres, the model reduces properly to the leading terms of the constitutive equation for dilute suspensions of spheroids. The model has three parameters: the aspect ratio R of the fibres, the volume fraction Φ, and A, which plays the role of the maximum-volume fraction of the fibres. Experimental shear data are used to estimate the parameter A, and the resulting model is used in a boundary-element program to study the flow past a sphere placed at the centre line of a cylinder for the whole range of volume fractions from 0.01 to near maximum volume fraction. The agreement with experimental data from Milliken et al. [1] is good.

The viscosity-volume fraction relation for suspensions of rod-like particles by falling-ball rheometry

The relative viscosities of suspensions of randomly oriented rods in a Newtonian fluid were measured using falling-ball rheometry. The rods were monodisperse and sufficiently large to render colloidal and Brownian forces negligible. Steel and brass ball bearings were dropped along the centreline of cylindrical columns containing the suspensions. The terminal velocities of the falling balls were measured and used to determine the average viscosities of the suspensions. The suspensions behaved as Newtonian fluids in that they were characterized by a constant viscosity. They exhibited a linear relative viscosity-volume fraction relationship for volume fractions less than 0.125, and, for volume fractions between 0.125 and 0.2315, the specific viscosity increased with the cube of the volume fraction. The relative viscosity was found to be independent of falling-ball size for a ratio of falling ball to fibre length greater than 0.3. It was found to be independent of the diameter of the containing cylindrical column for a ratio of column diameter to fibre length greater than 3.2. The value determined for the intrinsic viscosity is in good agreement with theoretical predictions for suspensions of randomly oriented rods.

Shear-induced particle migration in suspensions of rods

Shear‐induced migration of particles occurs in suspensions of neutrally buoyant spheres in Newtonian fluids undergoing shear in the annular space between two rotating, coaxial cylinders (a wide‐gap Couette), even when the suspension is in creeping flow. Previous studies have shown that the rate of migration of spherical particles from the high‐shear‐rate region near the inner (rotating) cylinder to the low‐shear‐rate region near the outer (stationary) cylinder increases rapidly with increasing sphere size. To determine the effect of particle shape, the migration of rods suspended in Newtonian fluids was recently measured. The behavior of several suspensions was studied. Each suspension contained well‐characterized, uniform rods with aspect ratios ranging from 2 to 18 at either 0.30 or 0.40 volume fraction. At the same volume fraction of solids, the steady‐state, radial concentration profiles for rods were independent of aspect ratio and were indistinguishable from those obtained from suspended spheres. On...

THE INERTIAL LIFT ON A RIGID SPHERE TRANSLATING IN A LINEAR SHEAR FLOW FIELD

Abstract The shear-induced inertial migration of a rigid sphere has been studied experimentally using a homogeneous shear flow apparatus. The experimentally measured migration velocities have been compared with those predicted by the asymptotic formulas previously given by Saffman and McLaughlin. The analysis of a sphere translating in a shear field in the presence of a single wall has been extended to the case of a sphere translating in a fluid undergoing a uniform shear between two parallel walls.

Antisymmetric stresses in suspensions: vortex viscosity and energy dissipation

When the individual particles in an otherwise quiescent suspension of freely suspended spherical particles are acted upon by external couples, the resulting suspension-scale fluid motion is characterized by a non-symmetric state of stress. Viewed at the interstitial scale (i.e. microscopic scale), this coupling between translational and rotational particle motions is a manifestation of particle–particle hydrodynamic interactions and vanishes with the volume fraction φ of suspended spheres. The antisymmetric portion of the stress is quantified by the suspension-scale vortex viscosity µv, different from the suspension’s shear viscosity µ. Numerical boundary element method (BEM) simulations of such force-free suspensions of spheres uniformly dispersed in incompressible Newtonian liquids of viscosity µ0 are performed for circumstances in which external couples (of any specified suspension-scale position-dependence) are applied individually to each of the suspended particles in order to cause them to rotate in otherwise quiescent fluids. In the absence of external forces acting on either the spheres or boundaries, such rotations indirectly, through interparticle coupling, cause translational motions of the individual spheres which, owing to the no-slip boundary condition, drag neighbouring fluid along with them. In turn, this combined particle–interstitial fluid movement is manifested as a suspension-scale velocity field, generated exclusively by the action of external couples. Use of this scheme to create suspension-scale particle-phase spin fields Ω and concomitant velocity fields v enables both the vortex and shear viscosities of suspensions to be determined as functions of φ in disordered systems. This scheme is shown, inter alia, to confirm the constitutive equation, T a =2 µve · [(1/2)∇ × v − Ω], proposed in the continuum mechanics literature for the linear relation between the antisymmetric stress T a and the disparity existing between the particle-phase spin rate Ω and half the suspension’s vorticity, ∇ × v (with the third-rank pseudotensor e the permutation triadic). Our dynamically based BEM simulations confirm the previous computations of the Prosperetti et al. group for the dependence of the vortex viscosity upon the solids volume fraction in concentrated disordered suspensions, obtained by a rather different simulation scheme. Moreover, our dynamically based rheological calculations are confirmed by our semi-independent, energetically based, calculations that equate the rates of working (equivalently, the energy dissipation rates) at the respective interstitial and suspension scales. As an incidental by-product, the same BEM simulation results also verify the suspension-scale Newtonian constitutive equation, T s = µ[∇v +( ∇v) † ],