Prabhu R. Nott

Pressure-driven flow of suspensions: simulation and theory

Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian suspension at zero Reynolds number were conducted using Stokesian Dynamics. The simulations are for a monolayer of identical particles as a function of the dimensionless channel width and the bulk particle concentration. Starting from a homogeneous dispersion, the particles gradually migrate towards the centre of the channel, resulting in an homogeneous concentration profile and a blunting of the particle velocity profile. The time for achieving steady state scales as (H/a)3a/[left angle bracket]u[right angle bracket], where H is the channel width, a the radii of the particles, and [left angle bracket]u[right angle bracket] the average suspension velocity in the channel. The concentration and velocity profiles determined from the simulations are in qualitative agreement with experiment. A model for suspension flow has been proposed in which macroscopic mass, momentum and energy balances are constructed and solved simultaneously. It is shown that the requirement that the suspension pressure be constant in directions perpendicular to the mean motion leads to particle migration and concentration variations in inhomogeneous flow. The concept of the suspension ‘temperature’ – a measure of the particle velocity fluctuations – is introduced in order to provide a nonlocal description of suspension behaviour. The results of this model for channel flow are in good agreement with the simulations.

Frictional-collisional equations of motion for particulate flows and their application to chutes

Measurements of the relation between mass hold-up and flow rate have been made for glass beads in fully developed flow down an inclined chute, over the whole range of inclinations for which such flows are possible. Velocity profiles in the flowing material have also been measured. For a given inclination it is found that two different flow regimes may exist for each value of the flow rate in a certain interval. One is an ‘energetic’ flow, and is produced when the particles are dropped into the chute from a height, while the other is relatively quiescent and occurs when entry to the chute is regulated by a gate. At some values of the inclination jumps in the flow pattern occur between these branches, and it is even possible for both branches to coexist in the same chute, separated by a shock. A theoretical treatment of chute flow has been based on a rheological model of the material which takes into account both collisional and fractional mechanisms for generating stress. Its predictions include most aspects of the observed behaviour, but quantitative comparison of theory and experiment is difficult because of the uncertain values of some parameters appearing in the theory.

An introduction to granular flow

Preface. 1. Introduction 2. Theory for slow plane flow 3. Flow through hoppers 4. Flow through wedge-shaped bunkers 5. Theory for slow three-dimensional flow 6. Flow through axisymmetric hoppers and bunkers 7. Theory for rapid flow of smooth, inelastic particles 8. Analysis of rapid flow in simple geometries 9. Theory for rapid flow of rough, inelastic particles 10. Hybrid theories A. Operations with vectors and tensors B. The stress tensor C. Hyperbolic partial differential equations of first order D. Jump balances E. Discontinuous solutions of hyperbolic equations F. Proof of the coaxiality condition G. Material frame-indifference H. The evaluation of some integrals I. Linear stability J. Pseudoscalars, vectors, and tensors K. Answers to selected problems References.

Experimental measurements of the normal stresses in sheared Stokesian suspensions

We present experimental measurements of the normal stresses in sheared Stokesian suspensions. Though the suspending fluid is Newtonian, dispersing rigid non-Brownian particles in it yields a suspension that is non-Newtonian, as it exhibits normal stress differences and an excess isotropic pressure in viscometric flows. At small to moderate concentrations, the normal stresses are very small in magnitude, and hence difficult to measure. This difficulty is compounded by the presence of noise due to unavoidable experimental artifacts. Owing to these limitations, most measurements reported earlier were carried out at relatively high particle concentrations, and some at shear rates large enough that the effects of particle and fluid inertia may have been significant. In our study, we have used a novel technique to measure the small stress levels. This was achieved by applying a sinusoidally varying shear rate with a fixed (low) frequency superimposed on a constant shear rate, and using a lock-in amplifier to measure the Fourier component of the same frequency in the stress signal. We have measured normal stresses in cylindrical-Couette and parallel-plate geometries, and combined these measurements to determine the two normal stress differences for particle volume fractions in the range 0.3–0.45. While the normal stresses are very small at low concentrations, they rise rapidly with increasing concentration. The normal stresses vary linearly with the magnitude of the shear rate, and are independent of its sign. In contrast to polymeric solutions, both normal stress differences are negative, and the first normal stress difference is significantly smaller in magnitude. We compare our data with the results of earlier studies, and observe good agreement.

The suspension balance model revisited

This paper addresses a fundamental discrepancy between the suspension balance model and other two-phase flow formulations. The former was proposed to capture the shear-induced migration of particles in Stokesian suspensions, and hinges on the presence of a particle phase stress to drive particle migration. This stress is taken to be the “particle stress,” defined as the particle contribution to the suspension stress. On the other hand, the two-phase flow equations derived in several studies show only a force acting on the particle phase, but no stress. We show that the identification of the particle phase stress with the particle contribution to the suspension stress in the suspension balance model is incorrect, but there exists a well-defined particle phase stress. Following the rigorous method of volume averaging, we show that the force on the particle phase may be written as the sum of an interphase drag and the divergence of the particle phase stress. We derive exact micromechanical relations for thes...

Normal stresses and microstructure in bounded sheared suspensions via Stokesian Dynamics simulations

We report the normal stresses in a non-Brownian suspension in plane Couette flow determined from Stokesian Dynamics simulations. The presence of normal stresses that are linear in the shear rate in a viscometric flow indicates a non-Newtonian character of the suspension, which is otherwise Newtonian. While in itself of interest, this phenomenon is also important because it is believed that normal stresses determine the migration of particles in flows with inhomogeneous shear fields. We simulate plane Couette flow by placing a layer of clear fluid adjacent to one wall in the master cell, which is then replicated periodically. From a combination of the traceless hydrodynamic stresslet on the suspended particles, the stresslet due to (non-hydrodynamic) inter-particle forces, and the total normal force on the walls, we determine the hydrodynamic and inter-particle force contributions to the isotropic ‘particle pressure’ and the first normal stress difference. We determine the stresses for a range of the particle concentration and the Couette gap. The particle pressure and the first normal stress difference exhibit a monotonic increase with the mean particle volume fraction \phi. The ratio of normal to shear stresses on the walls also increases with \phi, substantiating the result of Nott & Brady (1994) that this condition is required for stability to concentration fluctuations. We also study the microstructure by extracting the pair distribution function from our simulations; our results are in agreement with previous studies showing anisotropy in the pair distribution, which is the cause of normal stresses.

A frictional Cosserat model for the slow shearing of granular materials

A rigid-plastic Cosserat model for slow frictional flow of granular materials, proposed by us in an earlier paper, has been used to analyse plane and cylindrical Couette flow. In this model, the hydrodynamic fields of a classical continuum are supplemented by the couple stress and the intrinsic angular velocity fields. The balance of angular momentum, which is satisfied implicitly in a classical continuum, must be enforced in a Cosserat continuum. As a result, the stress tensor could be asymmetric, and the angular velocity of a material point may differ from half the local vorticity. An important consequence of treating the granular medium as a Cosserat continuum is that it incorporates a material length scale in the model, which is absent in frictional models based on a classical continuum. Further, the Cosserat model allows determination of the velocity fields uniquely in viscometric flows, in contrast to classical frictional models. Experiments on viscometric flows of dense, slowly deforming granular materials indicate that shear is confined to a narrow region, usually a few grain diameters thick, while the remaining material is largely undeformed. This feature is captured by the present model, and the velocity profile predicted for cylindrical Couette flow is in good agreement with reported data. When the walls of the Couette cell are smoother than the granular material, the model predicts that the shear layer thickness is independent of the Couette gap H when the latter is large compared to the grain diameter d p . When the walls are of the same roughness as the granular material, the model predicts that the shear layer thickness varies as (H/d p ) 1/3 (in the limit H/d p [dbl greater-than sign] 1) for plane shear under gravity and cylindrical Couette flow.

Structure and rheology of the defect-gel states of pure and particle-dispersed lyotropic lamellar phases

Abstract:We present important new results from light-microscopy and rheometry on a moderately concentrated lyotropic smectic, with and without particulate additives. Shear-treatment aligns the phase rapidly, except for a striking network of oily-streak defects, which anneals out much more slowly. If spherical particles several microns in diameter are dispersed in the lamellar medium, part of the defect network persists under shear-treatment, its nodes anchored on the particles. The sample as prepared has substantial storage and loss moduli, both of which decrease steadily under shear-treatment. Adding particles enhances the moduli and retards their decay under shear. The data for the frequency-dependent storage modulus after various durations of shear-treatment can be scaled to collapse onto a single curve. The elasticity and dissipation in these samples thus arises mainly from the defect network, not directly from the smectic elasticity and hydrodynamics.

Frictional–collisional equations of motion for granular materials and their application to flow in aerated chutes

Equations of motion and boundary conditions for a flowing granular material, developed in earlier publications, are here extended to allow for drag forces resulting from relative motion of the material and interstitial air. These are solved for fully developed flow down an inclined plane, through which a constant flow of air passes upward. The results are compared with measurements from an experimental aerated chute, in which the inclination of the chute, the flow rate of the granular material, and the flow of air are all varied. Using parameter values from independent measurements, as far as possible, the theory is found to give a good qualitative account of the observed behaviour. With a reasonable assigned value for the one parameter that cannot be determined independently the quantitative agreement is also satisfactory.

The collective dynamics of self-propelled particles

We propose a method for the dynamic simulation of a collection of self-propelled particles in a viscous Newtonian fluid. We restrict attention to particles whose size and velocity are small enough that the fluid motion is in the creeping flow regime. We propose a simple model for a self-propelled particle, and extended the Stokesian Dynamics method to conduct dynamic simulations of a collection of such particles. In our description, each particle is treated as a sphere with an orientation vector p, whose locomotion is driven by the action of a force dipole Sp of constant magnitude S0 at a point slightly displaced from its centre. To simplify the calculation, we place the dipole at the centre of the particle, and introduce a virtual propulsion force Fp to effect propulsion. The magnitude F0 of this force is proportional to S0. The directions of Sp and Fp are determined by p. In isolation, a self-propelled particle moves at a constant velocity u0 p, with the speed u0 determined by S0. When it coexists with many such particles, its hydrodynamic interaction with the other particles alters its velocity and, more importantly, its orientation. As a result, the motion of the particle is chaotic. Our simulations are not restricted to low particle concentration, as we implement the full hydrodynamic interactions between the particles, but we restrict the motion of particles to two dimensions to reduce computation. We have studied the statistical properties of a suspension of self-propelled particles for a range of the particle concentration, quantified by the area fraction φa. We find several interesting features in the microstructure and statistics. We find that particles tend to swim in clusters wherein they are in close proximity. Consequently, incorporating the finite size of the particles and the near-field hydrodynamic interactions is of the essence. There is a continuous process of breakage and formation of the clusters. We find that the distributions of particle velocity at low and high φa are qualitatively different; it is close to the normal distribution at high φa, in agreement with experimental measurements. The motion of the particles is diffusive at long time, and the self-diffusivity decreases with increasing φa. The pair correlation function shows a large anisotropic build-up near contact, which decays rapidly with separation. There is also an anisotropic orientation correlation near contact, which decays more slowly with separation. Movies are available with the online version of the paper.