Free-surface flow of concentrated suspensions

Free-surface flows of concentrated suspensions exhibit many interesting phenomena such as particle segregation and surface corrugation. In this work the flow structures associated with free-surface has been studied experimentally. The free-surface velocity for neutrally buoyant suspension of uniform spheres in a gravity driven inclined channel flow was determined by particle imaging velocimetry (PIV) technique. Experiments were carried out for concentrated suspensions with particle fractions ? ranging from 0.40 to 0.50. The measured velocities show blunted profile in the channel. The blunting of the velocity profile increases with the particle concentration. The rms velocity fluctuations measured at the free-surface progressively increase with particle fraction ? and are linear in shear rate γ. The surface roughness were characterized by analyzing the power spectral density of the refracted light from the free-surface. The characteristics observed are in support of earlier findings.     

Sedimentation of bidisperse suspensions

The average settling velocity of a suspension of identical particles through otherwise quiescent fluid is smaller than the settling velocity of a single particle in an unbounded fluid. When a suspension settles out to form a deposit, this hindered settling effect may lead to complicated sedimentation behaviour, even if the initial suspension is uniformly distributed. This study analyses the bulk sedimentation of bidisperse suspensions and calculates the evolution of the volume fraction of each species from an initially vertically uniform state through to the final steady state where both species have fully settled out of suspension and have formed a deposit. These calculations are analytical and employ the method of characteristics to reveal how both particle species evolve. The profiles often include ‘shocks’, across which discontinuous changes in volume fraction occur. Rarefaction fans may also be found across which the gradients of volume fraction are discontinuous. These new analytical solutions reveal the evolving composition of the suspension and the deposit and may be compared to experimental observations. They also provide test cases that can be used to verify recent numerical techniques for computing the bulk sedimentation behaviour of polydisperse suspensions.     

Viscosity of concentrated suspensions of spheres

Difficulties associated with the viscosity measurement of concentrated suspensions of particulate solids in a liquid solvent can effectively be overcome with the falling needle technique reported here. The comparison of the settling (terminal) velocity of a given needle in a Newtonian solvent, with its terminal velocity in a suspension, yields the suspension viscosity ratio directly. The van den Brule and Jongschaap constitutive model describes our high concentration data best. Falling sphere data (diameter of sphere/diameter of suspended particle 10) agree well with the falling needle data over the whole range (up to 40%) of solids concentrations used in our tests.In the opaque suspensions used, the passage of sedimenting needles and spheres was initially observed radiographically. Later tests used a more convenient technique using an inductance coil particle detector driven by a Colpitts oscillator.     

Boundary element simulations of spheres settling in circular,square and triangular conduits

Numerical simulations of a spherical particle sedimenting in circular, triangular and square conduits containing a viscous, inertialess, Newtonian fluid were investigated using the Boundary Element Method (BEM). Settling velocities and pressure drops for spheres falling along the centre-lines of the conduits were computed for a definitive range of sphere sizes. The numerical simulations for the settling velocities showed good agreement with existing experimental data. The most accurate analytic solution for a sphere settling along the axis of a circular conduit produced results which were almost indistinguishable from the present BEM calculations. For a sphere falling along the centre-line of a square conduit, the BEM calculations for small spheres agreed well with analytic results. No analytic results for a sphere falling along the axis of a triangular conduit were available for comparison. Extrapolation of the BEM predictions for the pressure drops, to infinitely small spheres, showed remarkable agreement with analytic results. For the circular conduit, the sphere's settling velocity and angular velocity were computed as a function of drop position for small, medium and large spheres. Excellent agreement with a reflection solution was achieved for the small sphere. In addition, end effects were investigated for centre-line drops and compared where possible with available experimental data and analytic results.Los Alamos National Laboratory, Los Alamos, New Mexico, USA.     

Numerical simulation of sedimentation of microparticles using the discrete particle method

A mathematical model has been formulated based on the combined continuous and discrete particle method for investigating the sedimentation behaviour of microparticles in aqueous suspensions, by treating the fluid phase as continuous and the particles phase as discrete, thus allowing the behaviour of individual particles to be followed and the evolution of the structure of the particle phase to be investigated as a function of time. The model takes into account most of the prevailing forces acting on individual particles including van der Waals attractive, electrostatic repulsive, gravitational, Brownian, depletion, steric, contact and drag forces. A code has also been developed based on the model. This paper reports some preliminary modelling results of mono-dispersed microparticles settling in aqueous suspensions under various conditions. The results show the short time dynamics of the fluid phase, which has a similar order of magnitude to the particle phase. Such short time dynamics could bear significance to processes such as particle aggregation when their size becomes very small. Preliminary analyses of the results have also been carried out on the evolution of particle settling based on a newly proposed parameter, local normalised volume fraction （LNVF）.     

The sedimentation of polydisperse suspensions in vessels having inclined walls

A theory is presented for describing the sedimentation of polydisperse suspensions in two-dimensional channels having walls that are inclined to the vertical. The theory assumes that the flow is laminar and that the suspension consists of spherical beads having small particle Reynolds numbers. The suspension may consist of either $\text{N}$ distinct species of particles or of a continuum of particle sizes and densities. For the sake of simplicity, the analysis is mostly confined to the case in which the hindered settling velocity of each particle is given by its Stokes settling velocity multiplied by a function of the total local solids concentration. Under these conditions, results are developed that are useful for the design of either batch or continuous settling devices. Experimental observations were found to be in good agreement with the predictions of the present theory.     

Boundary-enhanced sedimentation due to settling convection

Sedimentation rates are significantly enhanced when the process occurs in containers of certain shapes or orientations. For example, sedimentation in a conical vessel or a tilted tube may be several times faster than sedimentation in a vertical tube of the same height. This enhancement results from a naturally occurring convection.

Monodisperse particles were observed during settling from a viscous, incompressible, and Newtonian fluid contained beneath an upward-pointing cone. At particle concentrations of about 0.05 per cent by volume, convective velocities reach ten times the particle settling velocity, and sedimentation is complete in 40 per cent of the time necessary in a vertical tube.

Continuum fluid mechanics successfully models this settling convection. The mechanism is particle momentum transfer to fluid. The model requires numerical solution but its functionality can be investigated in terms of dimensionless parameters.     

Filtration in a Porous Granular Medium: 1. Simulation of Pore-Scale Particle Deposition and Clogging

This paper presents a numerical model for simulating the pore-scale transport and infiltration of dilute suspensions of particles in a granular porous medium under the action of hydrodynamic and gravitational forces. The formulation solves the Stokes’ flow equations for an incompressible fluid using a fixed grid, multigrid finite difference method and an embedded boundary technique for modeling particle–fluid coupling. The analyses simulate a constant flux of the fluid suspension through a cylindrical model pore. Randomly generated particles are collected within the model pore, initially through contact and attachment at the grain surface (pore wall) and later through mounding close to the pore inlet. Simple correlations have been derived from extensive numerical simulations in order to estimate the volume of filtered particles that accumulate in the pore and the differential pressure needed to maintain a constant flux through the pore. The results show that particle collection efficiency is correlated with the Stokes’ settling velocity and indirectly through the attachment probability with the particle–grain surface roughness. The differential pressure is correlated directly with the maximum mound height and indirectly with particle size and settling velocity that affect mound packing density. Simple modification factors are introduced to account for pore length and dip angle. These parameters are used to characterize pore-scale infiltration processes within larger scale network models of particle transport in granular porous media in a companion paper. Articlenote: Currently at GZA GeoEnvironmental Inc., 1 Edgewater Drive, Norwood, MA 02062, U.S.A.     

Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamic simulation of sedimentation of solid particles in an Oldroyd-B fluid

In this paper we present a two-dimensional numerical study of the viscoelastic effects on the sedimentation of particles in the presence of solid walls or another particle. The Navier-Stokes equations coupled with an Oldroyd-B model are solved using a finite-element method with the EVSS formalism, and the particles are moved according to their equations of motion. In a vertical channel filled with a viscoelastic fluid, a particle settling very close to one side wall experiences a repulsion from the wall; a particle farther away from the wall is attracted toward it. Thus a settling particle will approach an eccentric equilibrium position, which depends on the Reynolds and Deborah numbers. Two particles settling one on top of the other attract and form a doublet if their initial separation is not too large. Two particles settling side by side approach each other and the doublet also rotates till the line of centers is aligned with the direction of sedimentation. The particle-particle interactions are in qualitative agreement with experimental observations, while the wall repulsion has not been documented in experiments. The driving force for lateral migrations is shown to correlate with the pressure distribution on the particle's surface. As a rule, viscoelasticity affects the motion of particles by modifying the pressure distribution on their surface. The direct contribution of viscoelastic normal stresses to the force and torque is not important.     

Particle tracking for polydisperse sedimenting droplets in phase separation

When a binary fluid demixes under a slow temperature ramp, nucleation, coarsening and sedimentation of droplets lead to an oscillatory evolution of the phase-separating system. The advection of the sedimenting droplets is found to be chaotic. The flow is driven by density differences between two phases. Here, we show how image processing can be combined with particle tracking to resolve droplet size and velocity simultaneously. Droplets are used as tracer particles, and the sedimentation velocity is determined. Taking these effects into account, droplets with radii in the range of 4−40 μm are detected and tracked. Based on these data, we resolve the oscillations in the droplet size distribution that are coupled to the convective flow.     

Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method

An efficient immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for fully resolved simulations of suspended solid particles in viscoelastic flows. Stress LBM based on Giesekus and Oldroyd-B constitutive equation are used to model the viscoelastic stress tensor. A boundary thickening-based direct forcing IB method is adopted to solve the particle–fluid interactions with high accuracy for non-slip boundary conditions. A universal law is proposed to determine the diffusivity constant in a viscoelastic LBM model to balance the numerical accuracy and stability over a wide range of computational parameters. An asynchronous calculation strategy is adopted to further improve the computing efficiency. The method was firstly applicated to the simulation of sedimentation of a single particle and a pair of particles after good validations in cases of the flow past a fixed cylinder and particle migration in a Couette flow against FEM and FVM methods. The determination of the asynchronous calculation strategy and the effect of viscoelastic stress distribution on the settling behaviors of one and two particles are revealed. Subsequently, 504 particles settling in a closed cavity was simulated and the phenomenon that the viscoelastic stress stabilizing the Rayleigh–Taylor instabilities was observed. At last, simulations of a dense flow involving 11001 particles, the largest number of particles to date, were performed to investigate the instability behavior induced by elastic effect under hydrodynamic interactions in a viscoelastic fluid. The elasticity-induced ordering of the particle structures and fluid bubble structures in this dense flow is revealed for the first time. These simulations demonstrate the capability and prospects of the present method for aid in understanding the complex behaviors of viscoelastic particle suspensions.     

An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid

We present detailed experimental results examining “negative wakes” behind spheres settling along the centerline of a tube containing a viscoelastic aqueous polyacrylamide solution. Negative wakes are found for all Deborah numbers (2.43≤De(˙γ)≤8.75) and sphere-to-tube aspect ratios (0.060≤a/R≤0.396) examined. The wake structures are investigated using laser-Doppler velocimetry (LDV) to examine the centerline fluid velocity around the sphere and digital particle image velocimetry (DPIV) for full-field velocity profiles. For a fixed aspect ratio, the magnitude of the most negative velocity, U _min , in the wake is seen to increase with increasing De. Additionally, as the Deborah number becomes larger, the location of this minimum velocity shifts farther downstream. When normalized with the sphere radius and the steady state velocity of the sphere, the axial velocity profiles become self-similar to the point of the minimum velocity. Beyond this point, the wake structure varies weakly with aspect ratio and De, and it extends more than 20 radii downstream. Inertial effects at high Reynolds numbers are observed to shift the entire negative wake farther downstream. Using DPIV to investigate the transient kinematic response of the fluid to the initial acceleration of the sphere from rest, it is seen that the wake develops from the nonlinear fluid response at large strains. Measurements of the transient uniaxial extensional viscosity of this weakly strain-hardening fluid using a filament stretching rheometer show that the existence of a negative wake is consistent with theoretical arguments based on the opposing roles of extensional stresses and shearing stresses in the wake of the sphere. Received: 10 November 1997 Accepted: 1 May 1998     

Apparent wall slip velocity measurements in free surface flow of concentrated suspensions

In this work we have experimentally measured the apparent wall slip velocity in open channel flow of neutrally buoyant suspension of non-colloidal particles. The free surface velocity profile was measured using the tool of particle imaging velocimetry (PIV) for two different channels made of plane and rough walls. The rough walled channel prevents wall slip, whereas the plane wall showed significant wall slip due to formation of slip layer. By comparing the velocity profiles from these two cases we were able to determine the apparent wall slip velocity. This method allows characterization of wall slip in suspension of large sized particles which cannot be performed in conventional rheometers. Experiments were carried out for concentrated suspensions of various particle volume concentrations and for two different sizes of particles. It was observed that wall slip velocity increases with particle size and concentration but decreases with increase in the viscosity of suspending fluid. The apparent wall slip velocity coefficients are in qualitative agreement with the earlier measurements. The effect of wall slip on free surface corrugation was also studied by analyzing the power spectral density (PSD) of the refracted light from the free surface. Our results indicate that free surface corrugation is a bulk flow response and it does not arise from boundary problem such as development of slip layer.     

Effect of particle clustering on the gravitational settling velocity in homogeneous turbulence

Astatistical model of the gravitational settling velocity of clusters in homogeneous isotropic turbulence is developed. The effects of particle hydrodynamic interaction, inertia, size, and volume fraction on the particle settling velocity component attributable to particle clustering are analyzed.     

Sedimentation of particles in shear flows of viscoelastic fluids with fibers

We study the effect of fiber additives on rheology and sedimentation of particle suspensions in a base viscoelastic suspending fluid in the case when the suspension is subjected to shear flow. We found experimentally that fiber additives (3–6 mm in length and 8–12 μm in diameter at a mass fraction of 0–0.4%) increase the suspension viscosity and retard the particle sedimentation significantly. At the same mass concentration, long and thin fibers reduce the sedimentation velocity and increase the viscosity to a much greater extent than short and thick fibers. We revealed that both rheology and sedimentation are controlled by a single conformational parameter (overlap parameter) defined as the number of fibers per unit volume multiplied by fiber length cubed.     

The effect of particle size and migration on the formation of flow-induced structures in viscoelastic suspensions

Flow-induced structures in suspensions containing spheres in viscoelastic suspending media were investigated by microscopy and rheo-optical methods. Suspensions of monodisperse polystyrene spheres with diameters ranging from 1.2 to 2.8 μm and dispersed in aqueous solutions of hydroxypropylcellulose were studied in simple shear flows. Optical microscopy observations as well as small-angle light-scattering (SALS) experiments were performed using a parallel plate geometry. In agreement with previous work, necklaces of particles aligned in the flow direction were observed when shearing faster then a critical shear rate, which was found to be independent of particle size. In contrast to earlier work, however, the role of particle migration was found to be of prime importance. Particles were shown to migrate toward the plates where the particles assembled and aligned in strings running in the flow direction. For the smallest particles (1 μm diameter), the formation of particle doublets or short strings along the vorticity direction was observed at low shear rates, which flipped to an orientation into the flow direction and grew into longer strings at higher shear rates. SALS experiments were used to quantify the degree of alignment and its dependence on particle size, shear rate, and gap. For the system under investigation, the degree of alignment was found to increase with increasing shear rate and particle size and with decreasing gap. The present results suggest that, depending on the details of the suspending medium and the size and nature of the suspended particles, the formation of aligned structures is affected by the relative magnitude of the colloidal and hydrodynamic forces and the kinetics of string formation versus the kinetics of migration.     

Development and validation of a reduced order history force model

The history force model accounts for temporal development in fluid gradients in the viscous region surrounding a particle in point particle methods. The calculation of the history force typically requires storing and using relative velocity information during the life time of the particle. For a large number of particles integrated over large times, history force calculation can become prohibitively expensive. The current work presents a new modeling approach to calculate the history force in which a decay function is applied to a stored cumulative value of the history force. The proposed formulation is equivalent to applying the same function obtained from a constant acceleration assumption to a running average of the acceleration within the memory time of the particle. The new force model is validated with experimental measurements of settling spheres at Reynolds numbers ranging from around one to a few hundreds and at density ratios from 1.2 to about 9.32. More validation work was carried-out with experimental measurements of oscillating spheres at different frequencies and amplitudes, as well as bouncing spheres at different Reynolds numbers and density ratios. The model shows very good agreement with the experiments of settling spheres and reasonable/good agreement with oscillating and bouncing sphere experiments. The proposed model significantly reduces the computational resources required to calculate the history force especially when large number of particles need to be integrated over long times.     

Refractive-index and density matching in concentrated particle suspensions: a review

Optical measurement techniques such as particle image velocimetry (PIV) and laser Doppler velocimetry (LDV) are now routinely used in experimental fluid mechanics to investigate pure fluids or dilute suspensions. For highly concentrated particle suspensions, material turbidity has long been a substantial impediment to these techniques, which explains why they have been scarcely used so far. A renewed interest has emerged with the development of specific methods combining the use of iso-index suspensions and imaging techniques. This review paper gives a broad overview of recent advances in visualization techniques suited to concentrated particle suspensions. In particular, we show how classic methods such as PIV, LDV, particle tracking velocimetry, and laser induced fluorescence can be adapted to deal with concentrated particle suspensions.     

Determination of particle charge and size from the propagation of ultrasound in suspensions

The propagation of small perturbations in raulticomponent disperse media consisting of an uncharged dispersion fluid, positive and negative ions and charged particles or droplets of another fluid is investigated. When weak waves pass through emulsions and suspensions, because of the difference in the velocities of the ions and charged particles a non-uniform distribution of electric potential develops in the medium [1–3]. Expressions relating the amplitude of the electric potential and the amplitude of the fluid velocity in the wave, the particle charge and the parameters characterizing the medium are derived. Relations are obtained for the phase shift between the values of the electric potential and the fluid velocity. It is proposed to use the expressions obtained, which describe the propagation of ultrasound, for the experimental determination of the particle charge and other parameters of the disperse medium, in particular, the particle size.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1988.