# Fully coupled simulations of non-colloidal monodisperse sheared suspensions

**ABSTRACT** In this work we investigate numerically the dynamics of sheared suspensions in the limit of vanishingly small fluid and particle inertia. The numerical model we used is able to handle the multi-body hydrodynamic interactions between thousands of particles embedded in a linear shear flow. The presence of the particles is modeled by momentum source terms spread out on a spherical envelop forcing the Stokes equations of the creeping flow. Therefore all the velocity perturbations induced by the moving particles are simultaneously accounted for. The statistical properties of the sheared suspensions are related to the velocity fluctuation of the particles. We formed averages for the resulting velocity fluctuation and rotation rate tensors. We found that the latter are highly anisotropic and that all the velocity fluctuation terms grow linearly with particle volume fraction. Only one off-diagonal term is found to be non zero (clearly related to trajectory symmetry breaking induced by the non-hydrodynamic repulsion force). We also found a strong correlation of positive/negative velocities in the shear plane, on a time scale controlled by the shear rate (direct interaction of two particles). The time scale required to restore uncorrelated velocity fluctuations decreases continuously as the concentration increases. We calculated the shear induced self-diffusion coefficients using two different methods and the resulting diffusion tensor appears to be anisotropic too. The microstructure of the suspension is found to be drastically modified by particle interactions. First the probability density function of velocity fluctuations showed a transition from exponential to Gaussian behavior as particle concentration varies. Second the probability of finding close pairs while the particles move under shear flow is strongly enhanced by hydrodynamic interactions when the concentration increases.

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**ABSTRACT:**The shear-induced microstructure in non-Brownian suspensions is studied. The Pair Distribution Function in the shear plane is experimentally determined for particle volume fractions ranging from 0.05 to 0.56. Transparent suspensions made of PMMA particles (172µm in diameter) dispersed in a fluorescent index matched Newtonian liquid is sheared in a wide gap Couette rheometer. A thin laser sheet lights the shear plane. The particle positions are recorded and the pair distribution function (PDF) in the shear plane is computed. The PDF at contact is shown to be anisotropic, with a depleted area in the receding side of the reference particle. The angular position of the depleted zone, close to the velocity axis at low particle concentration, is tilted toward the dilatation axis as the volume fraction is increased. At high concentrations (larger than 0.45), the shape of the PDF changes qualitatively with a secondary depleted area in the compressional quadrant of the main flow and a probability peak in the velocity direction. These experimental results are in good agreement with numerical simulations in Stokesian Dynamics where the interaction force between particles has been tuned to reproduce the particle roughness effects.Journal of Rheology 11/2012; 57:273. · 3.28 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Homogenous oil in water dispersion has been investigated in a horizontal pipe. The mean droplet size is 25 μm. Experiments were carried out in a 7.5-m-long transparent pipe of 50-mm internal diameter. The wall friction has been measured and modeled for a wide range of flow parameters, mixture velocities ranging from 0.28 to 1.2 m/s, and dispersed phase volume fractions up to 0.6, including turbulent, intermediate, and laminar regimes. Flow regimes have been identified from velocity profiles measured by particle image velocimetry in a matched refractive index medium. It is shown that the concept of effective viscosity is relevant to scale the friction at the wall of the dispersed flow. Based on mixture properties, the friction factor follows the Hagen-Poiseuille and the Blasius' law in laminar and turbulent regimes, respectively. Interestingly, the transition toward turbulence is delayed as the dispersed phase fraction is increased. © 2010 American Institute of Chemical Engineers AIChE J, 2011AIChE Journal 06/2010; 57(5):1119 - 1131. · 2.58 Impact Factor - SourceAvailable from: Frédéric Blanc[Show abstract] [Hide abstract]

**ABSTRACT:**This paper reports experiments on the shear transient response of concentrated non-Brownian suspensions. The shear viscosity of the suspensions is measured using a wide-gap Couette rheometer equipped with a Particle Image Velocimetry (PIV) device that allows measuring the velocity field. The suspensions made of PMMA particles (31$\mu$m in diameter) suspended in a Newtonian index- and density-matched liquid are transparent enough to allow an accurate measurement of the local velocity for particle concentrations as high as 50%. In the wide-gap Couette cell, the shear induced particle migration is evidenced by the measurement of the time evolution of the flow profile. A peculiar radial zone in the gap is identified where the viscosity remains constant. At this special location, the local particle volume fraction is taken to be the mean particle concentration. The local shear transient response of the suspensions when the shear flow is reversed is measured at this point where the particle volume fraction is well defined. The local rheological measurements presented here confirm the macroscopic measurements of Gadala-Maria and Acrivos (1980). After shear reversal, the viscosity undergoes a step-like reduction, decreases slower and passes through a minimum before increasing again to reach a plateau. Upon varying the particle concentration, we have been able to show that the minimum and the plateau viscosities do not obey the same scaling law with respect to the particle volume fraction. These experimental results are consistent with the scaling predicted by Mills and Snabre (2009) and with the results of numerical simulation performed on random suspensions [Sierou and Brady (2001)]. The minimum seems to be associated with the viscosity of an isotropic suspension, or at least of a suspension whose particles do not interact through non-hydrodynamic forces, while the plateau value would correspond to the viscosity of a suspension structured by the shear where the non-hydrodynamic forces play a crucial role.Journal of Rheology 04/2011; 55:835. · 3.28 Impact Factor

Page 1

FULLY COUPLED SIMULATIONS OF

NON-COLLOIDAL MONODISPERSE

SHEARED SUSPENSIONS

M. Abbas1, E. Climent1,?and O. Simonin2

1Laboratoire de Ge ´nie Chimique, Toulouse, France.

2Institut de Me ´canique des Fluides, Toulouse, France.

Abstract: In this work we investigate numerically the dynamics of sheared suspensions in the

limit of vanishingly small fluid and particle inertia. The numerical model we used is able to

handle the multi-body hydrodynamic interactions between thousands of particles embedded in

a linear shear flow. The presence of the particles is modelled by momentum source terms

spread out on a spherical envelop forcing the Stokes equations of the creeping flow. Therefore

all the velocity perturbations induced by the moving particles are simultaneously accounted for.

The statistical properties of the sheared suspensions are related to the velocity fluctuation of

the particles. We formed averages for the resulting velocity fluctuation and rotation rate tensors.

We found that the latter are highly anisotropic and that all the velocity fluctuation terms grow lin-

early with particle volume fraction. Only one off-diagonal term is found to be non zero (clearly

related to trajectory symmetry breaking induced by the non-hydrodynamic repulsion force).

We also found a strong correlation of positive/negative velocities in the shear plane, on a

time scale controlled by the shear rate (direct interaction of two particles). The time scale

required to restore uncorrelated velocity fluctuations decreases continuously as the concen-

tration increases. We calculated the shear induced self-diffusion coefficients using two different

methods and the resulting diffusion tensor appears to be anisotropic too.

The microstructure of the suspension is found to be drastically modified by particle inter-

actions. First, the probability density function of velocity fluctuations showed a transition from

exponential to Gaussian behaviour as particle concentration varies. Second, the probability of

finding close pairs while the particles move under shear flow is strongly enhanced by hydrodyn-

amic interactions when the concentration increases.

Keywords: hydrodynamic interactions; force coupling method; dynamic simulation; shear

induced self-diffusion; particle velocity fluctuations.

INTRODUCTION

Understanding the dynamics of suspended

solid particles embedded in a viscous carrying

fluid is assuming greater importance in indus-

trial applications, driven by both environ-

mental issues and optimization of operating

costs in Chemical Engineering. Liquid–solid

dispersions may be fluid, paste or solid,

depending on the volumetric concentration

of solid and the surface interactions of par-

ticles. In moderately concentrated suspen-

sions (volumetric concentration up to 20%),

a relative motion of the particles is generally

always possible and the whole suspension

behaves as a fluid with effective properties.

Practical applications may be found in water

waste treatment, slurry flows and even in

biotechnologies. Even paste mixtures may

be subject to modification of rheological

properties, showing shear-thinning or shear-

thickeningbehaviourdependingon the

physicochemical treatment that controls par-

ticle interactions.

Sheared suspensions are sites for migration

and mixing of particles in various viscometer

flows.Weareconcernedonlywithsuspensions

of macroscopic particles leading to negligible

effect of interparticle forces through surface

interactions. Considering a suspension of

monodisperseparticlesembeddedinasheared

viscous fluid, the physics of interactions is

basically characterized by the following dimen-

sionlessparameters:

Archimedes and Pe ´clet numbers. The Stokes

number St ¼ rpGa2/9m evaluates the ratio

between the time required by the suspended

particlestorelaxafterasuddenlocalfluctuation

of the surrounding flow and the fluid time scale

G21(G being the shear rate) and the time

required by the suspended particles to relax

after a sudden local fluctuation of the surround-

ing flow (rpand a are respectively the density

and the radius of the particles). Particles and

Stokes,Reynolds,

Page 2

fluid usually have different densities. Then, the Archimedes

number (Ar ¼ ga3rf(rp–rf)/mf

buoyancy and viscous forces. Due to the buoyancy force,

particles may settle under gravity with a typical velocity V. The

Reynolds (Re ¼ rfaV/mf) compares the inertial to viscous

effects inthe flow, rfand mfbeing the fluid density and viscosity.

Inthecaseofsmallparticles,Brownianagitationmaycontribute

tothe evolutionofthe microstructure of the suspension. Hence,

thePe ´cletnumber(Pe ¼ 6pmfV/kBT)comparestheconvective

motion of the particles to their thermal agitation energy kBT.

Other non-dimensional parameters may be formed by combi-

nation of these numbers. As we neglect surface interactions,

the magnitude of attraction and repulsion forces does not

need to be compared to inertia or viscous effects.

The dependence of the dimensionless numbers on the par-

ticle size and liquid kinematic viscosity in uniformly sheared

suspensions (G ¼ 10 s21is a typical shear rate of classical

applications) is provided in Figure 1. Horizontal lines

correspond to a constant liquid viscosity varying between

1026m2s21(water) and 1024m2s21(oil). Constant particle

radii are marked by vertical dashed lines. Solid lines corre-

spond to dimensionless numbers equal to unity. Brownian

motion has to be considered when the suspension properties

are located on the left of the line Pe ¼ 1 and can be neg-

lected when Pe ?1. Particle trajectories are weakly influ-

enced by gravity and inertia effects on the left of Ar/Re ¼ 1

and St ¼ 1, respectively, while the particle behaviour is con-

trolled by buoyancy and inertial effects on the right of these

lines. For the line Re ¼ 1, suspensions standing on the left

are described by Stokes equations, whereas Navier–Stokes

equations describe the fluid motion on the right of this line.

In this paper, we aim at investigating numerically the beha-

viour of non-colloidal liquid–solid sheared suspensions of

neutrally buoyant particles in the limit of creeping flow. The

suspensions are macroscopically homogeneous and the par-

ticle inertia and buoyancy are neglected. Following the non-

dimensional analysis (Figure 1), such conditions are fulfilled

for a variety of suspensions characterized by particle diam-

eter in the range O (1 mm) suspended in a viscous liquid

(kinematic viscosity higher than 1024m2s21). Designing

processes where suspensions have to be fluidized and

2) compares the relative effect of

transported is often based on the knowledge of the macro-

scopic behaviour of this complex two-phase flow. Chemical

engineers need to model the effective physical parameters

that control the response of the system to various flow con-

figurations. Both effective viscosity in the bulk and dispersion

coefficient are often evaluated by semi-empirical law. The

simulations provide an efficient way to relate the local micro-

structure of the flow to the global quantities which control the

overall behaviour. In order to achieve a clear understanding

of the physics of suspension in flow with velocity gradients,

we propose to select the fundamental configuration of a pure

linear shear flow of monodisperse spherical particles. The

flow is unbounded and there are no geometry constraints,

such as the presence of walls, to modify the statistic homogen-

eity of the suspension. In that situation, the statistics are only

functions of the volumetric concentration of solid in the bulk.

The velocity gradient in a sheared suspension generates

relative motions of particles and therefore hydrodynamic

interactions. An analytical investigation of the hydrodynamic

interactions between a pair of particles began with the pio-

neering work of Batchelor and Green (1972). The flow can

be split in a far field interaction and a short range velocity per-

turbation accounting for the lubrication effect. Interactions

between three spheres were studied by Wang et al. (1996)

using far-field and near-field asymptotic velocity expressions.

Lately, Drazer et al. (2004) have investigated the microscopic

structure (pair probability distribution function) and macro-

scopic transport properties of sheared suspensions (transla-

tional and rotational velocity fluctuation tensors) from low to

high concentration in the case of homogeneous monodis-

perse suspensions.Although

method they used gave an accurate solution of the Stokes

equations, the authors were limited to simulating the simul-

taneous motion of a few hundred particles because of large

time requirement of the numerical approach. They found

that a sheared suspension where the particles are initially

randomly distributed remains homogeneous in time, but

pairs of interacting particles are promoted by the shear,

especially when the concentration increases. The fore-aft

symmetry of the pair probability distribution function is

broken by the effect of non-hydrodynamic repulsive forces

when approaching close particles. They highlighted the

difference between the statistics of dynamic simulations

where the particles driven by the shear flow are free to

move when time goes on and the static simulations where

the averages are formed over different frozen random particle

distributions. In both cases velocity fluctuations are anisotropic

and they follow the dilute limit theory estimate based on pair-

wise interactions up to a 15% volume concentration. However,

the fluctuations resulting from dynamic interactions are higher.

In a former work, Drazer et al. (2002) showed that the prob-

ability distribution function of the velocity fluctuations have an

exponential shape at low concentrations as a signature of

long term correlated structures. They appeared to be

Gaussian at high concentrations where hydrodynamic

screening is achieved by the random multi-body interactions.

Lagrangian velocity autocorrelation functions computed by

Marchioro and Acrivos (2001) and Drazer et al. (2002)

decay with time and have a negative loop located around a

characteristic time scale of order 1/G (where G is the flow

shear rate). This scaling shows that particle trajectories are

basically controlled by pair encounters at low concentration.

The velocity fluctuations remain correlated for a long time

theStokesian dynamics

Figure 1. Dependence of the dimensionless numbers on the suspen-

sion parameters (particle radius and kinematic viscosity). Each

oblique solid line delimits the regions where the dimensionless num-

bers are respectively lower and higher than unity.

Page 3

scale (of order 8/G). A chaotic evolution of the suspension

follows the loss of correlation, which is known to lead to the

shear induced diffusion even at high Pe ´clet number. Diffusion

in such conditions has been studied in many works, starting

from the reference experimental work of Leighton and

Acrivos (1986), to analytical calculations of Wang et al.

(1996) based on three particle interactions, to new experi-

mental techniques developed by Breedveld et al. (1998,

2001, 2002) and numerical simulations of Marchioro and

Acrivos (2001), Drazer et al. (2002), Sierou and Brady

(2004). Self-diffusion coefficients reported in these works

are anisotropic and strongly increasing with the suspension

concentration. Results obtained from different works achieve

only a qualitative agreement.

In this paper, we propose to use the force coupling method

as a numerical model to solve the Stokes equations account-

ing for the velocity perturbations induced by hydrodynamic

interactions of particles. This is a fully coupled numerical

model, simulating dynamic interactions between thousands

of particles under the approximation of Stokes flow. First,

we briefly describe the numerical model which has been vali-

dated in various configurations of Stokes flows. We propose

some quantitative tests which control the accuracy of the

model in the present configuration of a linear shear flow. In

the following sections, macroscopic quantities are investi-

gated by computing the velocity fluctuations, their probability

density function and Lagrangian autocorrelation functions.

Self-diffusion coefficients are then calculated. Aiming to look

at the suspension from a microscopic level, we computed

statistics highlighting the preferential orientation of interacting

pair of particles. We compare our results on statistical quan-

tities to the work of Drazer et al. (2002, 2004) which used

the Stokesian dynamics to compute interactions in the same

context of suspension dynamics. Finally, we conclude on the

ability of the force coupling method for simulating the flow of

suspensions and propose perspectives to this study.

THE FORCE COUPLING METHOD

The complexity of dispersed two-phase flows is related to

the numerous length scales that have to be resolved simul-

taneously. In the case of Stokes flows, the velocity disturb-

ance induced by a single particle falls off very slowly and

thenmulti-bodyhydrodynamic

evolution of the suspension. We propose to use a numerical

model, which is able to couple simultaneously the solution of

fluid flow equations and the Lagrangian tracking of the

particles. The force coupling method (FCM) is based on a

low order multipole expansion of the velocity disturbance

induced by the presence of particles. The equations of the

fluid motion are solved directly and the forcing term is mod-

eled by a spatial source of momentum added to the Stokes

equations. The accuracy of the model increases as we add

higher order terms in the multipole expansion, but solving

the equations becomes more time consuming.

interactions controlthe

Model Equations

Details on the theoretical background of the FCM can be

found in the paper of Maxey and Patel (2001). We describe

briefly the basic equations. We consider that the fluid is

incompressible [equation (1)], has a constant viscosity m,

and we neglect its inertia. The fluid velocity field u(x, t) and

the pressure p(x, t) are solutions of the Stokes equations

[equation (2)].

r ? u ¼ 0

0 ¼ ?rp þ mr2u þ f(x, t)

(1)

(2)

The forcing term on the right hand side of equation (2) is a

spatial distribution of momentum which is induced by the pre-

sence of the moving particles. This term is spatially and tem-

porally evolving while the particles are freely moving under

hydrodynamic interactions. Its expression is based on theor-

etical analysis of low Reynolds number flows. We consider

only two terms [see equation (3)] of the multipole expansion

of finite source terms, namely the force monopole (Stokeslet)

and the force dipole.

fi(x, t) ¼

X

þ G(n)

NB

n¼1

F(n)

iD½x ? Y(n)(t)?

ij

@

@xjD0½x ? Y(n)(t)?

(3)

The NBparticles are centered at locations Y(n)(t) and the

source terms are spread out on the flow field using finite

size envelopes [equation (4)]. The width of the Gaussian

envelopes [s for D(x) and s0for D0(x), respectively] are

related to the particle radius a by analytic expressions with

D(x) ¼ (2ps2)?3=2e(?jxj2=2s2)

(4)

The magnitude of the interaction force F(n)is directly related

to the force acting by the fluid on the considered particle. It is

a combination of buoyancy, inertia effect and an external

force Fext. [equation (5)], and it cancels for non-buoyant

particles driven by the shear flow at low Stokes number.

F(n)¼ (mP? mF) g ?dV(n)

dt

!

þ F(n)

ext

(5)

Gij

anti-symmetric parts (Gij

part (namely, the Stresslet) contributes to enforce a solid

body rotation within the fluid occupied by the particle. The

iterative scheme on a steepest descent scheme that was

used to enforce a zero strain rate within the particle volume

is described in details in the work of Dance and Maxey

(2003). The anti-symmetric part Aij

torque T(n)acting on the particle n [equation (6)] where A(n)

1

21ijkT(n)

(n)is a tensor which can be split into symmetric and

(n)¼ Sij

(n)þ Aij

(n)). The symmetric

(n)is related to the external

ij

¼

T(n)¼ ?(IP? IF)

dV(n)

dt

!

þ T(n)

ext

(6)

mP(resp. mF) is the mass of the particle (resp. fluid) volume

and IP(resp. IF) is the particle (resp. fuid) rotational inertia.

Throughout the paper, both the translational and rotational

inertia of particles will be neglected. Neglecting particle iner-

tia restricts the scope of our study to the important class of

problems related to solid–liquid suspensions with moderately

density ratio. The ratios a/s and a/s0are set as to match

respectively Stokes drag for an isolated sphere (radius a)

Page 4

and to ensure an average zero rate of strain [equation (7)]

within the volume occupied by the particle.

S(n)

ij

¼1

2

ð

@ui

@xjþ@uj

@xi

??

D0x ? Y(n)(t)

??

d3x ¼ 0 (7)

These requirements are fulfilled exactly in the limit of Stokes

approximation while a/s and a/s0are set analytically for a

Gaussian shaped envelope: a/s ¼pp and a/s0¼ (6pp)1/3

(see details in Maxey and Patel, 2001; Lomholt and Maxey,

2003). In a sheared suspension, the monopole is negligible

for non-buoyant spheres, and the Stresslet will be the major

contribution.

Particles move freely in a Lagrangian framework as their

trajectory equations are solved simultaneously. Particle vel-

ocities and rotation rates are obtained with a spatial filtering

of the flow velocity field based on the spherical Gaussian

envelop [equations (8) and (9)].

V(n)(t) ¼

ð

ð

u(x, t)D x ? Y(n)(t)

hi

d3x

(8)

V(n)(t) ¼ r ? u(x, t)D0x ? Y(n)(t)

hi

d3x

(9)

Then, the trajectory of each particle is computed by integrat-

ing equation (10).

dY(n)

dt

¼ V(n)(t)(10)

More details on the theoretical background and an extensive

validation of the method are available in Maxey and Patel

(2001), Lomholt et al. (2002), Lomholt and Maxey (2003).

Validation Tests and Accuracy

The multipole decomposition truncated to the first order

(force monopole) is not sufficient when near field interactions

have to be accounted for. Therefore, the flow resolution may

be improved by incrementing the order of the multipole

decomposition. The accuracy of the FCM is increased by

adding the force dipole in the source term. An iterative

scheme is used to enforce a zero rate of strain within the

volume occupied by the particles.

Neutrally buoyant particles moving in a shear flow experi-

ence only the symmetric dipole forcing. When a single

particle is seeded in a linear shear flow, it is driven by

the mean flow velocity (i.e., V1¼ Gx2). The presence of the

solid particle which is freely rotating induces a velocity pertur-

bation in the fluid flow. The perturbation velocity field has a

fore-aft symmetry. It is strained along the compression axis

of the mean flow (in the shear plane). The velocity pertur-

bation related to an isolated single particle in an unbounded

fluid was calculated analytically in the paper of Batchelor and

Green (1972). Comparing the velocity profiles with our

numerical simulations, we observed that the far-field approxi-

mation is perfectly reproduced and only slight differences

appear close to the particle surface. The agreement is

within few percent when the distance to the centre is larger

than 1.25 particle radius. Obviously the discrepancy is

caused by the representation of particles by distributed

momentum source terms without an actual separation

between the fluid and the solid body rotation. The presence

of the particles is enforced by the constraint [equation (7)]

on the local strain rate of the velocity perturbation.

The same authors have also determined analytical

expressions for the relative velocity [equation (11)] and

rotation rate [equation (12)] for an interacting pair of particles

in a shear flow. The velocities are expressed in terms of

three non dimensional scalar functions (A, B and C) which

depend only on the particle non-dimensional separation

distance r/a.

V(r) ¼ ?G

r2B=2 þ r2

r1B=2 þ r2

r1r2r3=r2(A ? B)

r1r3=r2

?r2r3=r2

(r2

>:

1=r2(A ? B)

2=r2(A ? B)

?

?

?

?

8

>:

><

(11)

V(r) ¼ ?CG

2

2? r2

1)=r2

8

><

(12)

r ¼ (r1, r2, r3) is the separation distance between the two

particle centres and G is the shear rate of the flow. The direc-

tion 1 is for the mean flow direction, 2 is for the direction of

shear and 3 is for the vorticity direction normal to the plane

of shear. The evolutions of A, B and C are shown in

Figure 2 for two equal spheres. The numerical simulations

are in good agreement with the analytical expressions

when the gap between the two particles is larger than 25%

of the particle radius. For extremely close particles, the

lubrication effects should be accounted for with more accu-

racy (short-range hydrodynamic interactions). Based on a

parameterization developed by Dance and Maxey (2003),

we showed in a former work (Abbas et al., 2006) that when

adding the lubrication forces to the FCM in a pair-wise

additive manner, the calculation of the interaction between

close particles is significantly improved. The accuracy of

the numerical approach limits the scope of our study to

moderately concentrated suspensions.

Figure 2. The dependence on the separation distance between

particle centers (j ¼ r/a) of the hydrodynamic parameters. Solid

lines: analytic solution; dashed lines: far-field approximation; symbols

are FCM results. Filled circles: A; empty circles: B; plus: C.

Page 5

Repulsion Barrier

The FCM is only an approximate model since fluid

occupies the whole domain and no-slip boundary conditions

on the particle surface are not strictly imposed. In Stokes

flow, actual contact between particles is very limited because

lubrication effects drastically slow down the approach of par-

ticles. A repulsive force Fbis added to the monopole coupling

term when the distance between the particle centres r is less

than a prescribed cut-off separation distance Rref. This repul-

sive barrier [equation (13)] is used to fix a numerical inaccur-

acy of the model by preventing particles from overlapping

(Figure 3).

Fb¼ ?Fref

2a

R2

R2

ref? 4a2

ref? r2

"#2

r

(13)

Once it is switched on, the repulsion barrier drives particles

rolling on the surface of each other preventing them from

overlapping (Figure 3). Different tests were carried out to

verify that the trajectories are only slightly modified by large

variation of the cut-off distance Rrefand the force scale Fref.

Whenthecut-offdistance

(Rref¼ 2.2a to Rref¼ 2.02a), the minimum gap between the

particle surfaces varied but no significant difference was

observed on the particle trajectories. The time step has to

be reduced (in a one-fifth ratio) so that the short range inter-

actions are well resolved. All the simulations analysed in this

paper were done with Rref¼ 2.2a leading to a minimum gap

smaller than one percent of the particle radius.

Although this force is used for preventing particle overlap, it

could be interpreted in terms of physical considerations. Even

when particles in solid–liquid suspensions are not charged

they generally experience a short range interparticle force due

to the double layer electric repulsion (DLVO-type) when the

gap is shorter than 1022a. The suspensions that we study

may be considered as stabilized suspensions while a strong

repulsive force prevent the formation of permanent cluster of

particles. Short range attraction forces, such as the Van der

isconsiderablyreduced

Waals potential, are screened by hydrodynamic lubrication

forceandtherepulsivebarrier.Thephysicalanalog ofoursimu-

lations would be the case where electrostatic repulsive forces

(proportionaltothefluidpermittivity1r10andtothesquareofpar-

ticle surface potential c2) are overcoming the Van der Waals

attraction force (related to the Hamaker constant L).

A practical example of suspension that fits the assumptions

of our simulations may be composed of polystyrene latex

spheres of 2 mm radius in a 50% glycerol-in-water mixture con-

taining 1023mol l21KCl, undergoing a flow shear rate of

1 s21. A DLVO-type interaction is characterized by the particle

surface potential c ? 4 mV and a Debye length k21¼ 88 A˚

giving 1r10? 5.9 ? 10210C2Jm21(Brady and Bossis, 1985).

The dynamic fluid viscosity and density are respectively

about4 ? 1023Pa s

suspension,thenon-dimensional

L ? 15 and 2p1r10c2a/mfGa3? 3.5 confirm that the electro-

static repulsive potential overcomes the attraction and shear-

induced energies. We calculated the Archimedes/Reynolds,

Stokes, Reynolds and Pe ´clet numbers and obtained respecti-

vely 8.5 ? 1023, 2.4 ? 1027, 1026and 160.

At these scales, the effect of roughness and residual

Brownian motion may be an issue because the particle

radius is O (1 mm). These phenomena would also behave

as a repulsive contribution and they can be simply modelled

by a repulsive force, although the precise magnitude and

form are unknown. When particles are far enough r . Rref,

the repulsion barrier [equation (13)] is switched off and the

dipole terms only control the motion of the particles. While

the gravity and the particle inertia are neglected, the Stokeslet

is always zero unless when the interaction forces F(n)

switched on: r , Rref. This rough treatment of the near-field

hydrodynamics of particles near contact may restrict the

accuracy of the model. We investigate only low to moderately

concentrated suspensions (volumetric concentration lower

than 20%). Careful tests in similar conditions have shown

that the impact of this repulsive force on the overall dynamics

of the suspension is weak (Dance et al., 2004; Da Cunha and

Hinch, 1996).

and 1.126 kg m23. For sucha

numbers2p1r10c2a/

ext is

Information on the Simulations

Simulations are performed in a cubic domain whose width

L is kept constant and equal to 2p. Various volumetric con-

centrations of the suspension are obtained by changing the

particle number in the domain, typically 3200 particles for a

12% concentration. The particle diameter is kept constant

for different concentrations, and extends over six grid

nodes. Stokes equations are solved with a spectral Fourier

algorithm and particle tracking is achieved through a fourth

order Adams-Bashforth scheme. In order to preserve the

homogeneity of the suspension flowing under a linear

shear, we impose periodic boundary conditions in the three

directions. Most simulations have been carried out with

L/a ¼ 48, with 1283mesh grids so that the influence of the

periodic images of any particle in the domain is considerably

reduced. The numerical scheme used for the solution of the

Stokes equations takes advantage of the periodic boundary

conditions by using Fast Fourier Transforms. We used a

domain decomposition algorithm to achieve scalable per-

formance on parallel supercomputers. Typical runs need

four processors and statistics are converged over a 30-h

computation.

Figure 3. Relative two particle trajectories in a shear flow for different

values of the repulsion barrier parameters. Filled circle: surface of the

reference particle. Dashed circle: overlapping limit. Thin solid line:

Fref¼ 0. Thick solid line: Fref¼ 0.3, Rref¼ 1.02, dt ¼ 2 ? 1023—

dashed line: Fref¼ 0.3, Rref¼ 1.2, dt ¼ 1022

Page 6

The Stokes equations are linear, so we only solve the flow

perturbation induced by the presence of the particles (Stress-

let contributions) and superimpose the linear shear flow

u1¼ Gx2(G is the shear rate) on the particle motions. Stati-

stical information can be obtained by means of two distinct

procedures: a ‘static’ one where the averages are formed

on independent sets of random seeding of the particles and

a ‘dynamic’ one where the averages are computed while

the suspension evolves in time. The static procedure may

seem unusual. Stokes equations are not intrinsically time

dependent (no inertia effect) because the velocity field

depends only on the relative positions of the particles.

Then, in a ‘static’ simulation, particles are randomly seeded

with non-overlapping positions in the sheared suspension.

For each independent configuration, the computation of

hydrodynamic interactions provides the flow velocity distri-

bution and consequently the particle velocities [see equations

(1), (2), (8) and (9)]. These simulations can be regarded as a

model of the perfectly random microstructure.

The ‘dynamic’ way to form averages accounts for the evol-

utionintimeofthesuspension.Inthatcase,particlesareinitially

seededatrandompositions.Then,trajectoriesarecomputedas

a sequence of fully coupled interactions between the fluid and

the particles [equations (1), (2), (8), (9) and equation (10)

additionally]. The trajectories are integrated with a constant

time step 5 ? 1023G21. This corresponds to a decrement of

a/100oftheseparationdistancebetweentwoapproachingpar-

ticles. When a particle exits the simulation domain from the

bottom (resp. upper) boundary, it appears on the opposite

side, and its velocity must be adjusted by adding (resp. sub-

stracting) the local flow velocity GL. This is equivalent to apply-

ing the shear in a dynamic way by means of the Lees-Edwards

boundary conditions (Allen and Tildesley, 1987). Ensemble

averages are formed over all the particles as time goes on.

The typical length, time and velocity scales are a, G21and

aG,whereaistheparticleradius.Typically,wesimulatethesus-

pension flow during a dimensionless time Gt proportional to

100 f21/3(f is the volumetric suspension concentration).

During this time, a particle is expected to experience enough

interactions with other particles to achieve its steady statistical

regime. It has been clearly pointed out that the determination

of self-diffusion coefficients needs very long time series for

reaching the diffusive behaviour (Sierou and Brady, 2004) of

the suspension and this point will be carefully checked later.

In this paper, velocity fluctuations and microstructure

organizations were calculated and compared for both static

and dynamic simulations. However, statistical quantities like

velocity autocorrelations and shear-induced diffusion are

obviously calculated only in dynamic suspensions after a

long time of shearing. We checked that the results do not

depend on the initial random seeding when a 1283grid is

used. Simulations performed with smaller resolution (and

consequently fewer particles) need to be averaged on distinct

initial seeding. This was a shortcoming of previous studies

using Stokesian dynamics which is more time consuming.

VELOCITY FLUCTUATIONS AND MICROSTRUCTURE

OF THE SUSPENSION

Velocity Fluctuations

Although the suspension is globally homogeneous, par-

ticles are not moving with the same instantaneous velocities.

At a scale related to a few particle radii, velocity perturbations

are not uniform in the suspension as they strongly depend on

the relative particle positions. If isolated, or far from each

other, the particles would be driven by the local unperturbed

velocity of the flow. However, in a sheared suspension the

particle velocities are subject to fluctuations induced by

their interaction with the fluid and with each other. The deter-

mination of the exact perturbations due to hydrodynamic

interactions between a pair of particles has been done ana-

lytically by Batchelor and Green (1972). But when more

than two particles are involved in the flow, the equations

become very complicated and no analytical solution exists

(only far-field approximations can be achieved) (Wang

et al., 1996). Based on the work of Batchelor and Green

(1972) and on the explicit relations given by Da Cunha and

Hinch (1996), Drazer et al. (2004) have predicted theoreti-

cally the evolution of the translational and rotational fluctu-

ation tensors [equations (14) and (15)] in the dilute regime.

They used two approximations for the microstructure of the

suspension. In the dilute limit, the purely random pair prob-

ability density function (derived from ‘static’ simulations)

models relative positions of the particles that are not corre-

lated. On the other hand, the pair probability density function

derived by Batchelor and Green (1972) is accounting for

hydrodynamic interactions in a shear flow (‘dynamic’ simu-

lations). Following symmetry arguments in a dilute suspen-

sion, the diagonal terms T11and T22of the dimensionless

translational fluctuation tensors (resp. w11 and w22 for

rotation) are equal, and different from T33(resp. w33). The

off-diagonal terms are strictly zero when the fore-aft sym-

metry is preserved (purely random static suspension).

(Ga)2Tij ¼,vivj. ? ,vi.,vj.

G2wij ¼,GiGj. ? ,Gi.,Gj.

(14)

(15)

In equations (14) and (15), v and G are, respectively, the

translational and rotational particle velocity perturbation (the

difference between the instantaneous velocity of the particle

V (resp. V) and the local unperturbed fluid velocity

u1¼ Gx2(resp. w3¼ (r ? u).e3). ,. stands for averages

in time and over all the particles. We verified that ,vi. and

,Gi. are vanishingly small when averages are formed

over long time series [,vi.,vj./,vivj. ¼ O (1025)].

We first simulate the evolution of the fluctuation tensors

with the concentration in a static configuration. Averages

are formed over more than 100 uncorrelated random seeding

of the particles. In Figure 4(c) and (d), translational and

rotational velocity fluctuations are compared to the theoretical

prediction of Drazer et al. (2004), based on pairwise inter-

actions for a purely random pair probability function. The fluc-

tuations scale linearly with the concentration (especially for

low suspension concentration) and they are highly anisotro-

pic. The fluctuations in the flow and shear directions (T11

and T22, resp. w11 and w22) are equal. The translational

(resp. rotational) fluctuation in the spanwise direction T33

(resp. w33) is nearly four times lower (resp. larger) than the

fluctuations in the other directions. The highest velocity fluc-

tuations take place in the plane of shear. All these results are

in good agreement with the theory.

The behaviour of all the diagonal terms of the velocity fluc-

tuations is similar in both dynamic and static simulations

when the concentration increases [Figure 4(a) and (b)].

Page 7

However, velocity fluctuations resulting from shear-induced

particle interactions are nearly two times larger than in the

static simulations. A linear scaling was expected to occur up

to moderately concentrated suspension (20%) as the velocity

perturbation induced by the Stresslet contribution decays like

1/r2. Fluctuations for extremely low volume fractions in the

dynamic simulations were not calculated since the corre-

sponding computations need a very long time to converge.

While the theoretical prediction assumes fore-aft symmetry

of the relative trajectory of a particle pair, zero off-diagonal

terms are expected. Figures 5(a) and (b) confirm the theory

for all off-diagonal terms except T12and w12in the case of

dynamic simulations. According to the work of Drazer et al.

(2004) we found that T12(resp. w12) is negative (resp. posi-

tive) and its magnitude increases with the concentration. As

off-diagonal terms are not zero, it suggests that a symmetry

breaking occurs following the particle interactions. This is

induced by the repulsive non-hydrodynamic force but also

due to multi-body interactions. The symmetry breaking is

clear in Figure 6 where the particle positions close to contact

(particle centers closer than 2.5a are recorded on the plot)

Figure 4. Diagonal terms of the (a) translational and (b) rotational

velocity fluctuation tensors versus concentration with dynamic

simulation conditions. Stars: T11 and w11—filled circles: T22 and

w22—filled triangles: T33 and w33. Solid lines: dilute limit theory

based on the pair probability function of Batchelor and Green

(1972). Dashed lines: dilute limit theory assuming a random distri-

bution in a static simulation. Diagonal terms of the (c) translational

and (d) rotational velocity fluctuation tensors versus concentration

with static simulation conditions. Superimposed filled circles and dia-

monds: (c) T11and T22, (d) w11and w22—Stars: (c) T33and (d) w33.

Solid and dashed lines: dilute limit theory assuming a random distri-

bution in a static simulation.

Figure 5. Off-diagonal terms of the (a) translational and (b) rotational

velocity fluctuation tensors versus concentration with dynamic simu-

lation conditions. Filled diamonds: T12and w12.

Page 8

appear to be highly anisotropic. The main contribution of the

shear flow is to enhance interactions along the compression

axis. Then, the repulsion barrier leads to depletion in the

receding side of the reference particle. Such a symmetry

breaking enhances the occurrence of negative (resp. posi-

tive) cross-products v1v2(resp. G1G2) of the velocity pertur-

bations [see equations (11) and (12)].

Pair Probability Density Function

Contrary to purely random seeding where particle pairs do

not have any angular preferential orientation (T12¼ 0,

w12¼ 0) it is clear from Figure 6 that an anisotropic angular

structure is developing in time in a sheared suspension

(T12, 0, w12. 0). A large number of particle pairs are

oriented along the compression axis of the flow compared

to the depletion of the receding side. Hydrodynamic inter-

actions increase when the particles approach from each

other driven by the flow velocity gradient, and particle pairs

are found to remain in close vicinity for an extended time

before separation occurs. The depletion of the particle pairs

on the receding side is enhanced by the non-hydrodynamic

repulsive force even if the lubrication force acting when par-

ticles are close to contact should restore the symmetry.

Such weak irreversible effects, which are present at a micro-

scopic scale have a measurable impact on the macroscopic

structure of the suspension and consequently on the effective

quantities such as velocity fluctuations T12 and shear-

induced self-diffusion as will be discussed later (Zarraga

and Leighton, 2001). The radial and angular pair probability

density function is not proposed in this paper since it needs

very long simulation time to converge. However, it is interest-

ing to quantify the microscopic structure organization, at least

by calculating the radial dependence of the pair probability

density function. It is a quantitative measure of the probability

of finding a particle at a separation distance r.

Based on the analytic expressions of the relative velocity of

particle pairs in a Stokes shear flow, Batchelor and Green

(1972) have theoretically calculated the particle trajectories

and the pair probability density function. They showed that

closed trajectories exist leading to a divergent evolution of

g(r) at short separation distances. But actually, the non-

hydrodynamic interparticle interactions such as surface

roughness, repulsive forces or even hydrodynamic inter-

actions between more than two particles, generate a transfer

of particles across the streamlines which cancels the prob-

ability of finding closed trajectories. Therefore the pair prob-

ability function always converges (Brady and Bossis, 1985).

We determined the evolution of the pair probability density

function with only the radial separation distance r of particle

pairs [equation (16)]. We computed, for each particle, the

number ni(r) of particles that can be found in an elementary

volume dV(r) at a separation distance r.

g(r) ¼4pa3=3

NfdV(r)

X

N

1

ni(r) (16)

As we have shown in Figure 6, the pair probability function is

not isotropic and g(r) is integrated over all angular orien-

tations. It is null for separation distance between particle cen-

ters corresponding to contact [g(r , 2a) ¼ 0] since there is no

particle overlapping. The probability function g(r) is normal-

ized usingÐ1

and for different suspension concentrations. Figures 7(a) and

(b) show that the pair probability functions have qualitatively

thesamebehaviourinbothcases.Theprobabilityoffindingpar-

ticlesnearcontactismaximumleadingtoapeakvalueatasep-

aration distance r/2a ¼ 1. The peak value increases with the

volume fraction f of the suspension. In the static simulations,

g(r) results only from the random seeding of particles and the

peak denotes only correlations owing to the excluded volume

effect.Itisclearthatcloseparticlepairshaveahigherprobability

to occur when the particles in the suspension are driven by the

shear flow. Similar results have been reported for Brownian

(Morris and Katyal, 2002) and non Brownian (Brady and

Bossis, 1985; Sierou and Brady, 2002; Drazer et al., 2004) sus-

pensions under shear flow.

0g(r)dr ¼ 1.

g(r)wascomputedusingbothstaticanddynamicsimulations,

Probability Distribution Function of Velocity

Fluctuations

Normalized probability density functions of the translational

velocity fluctuations in the shear direction are shown in

Figure 8(a) for four different volume fractions. We obtained

basically the same plots in other directions. The first obser-

vation is that for weak velocity fluctuations, all the probability

density functions have a Gaussian shape, but for higher mag-

nitude of velocity fluctuations the shape gradually changes as

the concentration increases. Intense velocity fluctuations are

more probable than the Gaussian estimate. For example, in

Figure 8(b) (5% concentration), a best fit of the pdf is com-

posed of a Gaussian behaviour for weak velocity fluctuations

jv/sj ,1 (where s is the standard deviation of the velocity

fluctuations) and an exponential tail for intense fluctuations.

Such a behaviour is related to the presence of persistent

small-scale structures, due presumably to the long-lasting

short-range hydrodynamic interactions of pairs of particles

at low volume fraction (Drazer et al., 2002). Whereas, when

the concentration is increased, the probability density func-

tion has a more pronounced Gaussian shape. At large con-

centrations, the mean separation distance between the

Figure 6. Relative positions of particle pairs in a shear flow. Filled

circle: reference particle. Dashed circle: limit of the overlapping

region (where r , 2a). þ: Location of particle centres close to contact

relatively to the reference particle (r , 2.5a). The figure is populated

with several independent time frames. The compression axis is mate-

rialized by a solid line.

Page 9

particles is reduced leading to multiple many-body inter-

actions with a weaker overall correlation. Similar changes

in flow statistics have been observed in a turbulent flow

(Verzicco and Camussi, 2001). For low Reynolds numbers

the distribution of vorticity fluctuations is almost Gaussian,

associated with weakly correlated fluid motions. Whereas

when the Reynolds number increases, coherent small scale

vorticity structures develop leading to stretched exponential

tails. Such behaviour is a common feature of many complex

physical systems [in fluidized beds for example, velocity fluc-

tuation distribution varies from Gaussian to exponential as

the particle concentration increases (Rouyer et al., 1999)].

VELOCITY AUTOCORRELATION AND

SELF-DIFFUSION

Lagrangian Velocity Autocorrelation

Multiple interactions of particles in the flow lead to a

chaotic motion in the suspension. Although the Stokes

equations are linear and deterministic, many-body hydro-

dynamic interactions and repulsion barrier effects force

the particles to move across streamlines. The system is

extremely sensitive to the initial conditions and even an

extremely weak perturbation introduced in the calculation

induces a complete loss of memory of the initial state.

Such a response is typically related to a diffusive behaviour

of the suspension at long times. Figure 9 shows an

example of the impact of cumulated numerical errors on

the temporal evolution of velocity fluctuations for an 18%

suspension concentration. Initial particle positions are iden-

tical for the two runs with only a very weak perturbation of

order 10215. The temporal evolutions of T22 are superim-

posed for non-dimensional times lower than 100. After-

wards, the two evolutions have distinct instantaneous

evolutions although statistics (mean and standard devi-

ation) are preserved.

The time required to get uncorrelated velocity fluctuations

along the particle trajectories is extracted from the normalized

Lagrangianvelocity autocorrelation function

Rii(t)

Figure 8. Normalized PDF of particle velocity fluctuations in the

(a) shear and (b) spanwise directions. Dashed line: Gaussian

distribution function. (a) Plus: f ¼ 1%—Circles: f ¼ 5%—Stars:

f ¼ 10%—Triangles: f ¼ 20%. (b) Circles: f ¼ 5%. Solid line:

bestfit byastretched

exp ?1:5 v=s

exponential PDF v=s

ð Þ ? 2=

ffiffiffiffiffiffi

2p

p

?? ??

??.

Figure 7. Pair probability density function g(r) calculated in the case

of (a) static simulations and (b) dynamic simulations. Symbols †, P

and B correspond respectively to the peak values of 1, 5 and 15%

concentration cases.

Page 10

[equation (17)] computed along the two transverse directions

(shear and spanwise direction).

Rii(t) ¼kvi(t)vi(t þ t)l

Tii

(17)

The product vi(t)vi(t þ t) is averaged over all the particles for

the different starting times t. In Figure 10 the velocity autocor-

relation functions are plot for different suspension concen-

trations up to 20%. We can observe that in all cases the

velocity fluctuation autocorrelations have a negative region

around a typical time Tc1of order 1/G suggesting that pair-

wise interaction is the major contribution to anti-correlated

motions. A two particle encounter has a life time of 1/G.

The negative region is more pronounced at low concentration

but is still prominent at moderate concentration in agreement

with Marchioro and Acrivos (2001). Another important

characteristic time scale Tc2is the time required to reach

uncorrelated fluctuations. Fully uncorrelated motions are

achieved around Gt ¼ 8 non-dimensional time units and

slightly shorter for more concentrated suspensions. The

third time scale TL1 allows the determination of the self-

diffusion coefficients [see equation (18)]. The time integral

of the autocorrelation function was found to be convergent

for all concentrations. It defines the Lagrangian time scale

TLi¼Ð1

summary of all these time scales is given in Table 1.

0Rii(t)dt which increases with the concentration,

since the negative loop is decreasing as f increases. A

Shear Induced Self-Diffusion

If the particles experienced a pure Brownian motion, they

would be subject to a short time self-diffusion related to the

local instantaneous particle mobility. This diffusion would

appear for a time scale longer than the relaxation time of

the particle in the fluid and shorter than the time necessary

to get uncorrelated motions. Another cause of diffusion

would develop if the suspension had a concentration gradi-

ent. Then a ‘collective’ diffusion (Fick’s law) would lead to a

macroscopic migration of the particles. However, although

the particles in this study are non-Brownian and the suspen-

sion is homogeneous, the loss of correlation after a long

simulation time indicates that a self-diffusion behaviour devel-

ops in spite of the deterministic and linear nature of creeping

flows. Indeed, the multi-body hydrodynamic interactions

(Wang et al., 1996) and the non-hydrodynamic repulsive

forces lead to a fore-aft symmetry breaking, and induce the

drift of the particles across the streamlines, enhancing the

overall chaotic evolution of the suspension or the so-called

shear-induced ‘self-diffusion’. Transverse self-diffusion coeffi-

cients have been estimated by multiple approaches, but the

determination of the diffusion coefficient in the flow direction

is more complicated due to the combination of the diffusion

and advection processes (Sierou and Brady, 2004). Acrivos

et al. (1992) predicted theoretically the self-diffusion coeffi-

cient parallel to the flow by introducing a mechanism of inter-

action with an additional pair of particles. Sierou and Brady

(2004) calculated this coefficient using a careful construction

of the advection-diffusion

equation.

The transverse diffusion coefficient can be determined in

two ways. Firstly, it can be evaluated from the integral of

the velocity autocorrelation function over a long period of

time [equation (18)].

namelythe Fokker-Planck

Dii¼ GTii

ð1

0

Rii(t)dt

(18)

The self-diffusion coefficient is calculated in the shear and

spanwise direction (i ¼ 2 or 3) and it is scaled by Ga2. It is

the product of the fluctuation level times the Lagrangian inte-

gral time scale TLwhich converges when the autocorrelation

function tends to zero. It is important to note that we found in

Figure 10. Lagrangian velocity autocorrelation function in the shear

direction for different suspension concentrations. —— f ¼ 1%, -.-.-

.f ¼ 5%,

f ¼ 10%,..... f ¼ 20%.

Table 1. Correlation times versus concentration.

f (%)15 101520

G Tc1

G Tc2

G TL1

1.16

8.8

0.14

1.18

8.8

0.2

1.16

8.8

0.24

1

6.5

0.26

1

6.5

0.3

Figure 9. Temporal evolution of the velocity fluctuation in the shear

direction T22(18% concentration). Solid and dashed lines show T22

obtained during simulations with the same initial conditions and a

weak noise (magnitude 10215).

Page 11

our simulations an enhancement of the velocity fluctuations

but also an increase of the integral diffusion time with increas-

ing concentration.

The second method is based on the long time behaviour of

the particle mean-square displacement [equation (19)].

Dii¼

1

2Ga2lim

t!1

d

dtk½xi(t) ? xi(0)?2l

(19)

In Figure 11 the temporal evolution of ,[xi(t) 2 xi(0)]2./t is

plotted for suspension concentration ranging from 1 to 20%.

This term has two different temporal regimes. It increases

at short time and reaches at long times a plateau providing

directly the diffusion coefficient. This representation of the

time evolution of the mean square displacement is equivalent

to the usual log-log representation used in the literature

(Drazer et al., 2002; Marchioro and Acrivos, 2001; Sierou

and Brady, 2004). At short time scales, the mean square dis-

placement has a quadratic growth rate, i.e., its square root is

linear and then the particle behaviour is not diffusive. The

coefficient a [equation (20)] characterizes the mobility of the

particles at short times.

ai¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Gat

k½xi(t) ? xi(0)?2l

q

(20)

The values of a2in Table 2 confirm that at short time scales

the particle moves in the transverse directions with a typical

velocity close to the particle averaged velocity fluctuation

ffiffiffiffiffiffiffi

The growth of the transverse mean-square displacement is

linear beyond the time necessary to reach uncorrelated vel-

ocity fluctuations. The diffusion coefficient Diiis evaluated at

long times, when the curves in Figure 11 reach a plateau.

These curves show that the simulation times were long

enough to reach the diffusion regime.

Transverse self-diffusion coefficients are strongly depen-

dent on the volume fraction of the suspension [Figure 12(a)

and (b)]. Da Cunha and Hinch (1996) and later Zarraga and

T22

p

turbulent diffusion.

(ballistic regime) in agreement with Taylor’s theory on

Leighton (2001) found that the shear-induced diffusion coeffi-

cient depends linearly on the concentration in the dilute

regime. In their work, the particles are mainly driven by the

non-hydrodynamic repulsive forces preventing particle over-

lap leading to a finite drift at each encounter. On the another

hand, Wang et al. (1996) have shown theoretically that self-

diffusion due to hydrodynamic interactions between more

than two particles shows quadratic growth with concentration.

Drazer et al. (2002) showed a transition between these

two limiting behaviours when the strength of the repulsive

force is substantially increased. The dependence of the

Figure 11. Temporal evolution of mean square displacement

,(x2(t) 2 x2(0))2./2 Ga2for different suspension concentrations.

From bottom to top, the concentration is respectively 1, 5, 10, 15

and 20%.

Table 2. Comparison of the local mobility coefficient a2 and the

average velocity fluctuations in the shear direction.

f (%)1510 1520

a2

(T22)1/2

0.0505

0.0503

0.096

0.12

0.133

0.169

0.141

0.206

0.199

0.242

Figure 12. Evolution of the shear induced self-diffusion coefficient (a)

in the shear direction D22and (b) in the spanwise direction D33.

A experimental work, Leighton and Acrivos (1986); þ: experimental

work, Breedveld et al. (2002);

(1992); o: Stokesian dynamics, Drazer et al. (2001); q: accelerated

Stokesian dynamics: Sierou and Brady (2004); P with dashed line:

force coupling method simulations.

?: analytic work, Acrivos et al.

Page 12

self-diffusion coefficient showed a cross over from a quadra-

tic to linear scaling which is controlled by the finite drift across

the streamlines induced by the non-hydrodynamic repulsive

force. Our simulations show that the diffusion process does

not depend only on the non-hydrodynamic effects, since the

transverse self-diffusion coefficient does not scale linearly

with the concentration. In an extremely dilute regime, we

should recover this linear scaling but statistics take much

longer time to converge as particle encounters become

very rare. Thus, we have tested the effect of the repulsion

barrier by varying the amplitude of the force scale at a con-

stant concentration of the suspension. We obtain a scattering

of less than 30% for large variations of the repulsion force.

Compared to former studies, the self-diffusion coefficient

computed with the FCM is slightly overestimated which is

essentially related to the inaccurate representation of the

local effects of viscous lubrication forces for small gap

widths. When we improved the local hydrodynamic inter-

actions by adding these lubrication forces (Abbas et al.,

2006), we noticed that the results obtained by simulations

using the simple barrier repulsion are not completely mislead-

ing. The velocity fluctuation intensities and their PDF were

well evaluated (with a 10% underestimation). The diffusion

coefficient, which is in general more critical since it depends

on the multi-body interactions and the final particle drift

across the streamlines, was reduced to nearly 35% and

had a fairly good agreement with the numerical work of

Sierou and Brady (2004).

The self-diffusion tensor is anisotropic, i.e., 1.2 , D22/

D33, 3 for concentrations between 1 and 20%, recalling

that the velocity fluctuations have the highest magnitude in

the plane of the shear. High anisotropy of self-diffusion was

also observed by Da Cunha and Hinch (1996) and Wang

et al. (1996) for very dilute suspensions (D22/D33? 10). As

we already mentioned, at low concentrations velocity fluctu-

ations and consequently the self-diffusion coefficient are basi-

cally controlled by pair encounters which are highly

anisotropic. When the concentration increases, multi-body

hydrodynamic interactions enhance the fluctuations and diffu-

sion of the particles in the direction perpendicular to the shear

plane, and the ratios T33/T22, as well as D33/D22, are

increasing (Figure 13). This trend is correct even for high con-

centrations up to 50% (Sierou and Brady 2004; Marchioro

and Acrivos, 2001).

We conclude that the results obtained with the FCM are in

the range of the former numerical and experimental investi-

gations where data are scattered due to experimental uncer-

tainties [a novel technique has been proposed by Breedveld

et al. (1998)], simulation limitations (discussed in Sierou and

Brady, 2004) and theoretical assumptions (Wang et al.,

1996).

CONCLUDING REMARKS AND PERSPECTIVES

The relation between macroscopic effective quantities and

the microscopic structure of non-colloidal sheared suspen-

sions under Stokes flow has been studied using the FCM.

We considered that particles have the same density as the

fluid and consequently we neglected inertia and buoyancy

effects. Typical suspensions satisfying such conditions con-

sist of solid micro-sized particles [a ¼ O (1 mm)] suspended

in a highly viscous liquid [n ¼ O (1024m2s21)]. Although

these conditions are restrictive, such suspensions are of

great interest in Chemical Engineering. The prediction of

flows of suspension following oil extraction or conveying

emulsions needs models accounting for multi-body hydro-

dynamic interactions. We chose the simple configuration of

a pure linear shear flow as a prototype of interactions induced

by local velocity gradients. In our simulations, fully periodic

boundary conditions avoid segregation effects and the sus-

pension remains homogeneous allowing the proper determi-

nation of statistics.

Interactions of particles occur because particles are

moving on streamlines at different mean velocity and close

interactions modifies the relative positions of the particle

centres. The far field hydrodynamic interactions are well

reproduced by the FCM. When the gap between the particle

surfaces is very short, lubrication forces and non-hydrodyn-

amic interactions may control the motion of particles. In a

suspension stabilized by surfactants the interaction force is

basically repulsive and can be approximated by DLVO-type

forces where double layer electric repulsion overcomes the

Van der Waals attraction. We were concerned with such sus-

pensions where aggregation phenomena are negligible. The

impact of these strong repulsive forces on the overall

response of the suspension was previously found to be

weak for moderately concentrated suspensions (20%), and

its role has been discussed by Da Cunha and Hinch (1996)

and Zarraga and Leighton (2000). The dynamics of stable

suspensions have been well documented using different

experimental and numerical methods (Breedveld et al.,

2002; Sierou and Brady, 2004; Drazer et al., 2004).

After validating our numerical approach on simple systems

of isolated or pairs of particles, we formed time averages on

the trajectories of particles embedded in suspensions. The

translational and rotational velocity fluctuations are found to

be anisotropic and monotonically increasing in the range of

solid volume fraction investigated. Particle pairs are formed

even in static suspensions due to excluded volume effects,

and their probability increases with concentration. When

shearing the suspension for a long time the velocity fluctu-

ations and the probability of finding close pairs of particles

are enhanced. We found that the radial distribution of pairs

close to contact is highly anisotropic. It is clearly related to

Figure 13. The evolution of anisotropy coefficients with concentration.

Squares: D33/D22—Stars: T33/T22.

Page 13

the repulsion force that prevents formation of pairs on the

receding side of the test particle. Consequently, the T12off-

diagonal velocity fluctuation term is different from zero due

to symmetry breaking of hydrodynamic interactions. The pair-

wise interactions played a dominant role at low concen-

trations. The shape of the probability distribution function of

velocity fluctuations gradually evolves from Gaussian to

exponential when the concentration decreases. Also, we

observed that velocity autocorrelation functions have nega-

tive regions related to anticorrelation of velocity correlation

on a time scale of order 1/G revealing that pair interactions

are prominent. The velocity fluctuations remain correlated

for a time of order 8/G before the lagrangian velocity autocor-

relation function tends to zero. This loss of velocity correlation

induced a hydrodynamic regime of diffusion. The shear-

induced self-diffusion coefficients are determined in the trans-

verse directions by two different methods showing that the

results are consistent. We obtained a good agreement with

previous numerical and experimental studies. The overall

dynamics is well predicted by our numerical approach and

this opens new fields of investigation.

The FCM is very flexible as we can implement very easily

various potentials for interparticle forces. Brownian motion

can also be modelled as a random force experienced by

each particle leading to a prescribed diffusion behaviour. If

interparticle forces are basically attractive, aggregation

occurs and the kinetics of formation of clusters can be fol-

lowed numerically. We successfully studied the formation kin-

etics of chains of paramagnetic particles (Climent et al.,

2004). We plan to use the same approach for seeking the

stability of colloids when both Brownian motion and shear

flow are simultaneously driving the motion of particles. Inter-

action potentials will be modelled using the DLVO theory with

interparticle forces ranging from purely attraction to strong

repulsion we already studied in the present paper.

NOMENCLATURE

a

Ar:

G

g

kB

Pe

Re

St

T

V

rp

rf

mf

nf

particle radius, m

Archimedes number

shear rate, s21

gravity constant, m s2

Boltzman constant, J K21

Pe ´clet number

Reynolds number

Stokes number

fluid temperature, K

particle velocity, m s21

density of the particle, kg m23

density of the fluid, kg m23

dynamic viscosity of the fluid, Pa s

kinematic viscosity of the fluid, m s22

FCM

A, B and C(r/a)

A(n)

ij

hydrodynamic quantities

anti-symmetric part of the dipole strength

tensor N m21

elementary volume, m3

monopole force strength due to the nth

particle, N

external force on the center to the nth particle,

N

repulsion barrier, N

scale of the repulsion force, N

dipole strength due to the nth particle, N m21

fluid inertia, kg m22

particle inertia, kg m22

width of the simulation domain, m

d3x

F(n)

Fext

Fb

Fref

Gij

If

Ip

L

(n)

mf

mp

p

r

fluid mass, kg

particle mass, kg

pressure in the fluid, Pa

separation distance between two particle

centres, m

cut-off distance of the repulsion barrier, m

symmetric part of the dipole strength tensor, N

m21

time, s

torque due to the nth particle, kg m22s22

external torque, kg m22s22

fluid velocity, m s21

position in the fluid, m

velocity of the nth particle, m s21

centre position of the nth particle, m

Gaussian envelop of momentum source

terms, m23

particle rotation of the nth particle, rad s21

Hamaker constant, J

fluid permittivity, C2J21m21

Electrical potential surface, V

width of the Gaussian envelops, m

Rref

S(n)

ij

t

T(n)

Text

u

x

V(n)

Y(n)

D and D0(x)

(n)

V(n)

L

1r10

c

s and s0

Suspension statistics

Dii

shear induced self-diffusion coefficient tensor,

dimensionless

radial pair probability distribution function

shear rate

direction index–1: for the flow, 2: for the shear,

3: for the spanwise direction

particle number density

particle number inside the elementary volume

dV(r)

Lagrangian autocorrelation function of the

velocity fluctuations

standard deviation of the velocity fluctuations,

m s21

autocorrelation time scales, s

translational velocity fluctuation tensor,

dimensionless

translational velocity fluctuation, m s21

rotational velocity fluctuation tensor,

dimensionless

particle position at time t, m

particle position at the initial time, m

local mobility coefficient of the particles at

short times

suspension concentration

rotational velocity fluctuation, s21

g(r)

G

i

ni

N

Rii

s

Tc1, Tc2, TLi

Tij

vi

wij

xi(t):

xi(0)

ai

f

Gi

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ACKNOWLEDGEMENTS

The initial version of the numerical code has been developed in the

group of Professor M.R. Maxey at Brown University. We gratefully

acknowledge his help in the first steps of this work. Most of the com-

putations have been carried out on the French regional and national

supercomputing centres: CalMip and IDRIS/CINES. We acknowl-

edge their support. Finally, we would like to thank the cooperative

research federation FERMaT for its support when funding this work.

The masnuscript was recevied 20 July 2006 and accepted for

publication after revision 23 November 2006.