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-
thickening behaviourdependingonthe
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-
persesuspensions.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
theStokesiandynamics
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
then multi-body hydrodynamic
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.
interactionscontrolthe
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.
Whenthe cut-off distance
(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
is considerablyreduced
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,the non-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 autocorrelationfunction
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:
bestfitbya stretched
exp ?1:5 v=s
exponentialPDF 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
ofthe 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)].
namely theFokker-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 1015 20
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 (%)15 1015 20
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.
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