Concentrated non-Brownian suspensions in viscous fluids.
Numerical simulation results,
a “granular” point of view
J.-N. Roux, with F. Chevoir, F. Lahmar, P.-E. Peyneau, S. Khamseh
Laboratoire Navier, Université Paris-Est, France
Granular materials :
• quasistatic response : gradually apply shear stress or
impose small shear rate on isotropic equilibrated state
• steady shear flow with inertial effects
Extension to dense suspensions :
Large hydrodynamic forces in narrow gaps between grains,
role of contacts
Approaches similar to dry granular case
Granular materials : initial density and critical state
Several samples with 4000 beads, prepared at different solid fraction 
Monodisperse sphere assembly, friction coefficient  = 0.3
Triaxial test
Similar response in
simple shear test !
Under large strain, critical state independent of initial state,
characterized by a ‘’flow structure’’ (density + nb of contacts,
anisotropy…)
Critical state of dry granular materials, 2D and3D results
Compilation of simulation literature, in Lemaître, Roux & Chevoir, Rheologica Acta 2009
c and c do not depend on contact stiffness if large enough
Internal friction coefficient
Critical solid fraction
(actually somewhat different between shear
tests and other load directions, e.g. triaxial,
see e.g., Peyneau & Roux, PRE 2008)
c = RCP for  = 0
What we know from simulations of dry grains (I)
Quasistatic rheology :
•interest of critical state concept (flow structure) c
minimum solid fraction in flow
•Friction coefficient  in contacts determines c and *c.
Small rolling friction also quite influential
• using contact law and applied pressure P, define stiffness
number (such that deflection is prop. to -1)2
, assess approach to
3
K
E


N
rigid grain limit

(
l
i
n
e
a
r
e
l
a
s
t
i
c
i
t
y
,
3
D
)
;
(
H
e
r
t
z
c
o
n
t
a
c
t
s
,
3
D
)
a

d
i
a
m
e
t
e
r


Inertial flows :

a
P

P


•Study shear flow under controlled normal stress rather
than fixed density : non-singular quasistatic limit
• use internal friction * as an alternative to viscosity
Simulation of steady uniform shear flow
Normal stress
is imposed
P   22
Fixed shear strain rate

What we know from studies on dry grains (II) :
Inertial number and constitutive relations
Material state in shear
flow ruled by one
dimensionless parameter,
the inertial number

m
m
I


I


2
D
3
D
P
a
P
Monodisperse spheres,
no intergranular friction
(Peyneau & Roux, Phys
Rev E 2008)
RCP

Generalization of critical state to I-dependent states with inertial effects 
Useful constitutive law, applied to different geometries (Pouliquen, Jop, Forterre…)
Steady shear flow of dry, frictionless beads at low I
number : normal stress-controlled vs. volume controlled
(P.-E. Peyneau)
Same behavior, very large stress fluctuations if  is
imposed (4000 grains)
Ratio 12/22 = * expresses material rheology
Simulations of dense suspensions
• in simulation literature : nothing for > 0.6 (spheres), ad-hoc
repulsive forces used to push grains apart, very few published results
with N >1000…
• sharp contrast with simulations of dry grains!
• lubrication singularities believed to lead to some dynamic jamming
phenomenon below RCP (related to possible origin of shear thickening)
Ball and Melrose, 1995-2004: no steady state (!?)
• Here : control normal stress rather than density, control lubrication
cutoff and contact interaction stiffness
Experiments on dense suspensions
• Attention paid to possible density inhomogeneities  local
measurements
• Boyer et al.  control of particle pressure !
Simulation of very dense suspensions
 Simplified modeling approach, fluid limited to near-neighbor gaps
and pairwise lubrication interactions (cf. Melrose & Ball)
 Lubrication singularity cut off at short distance. Contact forces (or
short–range repulsion due to polymer layer)
 Stokes régime : contact, external and viscous forces balance
Questions
Divergence of  at  < RCP ? Sensitivity to repulsive
forces ? To  alone ? Effective viscosity, non-Newtonian
effects, normal stresses…
Lubrication and hydrodynamic resistance matrix
Normal hydrodynamic force :
F N(h)
V
N
For 2 spheres of radius R, with
perfect lubrication :
3
R
N(h)
2 h
v
N
2
dominant forces transmitted by a network
of ‘quasi-contacts’
No contact within finite time !
Cut-off for narrow interstices h<hmin
(asperities), contact becomes possible
Without cutoff: both physically irrelevant
and computationally untractable
Choice of systems and parameters
D systems of identical
spheres, diameter a :
D systems of polydisperse
disks, diameter d, 0.7ada
tobeads
disks
 hmax/a = 0.1 or 0.3
 3D lubrication
 hmin/a = 10-4 ;  = 105
 hmax/a = 0.5
Vi from 0.1 down to 5.10-4
 hmin/a = 10-4 or 10-2 ;  = 104
  = 0 in solid contacts
Vi down to 10-4 or 10-6
 + alternative systems with
repulsive forces, no
lubrication cutoff
  = 0.3 in solid contacts
Model, computation method


• assemble hydrodynamic resistance matrix
(similar to stiffness matrix in elastic contact network)
• non-singular tangential coefficient
• add up ‘ordinary’ contact forces Fc (elasticity + friction) to
viscous hydrodynamic ones Fv when grains touch (simple
approximation)
F
F
F
0
c
v
ext

with
Fv V


Solve
and Fc depend on grain positions

VF
F
c
ext
Some technical aspects about simulations
Lees-Edwards boundary conditions
+ variable height, ensuring constant yy
Measurements in steady state :
 Check for stationarity of measurements
 Obtain error bar from ‘blocking’ technique
 Request long enough stationary intervals
t1
0a
tle
a
s
t...
Regression of fluctuations


N
-1/2
N -1/2
Constant volume and/or
constant shear stress
conditions should produce same
system state in large N limit
Choice of time step, integration
t such that matrix and r.h.s. do not change `too much’…
Euler (explicit) :
(error ~t2)

((
X
t
)
)

V
(
t
)

F
((
X
t
)
)

F
c
e
x
t
X
(
t


t
)

X
(
t
)


t
.(
V
t
)
‘Trapezoidal’ rule : (error ~ t3)

((
X
t
)
)

V
(
t
)

F
((
X
t
)
)

F
c
e
x
t
(
1
)

((
X
t
)

V
(
t
))

tV
 (
t
)

F
((
X
t
)

V
(
t
))

t
F
c
e
x
t
(
1
)
(
2
)
(
1
)

t (
1
)
(
2
)


X
(
t


t
)

X
(
t
)
.
V
(
t
)

V
(
t
)


2
A crucial test on the numerical integration of equations
of motion
Relative difference between variation of h and integration of normal
relative velocity in various interstices, 2 different numerical schemes :
Euler (dotted lines), trapezoidal (continuous lines)
Keep it below 0.05 !
Control parameter for dense suspensions : viscous
number Vi
Vi 2 D
Vi3 D
a
 
P

 
P
Plays analogous role to
inertia parameter
defined for dry grains
I2D
m
 
P
m
I 3 D  
aP
Vi = (decay time of h(t) in compressed layer within gap) / (shear time)
I = (acceleration time) / (shear time)
Acceleration (inertial) time replaced by a squeezing time in viscous layer
Cassar, Nicolas, Pouliquen, Phys. of Fluids 2005 (similar Vi, with drainage time)
Steady shear flow of lubricated beads at low Vi
number : normal stress-controlled vs. volume controlled
Vi = 10-3
Same behavior, large stress fluctuations if  is imposed (1372 grains).
Ratio 12/22 = * expresses material rheology
3D results : *and  as functions of Vi
N 1372 identical spherical grains
Vi
Vi
= 0 : difficult case ! Approach to * = 0.1, =0.64 …
Back to more traditional (constant ) approach
Effective viscosity
From





1
2
2
2
*
*
V
i
*
*

1


 



c
V
i
and


V
i one gets:
0 
*
*



if
0
(not satisfied in our case !)
Shear rate effects ?
*
 h




a
m
i
n

f

, ,
, 
  a K
N

 0
Exponent : 2.5 to 3 ?
In rigid grain limit replace 4th
argument by zero. No influence of
on effective viscosity

Effective viscosity, as a fonction of solid fraction

(hc = hmax)
Influence of repulsive force
Adsorbed polymer layer,
short-range repulsion, as
(Fredrickson & Pincus, 1991)
F
lb h 1]
0[
5
4
Polymer layer thickness
lb = 0.01 or 0.001
(open/filled symbols )
Force F0 ratio F0/a2P
(0.1, 1, 10) = (diamond, square, circle)
Shear-thinning within
studied parameter range

Vi 4.103
*
Vi  Vic
Vi  Vic

Ordered structure
polydispersity < 20%
Vic ~ 3.10-2
Pair correlations in plane yz
Shear thinning with repulsive forces
 lb
*
 a 2 
 f  , , ,


F0 
 a
 
*
 a
KN

Vi
F0

 a2 P
Shear thinning due to change
in reduced shear rate
Results correspond to

*
varying from 5.10-5 to 10-1
No shear thinning for smaller lb
Network of repulsive forces carry all shear stress as Vi decreases
 xy
 yy
Vi
Other ‘granular’ features: force
distribution, coordination number…
Viscosity of random, isotropic suspensions
•Assume ideal hard sphere particle distribution at given 
•Measure instantaneous shear viscosity
Comparison with Stokesian Dynamics results: encouraging agreement at
large densities, although treatment of subdominant terms not entirely
innocuous
3D frictionless spherical beads :
 Analogy with dry granular flow
 Difficulty to approach quasistatic limit
 No singularity below RCP density, a steady-state can be
reached
 Importance of non-hydrodynamic interactions
 Purely hydrodynamic model approached with stiff
interactions
 Effects of additional forces: 2-parameter space to be
explored
2D disk model
Easier system, introduction of tangential forces, friction, faster
approach to quasistatic limit
Show coincidence of quasistatic limits for dry
grains and dense suspension
2D results : internal friction coefficient versus Vi
(analogous to  function of I in dry inertial case)
Viscous
Dry, inertial


Viscous case hmin/a = 10-2 and hmin/a = 10-4 , =0,3
Dry inertial case (right) : =0.3 and =0
Coincidence I=0 / Vi=0
2D results : solid fraction versus Vi (in paste)
(analogous to function of I in dry inertial case)


Viscous
Dry, inertial
Viscous case hmin/a = 10-2 and hmin/a = 10-4 (896 grains), =0.3
Dry inertial case (right) : =0.3 and =0
Coincidence I=0 /Vi =0
Constitutive laws : dry grains (2D)  = 0 or 0.3
  cI
*

*
c
1 1
 eI
 
c
Constitutive laws, granular suspensions (2D), = 0.3
  cVi
*
*
c

1 1

 
e
(
V
i)
 
c
Same constant terms (quasistatic limit), different power laws
Effective viscosity, as a function of solid fraction
 Divergence of effective viscosity as critical solid fraction c is
approached
 (once again) no accurate determination of exponent (2 ? 2.5 ?)
 Little influence of roughness length scale
Importance of direct contact interactions
Pressure due to solid contact forces / total pressure
(Case hmin/a = 10-2)
Some conclusions on dense suspensions
 interesting to study dense suspensions under controlled
normal stress
 importance of contact interactions : solid friction,
asperities…) Contact or static forces dominate at low Vi
 relevance of critical state as initially introduced in soil
mechanics. Viscosity diverges as  approaches c
 With =0 in contacts, no jamming or viscosity divergence
or any specific singularity below RCP, except in small
systems

 If  is large enough, * independent of  with elastic
(-frictional) beads (but… Re ? Brownian effects ?)
 introduction of repulsive interaction with force scale 
shear-thinning (or thickening)
Perspectives
Improve performance of numerical methods
Continuum fluid !
(Keep separate treatment of lubrication singularities ?)
bridge the gap between real (and difficult) hydrodynamic
calculations and ‘conceptual models’