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Unifying Suspension and Granular Rheology
François Boyer, Elisabeth Guazzelli, Olivier Pouliquen
To cite this version:
François Boyer, Elisabeth Guazzelli, Olivier Pouliquen. Unifying Suspension and Granular Rheology.
Physical Review Letters, American Physical Society, 2011, 107, �10.1103/PhysRevLett.107.188301�.
�hal-01432411�
Unifying Suspension and Granular Rheology
Franc¸ois Boyer,
*
E
´
lisabeth Guazzelli, and Olivier Pouliquen
IUSTI, Aix-Marseille Universite
´
, CNRS (UMR 6595) 5 rue E. Fermi, 13453 Marseille cedex 13, France
(Received 29 July 2011; published 24 October 2011)
Using an original pressure-imposed shear cell, we study the rheology of dense suspensions. We show
that they exhibit a viscoplastic behavior similarly to granular media successfully described by a frictional
rheology and fully characterized by the evolution of the friction coefficient and the volume fraction
with a dimensionless viscous number I
v
. Dense suspension and granular media are thus unified under a
common framework. These results are shown to be compatible with classical empirical models of
suspension rheology and provide a clear determination of constitutive laws close to the jamming
transition.
DOI: 10.1103/PhysRevLett.107.188301 PACS numbers: 47.57.Gc, 83.80.Hj
The rheology of dispersions of solid particles in a fluid
has been extensively studied since the theoretical deriva-
tion of an effective viscosity in the dilute regime under-
taken by Einstein in 1905 [1]. Despite their relevance to
practical applications, even the simplest case of non-
Brownian hard spheres suspended in a Newtonian liquid
remains poorly understood, especially in the concentrated
regime [2]. Such mobile particulate systems are known to
undergo a jamming transition exhibiting a divergence of
their viscosity when the particle volume fraction reaches a
maximum value [3,4]. A detailed characterization of this
divergence is still missing and more generally the dense
regime, where both hydrodynamic and contact interactions
contribute to the suspension mechanics, lacks a unified
view.
In this Letter, we provide a new perspective on the
suspension rheology. Using an original experimental setup,
we analyze the results within the theoretical framework
that has recently led to express universal constitutive laws
for dense granular flows [5–8]. A continuous description of
flowing granular media has long remained a challenge [9]
but a major step was achieved by considering pressure-
imposed flows; see Fig. 1(a): when an assembly of hard
spheres (having diameter d and density
p
) is sheared at a
given shear rate
_
under a confining pressure P
p
, while
letting the medium free to dilate or to compact, there is
only one dimensionless control parameter I ¼ d
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
=P
p
q
_
[6]. This inertial number I can be seen as the ratio between
the inertial time of rearrangement t
micro
¼ d
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
=P
p
q
and
the time of strain t
macro
¼ 1=
_
. The granular rheology is
completely determined by two functions of the inertial
number: a friction law for the shear stress ¼ ðIÞP
p
and a volume-fraction law ðIÞ [5]. These scaling laws
have proved very successful for describing dense granular
flows in a wide range of flow configurations [8].
This granular paradigm can be applied to the case of
particles suspended in a fluid of viscosity
f
sheared at
constant particle pressure P
p
. Figure 1(b) depicts how P
p
can be imposed through a porous plate. If the Stokes
number St ¼
p
d
2
_
=
f
is small, viscous forces are domi-
nant at the particle scale and the internal time is now given
by a viscous scaling t
micro
¼
f
=P
p
. The system is no
longer governed by the inertial number and a dimension-
less viscous number should be used instead [10]
I
v
¼
f
_
P
p
: (1)
Following the approach used in the dry-granular case,
constitutive laws can be expressed as two functions of I
v
¼ ðI
v
ÞP
p
and ¼ ðI
v
Þ: (2)
Only a few measurements of the effective friction
have been performed, either indirectly [10] or in the quasi-
static limit [11–13], i.e., at vanishing I
v
. They suggest that
the medium whether dry or immersed in a liquid exhibits a
similar friction coefficient
1
0:3 when I
v
! 0.No
determination of the volume-fraction law ðI
v
Þ has been
reported so far. Therefore, a comprehensive description of
the suspension rheophysics is lacking. The objective of the
present work is to provide these two constitutive laws over
a wide range of I
v
.
Toward this goal, we have designed an original annular
shear cell in which pressure-imposed measurements are
performed as shown in Fig. 1(c). The suspension is con-
fined between a static truncated cone and a rotating annulus
having an internal (external) radius R
1
¼ 44 mm (R
2
¼
90 mm). The choice of a conical bottom plate ensures that
the shear rate is quasihomogeneous in the vertical and
radial directions. In the initial position ( ¼ 0, see below),
the suspension height hðrÞ varies linearly from H
1
¼
8:8mmto H
2
¼ 18 mm, and the shear cell volume is V
0
¼
255 cm
3
. Both the bottom and top plates are made bumpy
by gluing 0.5 mm steel bars every 5 mm. Two combina-
tions of particles and fluid are used in the experiments:
(i) poly(methyl methacrylate) (PMMA) spheres having
diameter d ¼ 1:10 0:05 mm in a Triton X-100/water/
zinc chloride mixture of viscosity
f
¼ 3:1Pa s;
PRL 107, 188301 (2011)
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(ii) polystyrene (PS) spheres (d ¼ 0:58 0:01 mm)in
polyethylene glycol-ran-propylene glycol monobutylether
(
f
¼ 2:15 Pa s). In both systems, the density of the
carrier fluid is closely matched to that of the suspended
particles so that sedimentation can be neglected. The large
size of the particles ensures that both colloidal forces and
Brownian motion are negligible.
The originality of the experimental system is twofold.
First, the interstitial fluid can pass through the top plate
which is porous having 5 mm holes covered by a 200 m
Nylon mesh. Second, the normal force F applied to the top
plate is imposed (but not its vertical position). The volume
fraction is thus not fixed, but adjusts to the imposed shear.
The experimental procedure is the following. The cell is
filled with the desired amount of fluid and particles. The
porous top plate is then lowered to the initial position for
which ¼ 0, see Fig. 1(c). The amount of particles is
chosen such that the volume fraction at this initial stage
is equal to
0
¼ 0:565 (for most of the experiments).
Changing
0
does not significantly affect the results as
seen by the good collapse of the data seen in Fig. 2. This
suggests the absence of shear-banding effect. The top plate
is driven by a MCR 501 (Anton Paar) rheometer at a given
torque M and normal force F, and the rotational speed !
and relative position are measured. This leads to the
determination of the shear stress ¼ 3M=2ðR
3
2
R
3
1
Þ,
the granular pressure P
p
¼
~
F=ðR
2
2
R
2
1
Þ [
~
F being the
normal force F compensated for the top plate buoyancy],
the mean shear rate averaged across the annulus
_
¼
h!r=hðrÞi, and the particle volume fraction ¼
0
=½1 þ fðÞ with fðÞ¼½3R
2
ðR
2
2
R
2
1
Þ=½2ðR
3
2
R
3
1
ÞH
2
. After a transient shown in Figs. 1(d)–1(f), where
both and
_
evolves in time, the system eventually
reaches a steady state (in typically 30 min which corre-
sponds to the slow vertical displacement of the top plate in
the viscous fluid). In this steady state, the applied normal
force
~
F is only balanced by the shear-induced granular
pressure P
p
exerted by the particles on the top plate and
this experiment then mimics the configuration of Fig. 1(b).
Note that no hysteresis is observed and that the long time
records show no slow variation. This suggests that particle
migration effects are negligible in the present system.
The flow curves ð
_
Þ obtained at different applied pres-
sures P
p
are shown in the inset of Fig. 2(a). When P
p
is
imposed, the shear stress is an increasing function of
_
and presents a yield stress
y
at vanishing
_
. Increasing P
p
drastically shifts the flow curves to higher values. In order
to test the relevance of the dimensional argument ex-
pressed by Eq. (1) and (2), the effective friction coefficient
¼ =P
p
is plotted against the viscous number I
v
¼
f
_
=P
p
in Figs. 2(a) and 2(b). All the data obtained for
different particle size, material, fluid viscosity, and initial
number of particles collapse on a single curve ðI
v
Þ. The
friction coefficient tends to a finite value
1
¼
y
=P
p
¼
0:32 0:03 at vanishing I
v
and increases with increasing
I
v
. The quasistatic value
1
is similar to values obtained
for dry-granular media. In contrast with the dry-granular
rheology, ðI
v
Þ does not saturate at large I
v
. As discussed
below, this behavior is consistent with the additional vis-
cous contribution to the total shear stress .
The second constitutive law relative to the volume frac-
tion Eq. (2) can also be tested. In Figs. 2(c) and 2(d), the
particle volume fraction is plotted versus I
v
for the same
experimental conditions. Again, the relevance of the vis-
cous number I
v
is demonstrated by the collapse of all the
data on a single curve ðI
v
Þ. The volume fraction is a
FIG. 1 (color online). Paradigmatic configurations of pressure-imposed shear of dry (a) or immersed (b) granular media;
(c) experimental setup; Variation of shear rate (e) and volume fraction (f) in response to a change in applied normal stress (d), the
shear stress being fixed (here, ¼ 150 Pa).
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decreasing function of I
v
since the medium dilates when
increasing the shear rate or diminishing the confining
pressure P
p
. From the semilogarithmic plot of ðI
v
Þ in
Fig. 2(d), we can precisely determine the maximum vol-
ume fraction
m
¼ 0:585 0:002 reached when I
v
! 0.
Note that this value of the maximum packing fraction of an
homogeneously sheared assembly of frictional spheres is
close to the one reported in dry-granular media (the so-
called critical volume fraction in critical state theory [14])
and significantly differs, as expected, from the random-
close packing fraction
rcp
’ 0:635. The inset of Fig. 2(d)
shows that the asymptotic behavior of close to
m
is
given by a power-law ð
m
Þ/I
1=2
v
. This greatly dif-
fers from the behavior ð
m
Þ/I observed for dry-
granular media [8].
We have shown that the rheology of a dense suspension
under imposed-pressure flow conditions is well described
by the constitutive laws (2) and shares similar features with
dry-granular rheology as long as the viscous number I
v
is
substituted for the inertial number I. This frictional behav-
ior of suspensions seems to differ from the classical view in
terms of an effective viscosity [2]. However, these two
formalisms can be reconciled as explained below. When
a suspension is sheared at a constant volume fraction, shear
and normal stresses scale viscously, i.e., /
f
_
and can be
expressed as two functions of
¼
s
ðÞ
f
_
and P
p
¼
n
ðÞ
f
_
; (3)
where
s
ðÞ and
n
ðÞ are the dimensionless effective
shear and normal viscosities respectively [2,15]. Relating
these constitutive laws to the above granular paradigm,
Eq. (2), is straightforward. The volume fraction ðI
v
Þ
being a monotonic function of the viscous number I
v
, the
inverse function I
v
ðÞ is defined unambiguously. Using
Eq. (1) and the frictional constitutive law, both the particle
pressure and the shear stress are found to scale viscously as
P
p
¼
1
I
v
ðÞ
f
_
and ¼
½I
v
ðÞ
I
v
ðÞ
f
_
: (4)
Identifying these later equations with Eq. (3) provides
s
ðÞ¼
½I
v
ðÞ
I
v
ðÞ
and
n
ðÞ¼
1
I
v
ðÞ
: (5)
The dimensionless effective shear and normal viscos-
ities can then be computed from the data of Fig. 2 by
plotting =I
v
and 1=I
v
as functions of , as shown in
Fig. 3. Figure 3(a) shows that
s
ðÞ¼=I
v
increases
with increasing and diverges when !
m
,as
expected from the vanishing of I
v
when !
m
.
Experimental data are in fairly good agreement with data
found in the literature and, in particular, classical empirical
correlations [2] such as those of Eilers (red dashed line)
and Krieger & Dougherty (green dashed line). It is impor-
tant to stress that data have been obtained very close to
m
(with values as close as
m
5 10
3
) reaching
huge values of
s
10
5
. Figure 3(b) also shows similar
increase and divergence of
n
ðÞ¼1=I
v
with . These
data are in good agreement with the scarce measurements
10 10 10 10
0
0.4
0.45
0.5
0.55
0.6
0 0.05 0.1 0.15 0.2
0.4
0.45
0.5
0.55
0.6
0 0.05 0.1 0.15 0.2
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5
0
50
100
150
200
10 10 10 10
0
0
0.5
1
1.5
10 10 10 10 10 10
0
10
10
10
10
0
1
2
0.585
0.32
FIG. 2 (color online). Top: Friction coefficient ¼ =P
p
as a function of the viscous number I
v
in linear (a) and semilogarithmic
(b) scales; inset: flow curves at different pressures P
p
(35, 53, 109, 142, 193, and 244 Pa), for different particles [1.1 mm PMMA
spheres (), 0.58 mm PS spheres (h)] at
0
¼ 0:565, and for the PS spheres at
0
¼ 0:433 (e). Bottom: Volume fraction as a
function of I
v
in linear (c) and semilogarithmic (d) scales; inset:
m
as a function of I
v
. Solid lines are given by Eq. (6) and (7).
PRL 107, 188301 (2011)
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of particle pressure [16] and greatly extends the range of
volume fraction investigated.
The effective viscosities
s
and
n
present the same
divergence ð
m
Þ
2
as stressed by the finite value of
when I
v
! 0 and clearly evidenced in the insets of Fig. 3.It
is worth noticing that these divergences are simply related
to the behavior of at vanishing I
v
and can be directly
inferred from the asymptotic form ð
m
Þ/I
1=2
v
.
We have shown that we can relate the frictional formal-
ism of dense suspensions with their classical viscous rheol-
ogy. We now propose constitutive laws which unify
suspension and granular rheology as far as the friction
law of dense suspensions ðI
v
Þ is modeled as the sum of
two contributions, coming, respectively, from contact and
hydrodynamic stresses
ðI
v
Þ¼
1
þ
2
1
1 þ I
0
=I
v
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
c
þ I
v
þ
5
2
m
I
1=2
v
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
h
: (6)
The contact contribution
c
ðI
v
Þ is chosen similar to that
reported in granular media, with
1
¼ 0:32 (as observed
here),
2
¼ 0:7 and I
0
¼ 0:005 in agreement with avail-
able literature [10]. The hydrodynamic contribution
h
ðI
v
Þ
is designed to recover Einstein viscosity at low (see
below) and gives the nonsaturating behavior at large I
v
.
This modeling fully captures experimental observation,
over the whole range of viscous number I
v
, regardless of
particle size, material, and interstitial fluid (see solid lines
in top Fig. 2).
A consistent model for the evolution of the volume
fraction as a function of I
v
should have the asymptotic
form
m
/ I
1=2
v
at vanishing I
v
, and stay positive for
all values of I
v
. The following function
ðI
v
Þ¼
m
1 þ I
1=2
v
; (7)
satisfactorily models the present measurements without
any fitting parameter (see solid lines in bottom Fig. 2).
Recasting these latter equations to obtain -dependent
constitutive laws of dense suspensions then leads to
s
ðÞ¼1 þ
5
2
1
m
1
þ
c
ðÞ
m
2
; (8)
n
ðÞ¼
m
2
; (9)
with
c
ðÞ¼
1
þð
2
1
Þ=½1 þ I
0
2
ð
m
Þ
2
.
Again these constitutive laws fit the experimental data
well, even for values very close to
m
(Fig. 3). In this
model, the effective shear viscosity
s
ðÞ comprises three
terms. The first two terms come from the hydrodynamic
contribution and tend to Einstein viscosity (1 þ 5=2)at
OðÞ. The third term is due to solid contacts and provides
the leading divergence in ð
m
Þ
2
. The effective nor-
mal viscosity
n
ðÞ presents the same divergence and
agrees well with proposed correlations [15,17].
The present study provides an alternative viewpoint of
dense suspension rheology. The granular paradigm is dem-
onstrated to successfully describe the behavior of suspen-
sions of hard spheres in a viscous fluid. An analytical
model consistent with both frictional and hydrodynamic
interactions is proposed. This can be easily generalized to
three-dimensional complex flows [7] and applications in
which the granular phase is subjected to gravitational
forces. Another important result is the examination of the
rheology close to the jamming transition. Using pressure-
imposed flows circumvents the divergences observed in
-imposed rheology at the jamming transition. Of funda-
mental importance is the finding of the critical (or maxi-
mum) volume fraction
m
¼ 0:585 0:002 at the
jamming point as well as the algebraic divergences of the
effective viscosities in ð
m
Þ
2
. The present study
opens the path to future comparison with theoretical and
numerical studies on the jamming transition of disordered
particulate systems [18].
*francois.boyer@polytech.univ-mrs.fr
[1] A. Einstein, Ann. Phys. (Berlin) 322, 549 (1905).
0.3 0.4 0.5
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10 10 10 10
0
10
0
10
1
10
2
10
3
10
4
-2
0.3 0.4 0.5
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10 10 10 10
0
10
0
10
1
10
2
10
3
10
4
-2
0.585
0.585
FIG. 3 (color online). Effective shear
s
(a) and normal
n
(b)
viscosities versus (same symbols as in Fig. 2). (a) Empirical
correlations of Eilers (red dashed line) and Krieger-Dougherty
(green dashed line); see [2]; (b) particle-pressure measurements
[16] (red down-triangles). Continuous lines are given by Eq. (8)
and (9). Insets: logarithmic plots of
s
and
n
versus
m
.
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![FIG. 3 (color online). Effective shear s (a) and normal n (b) viscosities versus (same symbols as in Fig. 2). (a) Empirical correlations of Eilers (red dashed line) and Krieger-Dougherty (green dashed line); see [2]; (b) particle-pressure measurements [16] (red down-triangles). Continuous lines are given by Eq. (8) and (9). Insets: logarithmic plots of s and n versus m .](/figures/fig-3-color-online-effective-shear-s-a-and-normal-n-b-58lm94y7.webp)
![FIG. 2 (color online). Top: Friction coefficient ¼ =Pp as a function of the viscous number Iv in linear (a) and semilogarithmic (b) scales; inset: flow curves at different pressures Pp (35, 53, 109, 142, 193, and 244 Pa), for different particles [1.1 mm PMMA spheres ( ), 0.58 mm PS spheres (h)] at 0 ¼ 0:565, and for the PS spheres at 0 ¼ 0:433 (e). Bottom: Volume fraction as a function of Iv in linear (c) and semilogarithmic (d) scales; inset: m as a function of Iv. Solid lines are given by Eq. (6) and (7).](/figures/fig-2-color-online-top-friction-coefficient-1-4-pp-as-a-3hntp9ud.webp)