This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics
1
Normal stress differences in a suspended
monolayer of rough spheres
Helen J. Wilson†,
Mathematics Department, University College London, Gower Street, London WC1E 6BT, UK
(Received xx; revised xx; accepted xx)
We study a model suspension consisting of a monolayer of identical spheres in a viscous
medium without Brownian effects, but with contact between particles due to microscopic
surface asperities. We use a combination of theoretical calculations, at low solids area
concentrations, and numerical simulations, at moderate area concentrations, to calculate
the effective steady rheology of the suspension. Though the suspension is a monolayer,
the stresses within the plane of flow are not matched to those in the neutral direction,
so both first and second normal stress difference can be induced by the particles.
We find that the first normal stress difference, N1 , is negative for area fractions up to
around 30% and then becomes positive for more concentrated suspensions, in contrast to
simulations in the literature for similar three-dimensional systems (for which it is always
negative). The induced second normal stress difference, N2 , is always negative. At low
concentrations it is smaller in magnitude than N1 , but for dense suspensions the second
normal stress difference dominates strongly over the first. Friction is found to increase
the magnitude of N1 (but only weakly, in contrast to recent results of Gallier et al. 2014)
and have almost no effect on the behaviour of N2 .
We find that the viscosity of the suspension is lowered by contact at all area fractions accessible to us; however, we have neglected hydrodynamic interactions between
asperities, which might reverse this trend if included.
Key words: low-Reynolds number flow; rheology; suspensions; normal stress; monolayer;
friction
1. Introduction
Non-colloidal suspensions – that is, suspensions of solid particles in which Brownian
motion can be neglected – are found in many industrial applications. It is important to
understand their rheology, both to predict their behaviour in flow and to understand
the evolution of their microstructure. The latter of these requires something beyond
experimental measurement – either pure theory or simulation – and there is a large
body of work on numerical simulations of non-colloidal suspensions (see, for example,
Bertevas et al. 2010; Gallier et al. 2014). These simulations are often tested in their early
stages on two-dimensional systems, which are much less computationally intensive than
a fully three-dimensional scenario. For this reason, in this paper we present a study of
the rheology of monolayer suspensions.
There is little experimental work on monolayers – and the few papers that do exist
(for example Belzons et al. 1981; Bouillot et al. 1982) report only on viscosity and not
† Email address for correspondence: helen.wilson@ucl.ac.uk
2
H. J. Wilson
on normal stresses. However, there has been a recent resurgence in bulk experiments on
non-colloidal sphere suspensions (in fully three-dimensional geometries), with a particular
focus on their normal stress behaviour in shearing flows. A recent review (Denn & Morris
2014) brings much of this work together. There is general consensus that the shear
viscosity η is an increasing function of concentration, well fit with a Krieger-Dougherty
law
−2.5φm
φ
η ≈µ 1−
φm
in which µ is the viscosity of the solvent fluid and φ the volume fraction. Here φm is the
maximum packing fraction, which may be determined empirically but is usually taken as
0.64, the random close packing fraction. There is also consensus that the second normal
stress difference N2 is negative, and much larger in magnitude than the first normal stress
difference N1 ; however, the first normal stress difference is more controversial. Some
authors (Dbouk & Lemaire 2013) claim that N1 is positive, others (Dai et al. 2013) that
it is negative, while yet others (e.g. Boyer et al. 2011; Couturier et al. 2011) are unable
to determine the sign because the stress difference is not significantly different from zero
in their experiments.
Finally, there is controversy over the behaviour at low concentrations. Experiments by
Tanner and coworkers (Tanner et al. 2013; Dai et al. 2013) find that both normal stresses
scale as φ3 for small φ, while those by the group of Pouliquen & Guazzelli (Boyer et al.
2011; Couturier et al. 2011) find no normal stresses at all for φ < 0.22. Finally, most
simulations (for example Morris & Boulay 1999) predict that the normal stresses should
scale as φ2 , in agreement with dilute theory for rough spheres (Wilson & Davis 2000).
Several questions of interest naturally arise as we move from three-dimensional to
monolayer suspensions. Does N2 remain negative? Can we determine the sign of N1 ?
And what is the behaviour in the dilute limit φ ≪ 1? We will address all these issues in
this paper.
It might seem at first sight that the second normal stress difference N2 has no place in
a monolayer flow, and we made this mistake in our earlier work (Wilson & Davis 2002),
but it is well-defined (as discussed in §2) and represents the difference in stress behaviour
between the flow-gradient and vorticity directions.
Stokes flow of a suspension of smooth particles is theoretically quasi-static and reversible: that is, a reversal of imposed forces should cause the suspension to exactly
reverse the flow it has just performed. One consequence of this is that the system can
never attain an “equilibrium” microstructure under flow: it will always retain memory
of how long the flow has persisted, as if the flow were reversed this is the length of
time it would take to regain its original microstructure. This means that simulations
of a pure Stokes system without Brownian motion are in some sense unphysical: the
symmetry of the governing equations will always be broken in practice, whether by
weak inertia, weak Brownian motion, or interparticle interactions. Indeed, experimental
evidence (Arp & Mason 1977; Zeng et al. 1996) shows that flow is not, in practice,
reversible.
In the current work, we consider the simplest symmetry-breaking interparticle interaction: contact caused by microscopic surface asperities. These microscopic surface
asperities can cause contact between particles when their nominal (smooth) surfaces are
still a small distance apart. This contact is an irreversible effect. As an example, consider
two identical spheres lying in the plane of a shearing flow. If their trajectory is such that
their closest approach is close enough for contact to occur, then at the contact point
Normal stresses in monolayers
3
they are pushed apart by the contact; if the shear flow driving the orbits is reversed, the
spheres will not return to their original positions but will have moved further apart.
The system we study in this paper is not a plane system of circular particles (as in
Lorenz & Hoekstra 2011), which would be subject to the Stokes paradox, but simply
the restriction of a real three-dimensional suspension to the special case in which the
suspended particles lie in a monolayer for all time. This is the natural test case for
simulations of three-dimensional suspensions. Much of the early work on Stokesian
Dynamics was on monolayers: in particular, we will compare our g(r) results with those
of Bossis & Brady (1984), who used DLVO-type repulsive potentials and did not present
any results for rheology; and our rheology results with those of Brady & Bossis (1985),
who considered spheres both without inter-particle forces and with a DLVO potential;
and finally, with the few monolayer results presented in the largely three-dimensional
work of Phung et al. (1996). The other planar study we will refer to is our own earlier
paper (Wilson & Davis 2002). In that work, it was stated that the second normal stress
difference N2 would be zero for a monolayer (interestingly, both of Brady & Bossis 1985;
Phung et al. 1996, also omitted N2 when reporting on monolayer simulations); this is
in fact not true, and in this work we present calculations of N2 , both analytically for
dilute suspensions, and computationally for denser concentrations. We also give corrected
results for the viscosity and first normal stress difference.
In section 2 we introduce the details of the physical system we are considering. Section 3
is concerned with the dilute theory, and in section 4 we describe our numerical simulations
at higher solids fractions. In section 5 we present the results of both sets of calculations.
We draw our final conclusions in section 6.
2. Physical System
We consider an infinite expanse of a Newtonian fluid of viscosity µ, in which are
suspended identical spherical particles of radius a. Throughout the motion, the centres
of the spheres lie in a single plane, described by convention as the plane z = 0. The
area fraction of that plane covered by sphere cross-sections is c. We impose an average
flow (for the whole suspension, fluid and spheres combined) which is simple shear at
shear-rate γ̇:
u = γ̇y 0 0
and report on the resultant rheological response.
Because the true volume fraction is zero, we normalise stresses by the effective volume
of the layer (Brady & Bossis 1985): that is, its area multiplied by a nominal width 2a.
The most general possible form of the deviatoric part of the stress tensor in this shear
flow (after taking into account flow symmetries) is:

  1

η γ̇
0
Σ11 Σ12
0
3 (2N1 + N2 )
1

≡
0
η γ̇
0
Σ =  Σ12 Σ22
3 (N2 − N1 )
0
0
−Σ11 − Σ22
0
0
− 31 (N1 + 2N2 )
(2.1)
The off-diagonal terms define the shear viscosity η of the suspension, while the diagonal
elements may be expressed in terms of normal stress differences: by convention, N1 =
Σ11 − Σ22 and N2 = Σ22 − Σ33 = Σ11 + 2Σ22 . In earlier work (Wilson & Davis 2002) we
stated that N2 ≡ 0 for a monolayer flow; this is not, in fact, the case, and we give values
for N2 later in this paper.
We have not made any calculations of the pressure term here (the term which was
subtracted from the total stress tensor to make it deviatoric). Nonetheless, there is a well-
4
H. J. Wilson
defined particle pressure in the suspension (Jeffrey et al. 1993) which would be modified
by the contacts between particles.
Our contact model (first presented by Davis 1992) treats the asperities as being
sufficiently sparse to have no effect on the hydrodynamics, but simply applying an
interparticle force. This force (equal and opposite on the two spheres in contact) is
specified in terms of components. The component parallel to the line of centres of the
two spheres is sufficient to exactly halt their approach, and has magnitude Fn . The
perpendicular component opposes the shearing motion between the sphere surfaces, and
has maximum magnitude νFn . The contact model has two parameters: the friction
coefficient ν and an effective roughness height h = aζ. This is the surface separation
at which the force comes into action.
3. Rheology Calculation: Dilute Suspensions
For dilute suspensions, we can express the macroscopic stress as an asymptotic expansion in small c; that is, an expansion about the state of no particles (which is, of course,
a Newtonian fluid having viscosity µ).
The leading-order correction to the suspension rheology is just an enhancement to the
viscosity η:
5
η =µ 1+ c .
(3.1)
3
This is the equivalent of the well-known (Einstein 1906, 1911) three-dimensional viscosity
correction η3D = µ(1+ 52 φ) for a suspension at volume fraction φ. The result given here is
for a monolayer of spheres, in a three-dimensional fluid volume, not a plane flow around
disks for which the result would be µ(1 + 2c) (Brady 1983).
When we continue the expansion to order c2 , the result can no longer be represented
by a single scalar viscosity. All three of the rheological functions are modified at this
order, and they can be expressed as
η = µ 1 + 35 c + kc2
(3.2)
N1 = µc2 γ̇ Ñ1
N2 = µc2 γ̇ Ñ2
(3.3)
(3.4)
in which all three of k, Ñ1 and Ñ2 are dimensionless quantities, independent of the values
of µ, c and γ̇. To find the values of k, Ñ1 and Ñ2 , we must calculate the average stress
tensor
c
S ij
(3.5)
Σij = 2µEij +
2πa3
where Eij is the symmetric part of the background flow gradient tensor and S ij is the
average stresslet:
Z
S ij =
Sij (r) P(r) d2 r
(3.6)
A
where Sij (r) is the stresslet induced by a pair of particles at positions x0 , x0 + r and
P(r) is the probability of finding two particles in that configuration given that our test
particle is at x0 (scaled such that P(r) → 1 as |r| → ∞). In section 3.1 we give the pair
distribution function, as first calculated by Wilson & Davis (2002), and in section 3.2 we
calculate the stresslet Sij . These are brought together to calculate the full rheology of
the suspension correct to order c2 , which is given in section 5.
Normal stresses in monolayers
5
y
x
Figure 1. Schematic of the pair distribution function in the case where contact breaks all closed
orbits. Particles which would otherwise have been in the white region are on trajectories which
come closer than contact. At close approach, they are moved onto the contact surface (high P,
thick-lined portion of the outer circle) which they then leave at x = 0 onto a sheet of high P.
Thus the black regions of very high particle density contain all the particles missing from the
white regions. Taken from figure 1 of Wilson & Davis (2002).
3.1. Pair distribution function
In the absence of Brownian motion, the pair distribution function P(r, t) evolves according to a simple advection equation (the high-Péclet number limit of the Smoluchowski
equation):
∂P
+ ∇ · [PV] = 0
∂t
where V(r) is the velocity of a sphere centred at x0 + r relative to one centred at x0 .
Thus the steady-state pair distribution P(r) satisfies
∇ · [PV] = 0.
(3.7)
This equation is first-order, and may be solved using the method of characteristics, which
essentially relates P at one point on a particle’s trajectory to its value at any other point
on the trajectory. If the trajectory comes from infinity, our imposed boundary condition
P(r) → 1 as |r| → ∞ is sufficient to specify the distribution function everywhere on the
particle path.
In a simple shear flow containing two solid spheres, some initial configurations cause
the spheres to perform orbits (Batchelor & Green 1972b): that is, they periodically move
around one another such that the trajectories of one relative to another are closed. These
trajectories never see the boundary condition at infinity, and so their pair distribution
function P is not defined†. So for perfectly smooth spheres, we cannot calculate the
suspension viscosity to order c2 . However, the largest of these orbits in the plane of shear
has a closest approach separation of 2.11 × 10−4 a, which is lower than many estimates of
the size of particle roughness. If the roughness height is larger than this value, then the
effect of contact in a monolayer shear flow is to break all closed trajectories. Particles
which would have carried out closed orbits are moved outwards by contact and eventually
become well separated. The pair distribution function is now well-defined everywhere,
and takes the form illustrated schematically in figure 1. (This would not happen in three
dimensions as there are arbitrarily large closed orbits outside the plane of shear.)
In the bulk (grey) region, the pair distribution is unaffected by contact, and is given
† The inclusion of weak Brownian motion would cause P to be defined everywhere, but at
the expense of the simplicity that allows us to calculate most of the quantities in this paper.
6
H. J. Wilson
by‡
P(r) = (1 − A(s))−1 φ−2 (s)
(3.8)
Z ∞
A(s′ ) − B(s′ ) ds′
φ(s) = exp
(3.9)
1 − A(s′ ) s′
s
in which we have defined the scalar separation s = |r|/a, and the functions A(s) and
B(s) are two-sphere hydrodynamic functions (Kim & Karrila 1991). Note that φ(s) is a
hydrodynamic mobility function, unrelated to the volume concentration φ; both of these
notations are conventional.
On the contact surface (black arc), the distribution function must be calculated
numerically; and since this provides the upstream boundary condition for the sheet
region (black curve) the contribution from the sheet, too, is dependent on one numerically
calculated data point. In the wake (white) region, we have P = 0 so this region does not
contribute to the stress integral of equation (3.6).
This schematic form of the pair distribution function was first described qualitatively
by Brady & Bossis (1985), though they did not describe the sheet structure. They used
a pure-repulsive interparticle force at a set separation, in three-dimensional simulations
which incorporated Brownian motion. However, their force is not simply our limit ν = 0,
because they imposed their force via a step-function potential, which limits the maximum
magnitude of the force (thereby permitting particles to approach within the nominal
contact height), whereas our forces can increase indefinitely if the hydrodynamic forces
driving particles together require it.
3.2. Stresslet calculation
There are two possible sources of stresslet for a given configuration of two particles.
There is the hydrodynamic stresslet, given by their direct interaction with the fluid:
3
H
2
Sij
= 20
3 πa µ (1 + K(s))Eij + L(s)(ni Ejk nk + nj Eik nk − 3 nk Ekl nl δij )
+ M (s)(ni nj − 31 δij )nk Ekl nl , (3.10)
and, if the particles are in contact, there is an additional stresslet induced by the contact
force Fc :
C
Sij
= 21 as(1 − A)Fkc nk (ni nj − 13 δij )
h
h
+ y12
))(Fic nj + ni Fjc − 2ni nj Fkc nk ),
+ 41 as(1 − B − 2(y11
(3.11)
h
h
where K, L, M , y11
and y12
are also two-sphere hydrodynamic functions (Kim & Karrila
1991). Since the two particles are in contact, their dimensionless separation is s = 2 + ζ
and all the hydrodynamic functions are evaluated at that separation. Here n is the unit
vector along the line joining the centres of the two particles.
The contact forces Fc can also be calculated in terms of two-sphere hydrodynamic
functions; they depend additionally on the friction coefficient ν, and are given in full in
equations (2.28) and (2.29) of Wilson & Davis (2002).
C
H
= 0 in these deviatoric
= S33
In our previous work we assumed (erroneously) that S33
forms of the stresslets. This led us to give incorrect values for N1 and not to calculate
N2 as it would not have been independent of N1 (our assertion that N2 would be zero
was a further error).
‡ There was a typographical error in the definition of φ(s) in equation (2.8) of Wilson & Davis
(2002); the form given here is correct.
Normal stresses in monolayers
7
4. Concentrated Suspensions
To study more concentrated suspensions, we use small Stokesian Dynamics simulations
(25 particles in a periodically-replicated box). We experimented with two different
roughness heights, motivated by experimental measurements: ζ = 10−3 and ζ = 10−2 ;
however (as explained in §5.2) we present our main results here only at ζ = 10−2 . Each
simulation is run for 200 shear units total, from an initially pseudo-random distribution
of particles. The results from the first 20 units are discarded (to allow microstructure to
build up); each data point reported here is a time-average over the remaining 180 shear
units, averaged across two independent simulations.
The contact forces are not just a simple repulsive force based on separation, such as
would be used to prevent overlap (or imitate Hertz contact) in a standard Stokesian
Dynamics implementation (e.g. Brady & Bossis 1985). Rather, the contact force necessary to halt the approach of a pair of particles is calculated, using the lubrication
approximation, from the free velocities; the velocities those particles would have under
pure hydrodynamics with no contact. The tangential component required for rolling
contact (i.e. no relative motion at the point of contact) is also calculated using the
lubrication approximation, but of course the constraint that the magnitude of the
tangential force is no larger than ν times the magnitude of the normal force is applied
at this point. We track which pairs of particles are in contact: a pair enters the contact
state if the free velocities of the two particles indicate they are approaching and their
surface separation falls below 1.001aζ; the pair only leaves contact when the free velocities
indicate that the particles are separating. At each timestep we reset all contacting pairs
which are closer than 0.999aζ or further than 1.001aζ apart, back to a separation aζ.
Similar calculations were carried out in our earlier work (Wilson & Davis 2002) but
with a fundamental coding error related to our belief that N2 must be zero for a
monolayer. Essentially, in reducing each tensor to a linear array of independent values
(as one must do in implementing Stokesian Dynamics) we built in the assumption that
each deviatoric stresslet would have an identically zero out-of-plane component S33 .
5. Results
5.1. Computational results: general observations
We have not included a short-range repulsive force other than the contact force in our
simulations, as is usually done with Stokesian Dynamics simulations. This is because we
felt that a study, like this one, which focuses on the effect of changing the inter-particle
force, should not have an extra inter-particle force beyond the one being studied. This
does, however, mean that we have potential problems with overlapping particles (as seen
in early works without repulsive forces, e.g. Bossis & Brady 1984). When the roughness
height is large (ζ = 10−2 ) we have no trouble; but for our smaller roughness height (ζ =
10−3 ) particles begin to overlap (and the simulations halt) at an area concentration c =
0.6, and for smooth particles ζ = 0 we have overlap problems even at low concentrations.
This means we cannot address the issue of jamming. In particular, the recent predictions by Mari et al. (2014) that the viscosity curves should diverge at a different φm
depending on the friction coefficient are beyond the scope of this paper to verify (or
not). When we talk in §5.3 about the dependence of viscosity on friction coefficient in
concentrated suspensions, the whole discussion should be read with the caveat well away
from the jamming transition.
8
H. J. Wilson
Figure 2. Snapshot of the positions in a “crystallised” simulation at c = 0.55, ζ = 10−3 , ν = 0.5,
time t = 120. We have extended the size of the plot to illustrate 9 copies (and a little more in
the width) of the central simulation cell (shown in outline). Each crystal moves as a rigid body,
horizontally with a different velocity, as shown schematically with the large arrows.
5.2. Crystallisation and system size
These are small simulations, and it is natural to ask how important is our system size.
Realistically, we are not expecting such a small simulation to give a good quantitative
measure of the rheological properties; but we hope that trends we observe will be robust
to changes in size of system.
In several simulations we observe crystallisation occurring at long times. An example
crystalline structure is shown in figure 2: the layers shown here, which are close to
hexagonal close packing, move past one another but the particles do not move within
each layer. This behaviour is clearly unphysical in the sense that it would not occur in
a truly macroscopic system without periodic boxes; and it can be used as an indicator
that our system is too small.
The hallmark of crystallisation can be seen in the plot of any of the rheological functions
against time. From a noisy signal with large oscillations, the rheology settles down
(sometimes quite suddenly) to a periodic signal which oscillates with a period of one
shear unit, corresponding to the crystal moving past one image of itself to the next.
It is tempting to think that one should be able to quickly derive the approximate
viscosity of such a suspension, by working out the volume of the crystal and thus of the
free solvent, and constraining the shear flow to lie within the solvent layer (treating the
crystal as an impermeable solid). But this is a monolayer crystal in a three-dimensional
solvent shear flow, and the corresponding calculation is too involved to be worth the
effort for such a crude approximation.
This crystallisation phenomenon was never seen in our simulations at the larger
roughness height ζ = 10−2 , nor at low to moderate concentrations c 6 0.4. However,
at the smaller roughness height ζ = 10−3 , it occurred in almost every simulation having
c > 0.5 (only for the case c = 0.5, ν = 0.5 was there one simulation which did not
crystallise within 200 shear units). This suggests that the more frequent contacts which
occur with a larger roughness height act as a source of noise, or perturbations, on the
system, keeping it away from the crystal state.
9
Normal stresses in monolayers
80
70
Viscosity η/µ
60
50
40
30
20
10
0
0
20
40
60
80
100
120
140
160
180
200
Time, t
Figure 3. Time-trace of the viscosity for one simulation having c = 0.5, ζ = 10−3 and ν = 0.5.
In the range 100 < t < 120 we see a nearly-steady viscosity showing oscillations of period 1 (i.e.
one shear unit). This is a quasi-crystallised state which eventually breaks down and the system
returns to whole-system fluctuations.
In one of the simulations at c = 0.5, ζ = 10−3 and ν = 0.5, the crystallisation is
transient. Its viscosity plot against time is given in figure 3. We can see that for times
between 100 and 120 shear units, we have small oscillations of period 1 in the viscosity
trace, the hallmark of the crystallised state. However, in this case (confirmed by a video
of particle motion) there is some small deformation at the edge of the close-packed region.
This eventually causes the phase separation to break down, and the simulation returns
to whole-system dynamics.
In all other runs, a visual inspection of the time-traces indicates that over time there
are what looks like statistically homogeneous fluctuations around a steady rheology, and
that 20 shear units is more than ample for this state to be attained (indeed, 10 shear
units is enough in the vast majority of cases).
In the crystallising suspensions, the steady state reached is clearly a box-size artefact,
so we are not presenting results from these runs. This means that we have no results
for the rheology of systems at ζ = 10−3 and c > 0.5. In all other runs, we will
present the means (for all three rheological measures) and standard deviation (at higher
concentrations only) in sections 5.3, 5.4 and 5.5.
Finally, we have tried to give a feel for the importance of box size by running one
simulation with 100 particles: still small, but on the limit of what can be done with the
current technique and the computational power available to us. Figure 4 shows the timetrace of each rheological function for these parameters (c = 0.5, ζ = 10−3 , ν = 0) over
100 time units. Beyond 100 time units, both the simulations having 25 particles have
crystallised. We can see that the fluctuations are (as one might expect) rather smaller
for a system having more particles; and (in this case at least) the “steady state” value
of a crystallised system is neither reproducible over different realisations, nor a good
predictor of the mean value from the larger system (except, in this case, for N2 ; this is
probably just coincidence).
5.3. Viscosity
For very dilute suspensions, the pair-based calculation to order c2 is not possible for
roughness heights less than the closest approach of a closed orbit, ζ < 2.11 × 10−4 ,
because of the indeterminacy of P, as discussed in section 3.1. (This difficulty does not
occur in normal stress calculations because a closed orbit does not contribute to the
normal stress differences, by symmetry.)
10
H. J. Wilson
120
20
0
(b)
15
(c)
-20
-40
80
60
40
N2 /µγ̇
10
N1 /µγ̇
Viscosity η/µ
(a)
100
5
0
-5
-60
-80
-100
-120
-10
-140
-15
-160
20
-20
0
-180
-25
0
20
40
60
80
100
-200
0
20
40
60
80
100
0
20
40
60
80
100
Time, t
Time, t
Time, t
Figure 4. Time-traces of (a) viscosity, (b) first normal stress difference and (c) second normal
stress difference, at c = 0.5, ζ = 10−3 , ν = 0. Grey lines: two independent simulations with
25 particles. Black line: simulation with 100 particles. Both the smaller simulations end in a
crystallised state whose stress oscillates with period one shear unit, while the larger simulation
continues to demonstrate whole-system fluctuations (the amplitude of which reduces for t > 40).
The observation that N2 is large and negative is robust across all three simulations, even when
crystallisation occurs.
In figure 5 we show the O(c2 ) coefficient of viscosity, k, for a range of roughness heights
ζ > 2.11 × 10−4 , for the two extreme cases of friction coefficient ν = 0 and ν → ∞. At
all roughness heights, k is found to be a monotonic increasing function of ν so these
two cases capture the whole dependence on friction coefficient: the viscosity increases
very weakly with increasing friction. More significant is the change in viscosity with
increasing roughness height. It is clear that increasing the size of the roughness decreases
the suspension viscosity. This is because the particles are excluded from configurations
with very small interparticle gaps, which are precisely those configurations in which
lubrication interactions cause strong dissipation. The viscosity values given here include
a direct contribution from the contact force, which does of course act to increase the
viscosity; but its effect is dwarfed by the reduction in the hydrodynamic shear stress.
However, we should note that we have not allowed for the hydrodynamic contribution
from a narrow lubrication region between the asperities, simply because to do so would
be to re-introduce the reversibility which the contact was designed to break. If these
lubrication stresses were included they might well reverse the trend for roughness to
lower the macroscopic viscosity.
The dilute results presented here are qualitatively unchanged from our earlier work
(Wilson & Davis 2002) but have been quantitatively changed following the removal of
the assumption that N2 ≡ 0.
In figure 6 we give the results of our simulations at higher concentrations, viscosity
plotted against concentration. It should be noted that we cannot approach jamming in
these simulations due to the nature of our contact implementation (as discussed in §4) so
we are not capturing the shear-thickening recently explained by, for example, Mari et al.
(2014). In this figure we plot comprehensive results for each roughness height we have
studied, along with data from the literature for comparison.
We can see that introducing contact seems to lower the viscosity, at least at moderate
concentrations, in a continuation of the trend for dilute systems, in which the forced
increase of the interparticle gap reduces the magnitude of lubrication stresses. At higher
concentrations, there is only one data point to compare with from simulations without
interparticle forces: the point at c = 0.675 taken from Phung et al. (1996). Those
simulations included Brownian motion, and we have taken their result at a large, but still
finite, Péclet number, which may affect comparability between their results and ours; we
would appear to be predicting a higher viscosity than they do.
In all cases, the experimental results predict a lower viscosity than the simulations.
11
Normal stresses in monolayers
Viscosity coefficient k
5.1
5
4.9
4.8
4.7
4.6
4.5
4.4
0.001
0.01
0.1
Roughness height ζ
Figure 5. Viscosity in a dilute suspension: plot of the O(c2 ) contribution to viscosity k, against
roughness height ζ. The solid curve represents the case of no friction, ν = 0, and the dotted
curve the opposite extreme, ν → ∞.
This is expected to be because of the limitations of the monolayer idealisation. In the
experiments of Belzons et al. (1981), the particles lay in a layer of oil of width 2a. The
forces constraining them to the layer are moderate at best, so when extreme lubrication
forces occur as two particles come close together, it is expected that some particles will
move out of the strict monolayer to find an “easier” way past (Brady & Bossis 1985).
Experimental results of Bouillot et al. (1982) (not shown here) give even lower viscosities,
with η/µ 6 5 even at c = 0.7.
We can clearly see that increasing friction increases the viscosity here. This generic
picture is in accord with the recent simulations by Gallier et al. (2014) (and with our
earlier results Wilson & Davis 2002). However, it should be noticed that increasing
friction only changes our viscosity measures by around 10–20% whereas the results in
Gallier et al. (2014) (which are for three-dimensional suspensions with ζ = 5 × 10−3 )
suggest that increasing ν from 0 to 0.5 can increase the viscosity by around a factor of 2
at a volume fraction φ = 0.4. If we scale relative to the volume fraction at which jamming
would occur in the absence of contact, we can roughly convert their volume fraction to
an area concentration as
c
φ
≈
cm
φm
φ
c
≈
0.78
0.64
c[φ=0.4] ≈ 0.5.
At an area concentration of c = 0.5, changing friction from ν = 0 to ν = 10 increases our
viscosity by only 5%.
It is possible that the behaviour for intermediate values of ν is outside that of the
two extreme values, which could explain the difference between our work and that of
Gallier et al. (2014); so in table 1 we present the average viscosity at four different values
of ν, at each of the higher concentrations we have used. We can see that for the specific
case in point, increasing the friction coefficient from zero to 0.5 does indeed cause a
greater increase in viscosity than using a much higher friction coefficient; but the viscosity
increase is still much smaller than that seen by Gallier et al. (2014), at only 8%. We can
only conclude that the strong increase in viscosity due to friction is an essentially threedimensional effect. Given the geometrical constraints involved, this does not seem an
unreasonable conjecture.
In table 1 we have also given the standard deviation of each viscosity measure, to give
12
H. J. Wilson
4
11
(a)
9
8
Viscosity η/µ
3
Viscosity η/µ
(b)
10
3.5
2.5
2
1.5
7
6
5
4
3
2
1
1
0
0.1
0.2
0.3
0.4
0
Area concentration c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Area concentration c
Figure 6. Dependence of viscosity η on area fraction c. Roughness heights (a) ζ = 10−3 , (b)
ζ = 10−2 . For the smaller roughness height we could not obtain results for c > 0.4. In each
graph, the thick curves represent the relevant dilute theory (at both ν = 0 and ν → ∞) as
shown in figure 5; the two curves on each graph are indistinguishable from one another. The
lower thin curve (straight line) in each plot is the Einstein viscosity value η/µ = 1+5c/3, correct
to order c. The upper thin curve is the Batchelor & Green (1972a) value η/µ = 1 + 5c/3 + 7.4c2 ,
calculated for smooth spheres using the assumption that the pair distribution function is given
by equation (3.8) everywhere (when in fact it is indeterminate on the closed trajectories). The
filled circles (identical on the two plots) are simulation results from the literature, for purely
hydrodynamic forces between the particles. Most points here are from Brady & Bossis (1985)
but the result at c = 0.675 is from Phung et al. (1996). The open circles are experimental results
by Belzons et al. (1981) (extracted from Brady & Bossis 1985). The remaining symbols are our
simulation results at ν = 0 (+) and ν → ∞ (×). For c > 0.4 we found we could not simulate in
the true limit ν → ∞ so these results are for ν = 10.
c
ζ
0.40
0.40
0.50
0.55
0.60
0.001
0.010
0.010
0.010
0.010
Average viscosity
ν = 0 ν = 0.5 ν = 1 ν = 10
Standard deviation
ν = 0 ν = 0.5 ν = 1 ν = 10
2.209
2.137
3.237
4.043
5.282
0.066
0.067
0.074
0.179
0.131
2.292
2.236
3.509
4.441
5.828
2.238
2.229
3.516
4.388
5.879
2.253
2.233
3.399
4.450
5.956
0.115
0.068
0.136
0.148
0.160
0.126
0.091
0.103
0.126
0.105
0.077
0.083
0.135
0.110
0.141
Table 1. Values of the time-averaged suspension viscosity (normalised by the viscosity of
the suspending fluid) as it depends on area concentration c, roughness height ζ, and friction
coefficient, ν.
a feel for the size of fluctuations in the time signal. For all the results given here, the
standard deviation is less than 5% of the mean; however, this does weaken further our
confidence in the dependence on friction coefficient.
The dependence of viscosity on roughness (either height or friction coefficient) is,
for our simulations, much less important than the dependence on area concentration;
essentially, away from jamming concentrations at which friction becomes very important (Mari et al. 2014) its effects are not critical, though the conclusions of our previous
work, that the viscosity is reduced by surface roughness at these concentrations, still
hold.
13
Normal stress coefficient Ñ1
Normal stresses in monolayers
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
0.001
0.01
0.1
Roughness height ζ
Figure 7. First normal stress difference in dilute suspensions: Plot of the normalised first normal
stress difference Ñ1 , against roughness height ζ. The solid curve represents the case of no friction,
ν = 0, and the dotted curve the opposite extreme, ν → ∞.
At the concentrations given in table 1 the fluctuations in viscosity look like symmetric
noise around a steady signal; at lower concentrations, the pattern shifts, showing instead
sporadic fluctuations upwards from a low minimum viscosity level.
5.4. First normal stress difference
For dilute suspensions, figure 7 shows N1 /(µc2 γ̇) plotted against roughness height ζ
for the two extreme values of friction coefficient ν = 0 and ν → ∞. Note that for very
small roughness heights ζ < 2.11 × 10−4 , the normal stress differences are identically zero
because of the symmetry of particle orbits.
Roughness is shown to induce a negative value of N1 . The two curves (for zero friction
and infinite friction) are almost indistinguishable, as are curves we have calculated for
intermediate values of the friction coefficient ν (the largest variation at fixed ζ is around
1%), indicating that friction is a negligible component of the first normal stress difference
for dilute suspensions. Again, these results are qualitatively similar to those reported by
Wilson & Davis (2002); the magnitude of N1 is slightly larger than was reported then.
At more realistic concentrations, our simulation results are shown in figure 8. Here there
is a qualitative difference from our earlier predictions. Above a critical area concentration
around c = 0.4 (the concentration at which each particle becomes “close” to more than
one neighbour on average), the first normal stress difference becomes positive, in contrast
to the dilute results: and in contrast to earlier simulations by Phung et al. (1996), which
incorporated weak Brownian motion, the recent work of Gallier et al. (2014), and our
own erroneous earlier work (Wilson & Davis 2002), all of which give N1 negative at all
concentrations. The observation is, however, in accord with the experimental results of
Dbouk & Lemaire (2013).
It would seem from figure 8 that the effect of friction is to reduce the magnitude
of N1 at high concentrations; however, we have carried out calculations at the higher
concentrations with intermediate values of the friction coefficient ν: these are given in
table 2. There are cases where the value of N1 at an intermediate value of ν does not
lie between those at the extremes of ν: for instance, at c = 0.4 and ζ = 10−3 , the value
at ν = 0.5 is larger than those at the extremes of ν, and the value at ν = 1 is smaller
than those at the extremes of ν. However, in all such cases N1 remains positive and of
roughly the same order of magnitude as the extreme values; and there is no coherent
14
H. J. Wilson
0.3
1.5
(a)
(b)
1
Normal stress N1 /γ̇µ
Normal stress N1 /γ̇µ
0.25
0.2
0.15
0.1
0.5
0
-0.5
0.05
0
-1
-1.5
-0.05
-2
0
0.1
0.2
0.3
0.4
0
Area concentration, c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Area concentration, c
Figure 8. Dependence of first normal stress difference N1 on area fraction c. Roughness heights
(a) ζ = 10−3 , (b) ζ = 10−2 . For the smaller roughness height we could not obtain results for
c > 0.4. In each graph, the curves represent the relevant dilute theory (at both ν = 0 and
ν → ∞) as shown in figure 5; the two curves on each graph are indistinguishable from one
another. The filled circle is the only simulation result available from the literature, for purely
hydrodynamic forces between the particles: Brady & Bossis (1985) predict that N1 should be
zero in the absence of non-hydrodynamic forces, but Phung et al. (1996) give one nonzero value
at c = 0.675. The remaining symbols are our simulation results at ν = 0 (+) and ν → ∞ (×).
For c > 0.4 we found we could not simulate in the true limit ν → ∞ so these results are for
ν = 10.
c
ζ
0.40
0.40
0.50
0.55
0.60
0.001
0.010
0.010
0.010
0.010
Average N1
ν = 0 ν = 0.5 ν = 1 ν = 10
Standard deviation
ν = 0 ν = 0.5 ν = 1 ν = 10
0.253
0.208
0.831
1.105
1.103
0.107
0.080
0.087
0.280
0.137
0.391
0.155
0.649
0.866
1.018
0.102
0.121
0.587
0.837
0.856
0.114
0.087
0.504
0.784
0.952
0.535
0.125
0.051
0.075
0.136
0.126
0.097
0.112
0.074
0.195
0.091
0.050
0.100
0.054
0.075
Table 2. The first normal stress difference N1 (normalised by the solvent viscosity µ and the
shear rate γ̇) as it depends on area concentration c, roughness height ζ and friction coefficient
ν.
trend here. These are noisy results (the normal stress difference fluctuates over time and
is often instantaneously negative) and we should not read too much into small differences
between runs.
For the small roughness height ζ = 10−3 , the standard deviation of N1 is the same
order of magnitude as the stress itself, so we cannot be sure that we are predicting a
positive value for N1 ; however, at ζ = 10−2 the standard deviation is closer to 20% of
the mean value and we can attribute some weight to our claim that N1 > 0 for c > 0.5
in a monolayer suspension.
We have only given one data point from the literature here: the value −1.75 at c = 0.675
given by Phung et al. (1996). We have not been able to reproduce this even qualitatively.
However, there is more information available. Brady & Bossis (1985) found nonzero
values for N1 when they used a DLVO-type repulsive potential. The nature of their
force is such that the normal stress coefficient N1 /γ̇µ is not independent of γ̇; the actual
15
Normal stresses in monolayers
Normal stress coefficient Ñ2
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
0.001
0.01
0.1
Roughness height ζ
Figure 9. Second normal stress difference in dilute suspensions. Plot of the normalised first
normal stress difference Ñ2 against the roughness height ζ. The lower (solid) curve represents
the case of no friction, ν = 0, and the upper (dotted) curve the opposite extreme, ν → ∞.
value of N1 /γ̇µ can range over several orders of magnitude as γ̇ is varied. However, they
do report that N1 is positive at both c = 0.4 and c = 0.5, in agreement with the current
results.
5.5. Second normal stress difference
In this section we present new results for the second normal stress difference N2 ,
which were not calculated in our previous work. For dilute suspensions, figure 9 plots
the normalised second normal stress difference Ñ2 = N2 /µc2 γ̇ against roughness height.
Roughness is shown to induce a negative value of N2 , and again we find that the friction
coefficient is rather unimportant; however, the dependence of N2 on ν is not simple; the
maximum magnitude of N2 occurs for ν = 0 and the minimum in the limit ν → ∞ but
the curve has two turning points in the range 0 < ν < 1. N2 is smaller in magnitude
than N1 by a factor of around 3–5.
At higher values of the concentration, our simulation results are plotted in figure 10.
They indicate that N2 continues to be negative, but with a much larger magnitude than
that predicted by the dilute theory. This is in agreement with all the literature for threedimensional suspensions, indicating that the second normal stress difference does have
the same physical meaning for a monolayer as a full suspension.
Though our simulations for the small roughness height are limited by crystallisation,
even these limited results make it clear that the magnitude of N2 is larger for small
roughness heights. This may seem counter-intuitive, but can perhaps be justified as
follows: when the roughness height is larger, particles cannot approach one another
closely, which limits the size of local lubrication stresses. Of course the contact portion
of the normal stress difference will be increased by increasing the roughness height, but
our results suggest that the hydrodynamics dominate here.
If, as reversibility would suggest, both normal stresses are zero for a suspension of
perfectly smooth spheres, then there must be a critical roughness height which has
maximum effect; the results reported here suggest that this critical roughness height
is probably within or even below the physical range.
The second normal stress difference given in figure 10 appears to be almost completely
independent of the friction coefficient ν. For completeness, we give results at intermediate
values of ν in table 3; any marginal dependence on ν is completely dwarfed by the
16
H. J. Wilson
0
0
(a)
-5
(b)
-1
Normal stress N2 /γ̇µ
Normal stress N2 /γ̇µ
-10
-15
-20
-25
-30
-35
-40
-2
-3
-4
-5
-6
-7
-8
0
0.1
0.2
0.3
0.4
0
Area concentration, c
0.1
0.2
0.3
0.4
0.5
0.6
Area concentration, c
Figure 10. Dependence of second normal stress difference N2 on area fraction c. Roughness
heights (a) ζ = 10−3 , (b) ζ = 10−2 . For the smaller roughness height we could not obtain results
for c > 0.4. Points are our Stokesian Dynamics simulation results at ν = 0 (+) and ν → ∞ (×).
For c > 0.4 we could not simulate in the true limit ν → ∞ so these results are for ν = 10. In
both cases the results from dilute theory are indistinguishable from zero on this scale.
c
ζ
0.40
0.40
0.50
0.55
0.60
0.001
0.010
0.010
0.010
0.010
Average N2
ν = 0 ν = 0.5 ν = 1 ν = 10
-35.127 -39.503 -27.939 -29.365
-2.665 -2.633 -2.547 -2.479
-4.877 -5.089 -4.982 -4.629
-5.581 -6.218 -6.225 -6.268
-6.938 -7.880 -7.918 -7.972
Standard deviation
ν = 0 ν = 0.5 ν = 1 ν = 10
3.744
0.261
0.392
0.262
0.142
15.065
0.324
0.164
0.336
0.308
4.345
0.293
0.315
0.303
0.159
3.040
0.313
0.349
0.212
0.251
Table 3. The second normal stress difference N2 (normalised by the solvent viscosity µ and the
shear rate γ̇) as it depends on area concentration c, roughness height ζ and friction coefficient
ν.
dependence on c and ζ. This is, again, in contrast to the strong frictional dependence
reported by Gallier et al. (2014); at φ = 0.4 and ζ = 5 × 10−3 they saw that changing
from ν = 0 to ν = 0.5 increased |N2 | by a factor of 6.
In table 3 we also show the standard deviation of the time series of N2 . In all cases the
magnitude of the fluctuations is less than 20% of the time-averaged |N2 |; at ζ = 10−2
the fluctuations are usually less than 10% of the mean. This means that our result that
N2 is negative and much larger in magnitude than N1 is robust despite our small system
size.
In common with all other simulations, and the theory, we find that for dilute systems
|N2 | ∝ c2 for small c. This contradicts experimental results of Tanner and coworkers (Tanner et al. 2013; Dai et al. 2013), who found |N2 | ∝ φ3 , and the group of Pouliquen
& Guazzelli (Boyer et al. 2011; Couturier et al. 2011), who claim N2 = 0 for φ < 0.22.
In the dilute theory, we do not see a minimum concentration below which the normal
stresses cease to exist; but we may have an explanation for the φ3 dependence seen by
Tanner. Over the range 0.1 6 c 6 0.6, our results for N2 at ζ = 10−2 are well fit by the
curve N2 ∼ −32c2.8 ; the O(c2 ) behaviour is only seen for truly dilute suspensions having
very small values of c.
17
Normal stresses in monolayers
8
(a) 10
7
30
10
(b)
25
6
5
5
20
5
0
4
0
15
3
-5
10
-5
2
5
-10
1
-10
0
-10
-5
0
5
10
0
-10
-5
0
5
10
Figure 11. Pair distribution function P(r) for two simulations. (a) Dilute suspension: c = 0.1,
ζ = 10−2 , ν = 0, 25 particles, 20 < t < 200; (b) Concentrated suspension: c = 0.5, ζ = 10−3 ,
ν = 0, 100 particles, 20 < t < 100. In each case P(r) is normalised so that P → 1 as |r| → ∞;
and the axes are measured in units of the particle radius a.
5.6. Pair distribution function
It is of interest to see how closely the true pair distribution function in the simulations
follows the schematic form shown in figure 1. In this section we calculate the pair
distribution function directly from the particle positions in the simulations. In each
case, we consider a single simulation, and average over all particles and also over many
timesteps (we discard the first 20 shear units and sample every 0.1 shear unit thereafter).
In figure 11 we give two pair distribution functions: one at a low concentration (c = 0.1)
for comparison with the dilute theory and the second at a higher concentration (c = 0.5).
In the dilute case, the schematic form illustrated in figure 1 is clearly visible. There is, of
course, an excluded region s 6 sc = 2 + ζ. At particle separations just above sc , we see a
dense region in which particles are in contact; and along the horizontal line at y = 0 there
is a depleted (white) wake region. The sheet region is largely obliterated by multi-body
effects, but it is just possible to make it out as a darker “smudge” to the right of the top,
and to the left of the bottom, of the excluded region. Figure 11(b) looks qualitatively
very similar to the top-right image in figure 1 of Brady & Morris (1997), which is the
projection of g(r) for a three-dimensional suspension onto the plane of shear at a Péclet
number of 104 and φ = 0.45.
At the higher concentration, the pair distribution function is completely different, as
the particles spend much of their time close to a hexagonal close-packed arrangement.
There are strong peaks in P(r) where a crystal would form aligned on the compressional
axis, and circular arcs around these positions, indicating that the semi-formed crystals
are rotated by the flow before breaking up.
To capture the variation in these functions in more detail, in figure 12 we show
the radial pair correlation function g(r) (averaged over all angles) for several different
simulations. We see that as c increases, the strength
√ for each of the HCP peaks
√ of signal
increases. These peaks are the positions (s = 2, 2 3, 4, 2 7 etc.) of the centres of the
nearest neighbours of a test sphere in a hexagonal close packed array, which is (as one
might expect) more closely approached at higher concentrations. At c = 0.1 there is
practically no radial structure apart from the close-contact region; as c increases the first
secondary peak to emerge is at √s = 4 (three particles lined up) and then s = 6 (four
particles lined up); the peak at 2 3 only begins to emerge at high concentrations and the
18
H. J. Wilson
10
8
6
4
2
0
2
3
4
5
6
7
8
Figure 12. Angle-averaged pair correlation function g(r), plotted against dimensionless
separation s = r/a, for simulations at ζ = 10−2 and (bottom to top) c = 0.1, 0.2, 0.3, 0.4, 0.5
and 0.6. 1 has been added to each successive plot for clarity: all the functions satisfy g(r) → 1
as r → ∞; the curve shown at level n comes from the simulation at c = 0.1n. The points show
the position of the expected peaks for a hexagonal close-packed arrangement.
c
0.1
0.2
0.3
0.4
0.5
rmax /a g(rmax )
2.11
2.09
2.06
2.04
2.02
12.9
9.5
8.3
8.6
8.0
Table 4. Position and strength of the secondary maximum in g(r) (after the contact peak) at
each concentration c, for a roughness height ζ = 10−2 . In each case g(r) is normalised such that
g(r) → 1 as r → ∞.
other out-of-line peaks are not visible. This is because of the influence of the shear flow
(a close-packed structure cannot flow) but stable strings of close particles which form
along the flow direction can persist without being affected directly by the flow.
In figure 12 we have truncated the y-axis to allow the detail of the structure at large
s to be seen. This necessarily removes the information about the distribution close to
contact. All plots have a large peak at s = sc = 2 + ζ. All plots except c = 0.6 also have
a secondary peak in the near field: we give the position and value of these maxima in
table 4.
Bossis & Brady (1984) presented g(r) in great detail for monolayer simulations with
an interparticle force of the form
F=λ
τ e[−τ h]
n,
1 − e[−τ h]
(5.1)
in which h = s − 2 and λ and τ are scalar parameters. They consider the cases of no
interparticle force (in which their particles often overlap), and τ = 10 or τ = 227. For
τ = 10 the peak in the averaged g(r) occurs at s ≈ 2.1, which is much larger than
the contact distances we are using; so we will use their results at τ = 227 (for which
the peak is around s = 2.02) to compare with ours. In figure 13 we present a detailed
comparison between our results and figure 8 of Bossis & Brady (1984), for which c = 0.4.
In this figure, the function g(r) is sampled over four different angular “slices”, each of
width 18◦ . These are denoted by their mean angle θ, where θ = 0 describes a pair
19
Pair distribution function g(r, θ)
Pair distribution function g(r, θ)
Normal stresses in monolayers
4
(a): θ = 9◦
3.5
3
2.5
2
1.5
1
0.5
0
2
2.5
3
3.5
4
(c): θ = 99◦
3
2.5
2
1.5
1
0.5
0
2
2.5
3
3.5
4
3
2.5
2
1.5
1
0.5
0
2
4.5
Particle centre separation s = r/a
2.5
3
3.5
4
4.5
Particle centre separation s = r/a
Pair distribution function g(r, θ)
Pair distribution function g(r, θ)
4
(b): θ = 27◦
3.5
4.5
Particle centre separation s = r/a
3.5
4
4
(d): θ = 171◦
3.5
3
2.5
2
1.5
1
0.5
0
2
2.5
3
3.5
4
4.5
Particle centre separation s = r/a
Figure 13. Radial dependence of the pair distribution function g(r, θ) at different values of θ
for c = 0.4 and ζ = 10−2 . (a) θ = 9◦ ; (b) θ = 27◦ ; (c) θ = 99◦ ; (d) θ = 171◦ . Lines are from the
current work, crosses (×) from Bossis & Brady (1984), figure 8, taking their parameter τ = 227.
of particles lined up in the shear direction, and the angle increases anticlockwise. The
largest discrepancy between the current work and that of Bossis & Brady (1984) is in
the downstream region (θ = 27◦ and θ = 9◦ ), where they saw a complete obliteration
of the peak at s = 4 corresponding to three particles contacting in a straight line. We
believe this is because the diffuse nature of their interparticle force causes particles which
are lined up along the flow to be continuously driven apart, while our force ceases to act
once the particles leave contact and they can remain just outside the contact height
indefinitely. This is less important in the compressive quadrant (θ > 90◦ ) as the flow acts
against the repulsive force there.
Again, the graphs in figure 13 truncate the vertical scale of g(r, θ), which loses
information about the distribution close to contact. In figure 14 we show the angular
dependence of g(r, θ) considering only particles which are in contact, at three indicative
concentrations c = 0.1 (dilute), 0.4 (comparing with Bossis & Brady 1984) and 0.6
(concentrated).
Here we find something genuinely new. Even for our dilute system, there is a preponderance of contacting pairs in the quadrant having positive slope (0 < θ < 90◦ ). This
is in contrast to the dilute theory of section 3 and also (at c = 0.4) to the results of
Bossis & Brady (1984).
This counter-intuitive behaviour led us at first to suspect a simple coding error in
the software to extract g(r) from the particle positions. However, producing videos of
20
H. J. Wilson
1
0.8
0.6
0.4
0.2
0
0
45
90
135
180
Figure 14. The pair distribution function of contacting pairs, g(rc , θ) plotted against θ, at
different area concentrations. Points joined with lines are our results: c = 0.1 (∗), 0.4 (+) and
0.6 (). The cross symbols (×) are results from Bossis & Brady (1984) at c = 0.4 (counting all
pairs with 2 < s < 2.03), which are quite different. All these functions have been normalised by
their maximum values.
the particle dynamics, it is clear from watching the dynamics at c = 0.1 that this is
not the case. Some isolated pairs of particles are seen to behave exactly as predicted
in our dilute theory: they are pushed together in the compressive quadrant of the flow
180◦ > θ > 90◦ and separate shortly after θ passes through 90◦ ; but other pairs, in
which a third particle is close, stay in contact throughout a Jeffery orbit, and spend far
longer in the supposedly extensional quadrant than they do in the compressive quadrant,
thereby contributing much more to g(r) in the extensional quadrant. We do not have a
good explanation for this behaviour but it is robust.
6. Conclusions
We have assessed the rheology and microstructure of sheared monolayer suspensions
with frictional contact between spheres.
We find that viscosity is lowered by contact, for both dilute and moderately concentrated systems, but our neglect of the lubrication stresses at the contact point is likely
to mean we are underpredicting the viscosity. Friction does increase viscosity, but only
marginally, unlike the recent three-dimensional simulations of Gallier et al. (2014), in
which friction can easily double the suspension viscosity. We are unable to approach
jamming, so we cannot comment on the recent results of Mari et al. (2014), who suggest
that the jamming volume fraction is reduced by the addition of friction.
Non-zero normal stress differences are induced by contact, and for dilute systems,
these both scale as c2 , in agreement with many three-dimensional simulations (such as
Morris & Boulay 1999); however, our simulations at moderate concentrations show that a
rough fit for N2 can be obtained with a scaling of c2.8 , which might explain the power-law
of c3 suggested by some experiments (Tanner et al. 2013; Dai et al. 2013).
There is agreement in the literature that N2 is negative and larger in magnitude than
N1 , and we have reproduced this; but our dependence on friction is again far weaker than
that seen by Gallier et al. (2014).
Even the sign of the first normal stress difference is controversial in the literature. We
Normal stresses in monolayers
21
find that N1 is negative for dilute systems, and becomes positive at higher concentrations,
with a crossover at an area concentration c ≈ 0.4.
Finally, our microstructure at contact has produced some surprising observations. Even
for reasonably dilute systems (c = 0.1), the contact value of g(r) has a higher density of
pairs in the downstream quadrant than the upstream quadrant. We observe within the
simulations that an isolated pair behaves as dilute theory predicts, but a pair with a third
particle close to it undertakes a much slower rotation and can stay in close proximity
through a whole orbit. This three-body behaviour is not understood and may warrant
further attention.
Acknowledgement
This work builds on initial studies performed while the author was at the University
of Colorado, Boulder.
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