J.
Fluid
Mech.
(1994),
1101.
275,
pp.
157-199
Copyright
0
1994 Cambridge University Press
157
Pressure-driven
flow
of
suspensions
:
simulation
and theory
By PRABHU R. NOTTT
AND
JOHN
F.
BRADY
Division
of
Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, CA
91
125,
USA
(Received
27
December
1992
and in revised form
3
April
1994)
Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian
suspension at zero Reynolds number were conducted using Stokesian Dynamics. The
simulations are for a monolayer of identical particles as a function of the dimensionless
channel width and the bulk particle concentration. Starting from a homogeneous
dispersion, the particles gradually migrate towards the centre of the channel, resulting
in an homogeneous concentration profile and a blunting of the particle velocity profile.
The time for achieving steady state scales as
(H/~)~a/(u),
where
H
is the channel
width,
a
the radii of the particles, and
(u)
the average suspension velocity in the
channel. The concentration and velocity profiles determined from the simulations are
in qualitative agreement with experiment.
A
model for suspension flow has been proposed in which macroscopic mass,
momentum and energy balances are constructed and solved simultaneously. It is
shown that the requirement that the suspension pressure be constant in directions
perpendicular to the mean motion leads to particle migration and concentration
variations in inhomogeneous flow. The concept of the suspension ‘temperature
’
-
a
measure
of
the particle velocity fluctuations
-
is introduced in order to provide a non-
local description of suspension behaviour. The results of this model for channel flow
are in good agreement with the simulations.
1.
Introduction
The behaviour
of
flowing suspensions has been a matter of interest for many years
and there has recently been a considerable increase in research activity. From a
practical point of view, the importance of this subject stems from the many
applications in industry where the processing and transport of suspensions is an
important operation. Of equal importance is the need for a fundamental understanding
of the physical phenomena occurring in suspensions. Though there is as yet no
complete theory that fully accounts for all the interactions and resultant phenomena
in concentrated suspensions, recent advances in experimental and analytical techniques
have led to some progress in the understanding of these systems.
The present work focuses on the migration
of
particles in situations where
inhomogeneous stress or shear fields are present, specifically in the pressure-driven flow
of a suspension in a channel. This phenomenon is distinct from the migration of
particles due to inertial forces, first observed by SegrC
&
Silberberg
(1962).
Here, the
motion of the suspension
is
in the Stokes flow regime, i.e.
Re,
<
1,
where
Re,
is the
t
Current address: India Institute of Science, Bangalore, India.
6
FLM
275
158
P.
R.
Nott and
J.
F.
Brady
Reynolds number based on the size of the suspended particles. This restriction to low
Reynolds numbers
is
not an overly severe assumption, however, because it is satisfied
in many practical situations of suspension processing. Furthermore, particle migration
in inhomogeneous flows represents a new fundamental process and therefore deserves
study in all regimes of Reynolds number.
Most previous theoretical and experimental studies of suspension flow have been on
situations of uniform stress (or rate of strain) fields, e.g. narrow-gap Couette flow and
shear flow between parallel plates. In such situations, the ‘state’ of the suspension is
independent of position and there is thus no preferred position to be sought by the
particles
-
an average particle moves about randomly as it follows along the mean
suspension motion. This random motion is due to the chaotic nature of the particle
evolution equations in concentrated suspensions. This random motion can give rise to
a diffusive behaviour, and the shear-induced self-diffusivity of non-Brownian particles
has been measured experimentally by Eckstein, Bailey
&
Shapiro (1977) and Leighton
&
Acrivos (1987a), and has been studied via Stokesian dynamics simulations by Bossis
&
Brady (1987), Phung
&
Brady (1992) and Phung (1993).
The problem under consideration in the present work is different in that the concern
here is on
net migration
of particles in a suspension exposed to inhomogeneous stress
or shear fields. Inhomogeneous suspension flows are,
of
course, common in practical
situations-flow in tubes or channels is probably the most common example.
Moreover, most experimental measurements of suspension rheology are made in
Couette viscometers, and this flow geometry may also give rise to an inhomogeneous
flow field, unless the gap between the cylinders is small compared to their radii.
Experiments on suspension flow in these geometries have been conducted by several
investigators in recent years (Karnis, Goldsmith
&
Mason 1966; Leighton
&
Acrivos
1987a; Hookham 1986; Koh 1991; Koh, Hookham
&
Leal 1994; Sinton
&
Chow
1991; Abbott
el
al.
1991).
The first evidence of shear-induced migration was provided by the experiment of
Gadala-Maria
&
Acrivos (1980), where they observed the steady decrease
of
the
viscosity of a concentrated suspension in a Couette viscometer. Later, Leighton
&
Acrivos (19873) showed by further experiments that this was due to migration of
particles from the Couette gap into the reservoir at the bottom, which was at a
considerably lower shear rate. Migration of particles in this case was perpendicular to
the plane of shear, but in the direction of the gradient in shear rate; in all other studies
(on pressure-driven flow and wide-gap Couette flow), the gradient in shear rate was in
the same direction as the shear rate. Among the studies on pressure-driven suspension
flow, Hookham (1986) and Koh
et
al.
(1994) have observed substantial particle
migration accompanied by blunting (or flattening) of the velocity profile, while Karnis
et
al.
(1966) and Sinton
&
Chow (1991) report velocity blunting but no detectable
migration of particles. The experiments of Abbott
et
al.
(1991), in a wide-gap Couette
device, show clearly the movement of particles away from the rotating inner rod, i.e.
from regions of high to low shear rates.
Theoretical models for the migration process have been proposed by Leighton
&
Acrivos (19873) and Jenkins
&
McTigue (1990). The former authors proposed a
diffusion equation for particles, to be solved in conjunction with the continuity and
momentum equations for the entire suspension. Jenkins
&
McTigue on the other hand
considered only the particle phase, and, in a manner akin to the treatment of dry
granular materials, they propose conservation equations for mass, momentum and
fluctuational energy for the particular phase. Thus, in their one-component system,
there is no diffusion equation, but the particle concentration, ‘temperature
’
(which is
Pressure-driven
Jlow
of
suspensions
159
a measure of the fluctuational motion of the particles), and bulk velocity fields are set
up
so
as to satisfy the continuity, momentum and energy equations. In this paper we
develop a
‘
suspension balance’ model which phenomenologically has similarities with
model Jenkins
&
McTigue propose, although the physics underlying the model is quite
different.
From a fundamental standpoint, the mechanism for shear-induced migration has
been controversial. As recently as in 1991, Abbott
et
al.
attributed it to ‘the existence
of forces not described by Stokes equations
’,
because it was implicitly assumed that the
reversibility of the Stokes flow cannot produce irreversible motior,. The mechanism put
forward by Leighton
&
Acrivos is that small-scale surface roughness of the particles
leads to irreversible motion during inter-particle interactions; in a flow field with a
gradient in shear rate, shear stress or particle concentration, net migration arises due
to a greater number of interactions on one side
of
a particle than the other. However,
surface roughness is not the sole mechanism and, as we demonstrate in this work,
irreversible migration is produced even when the suspended particles are perfect hard
spheres.
The serious disparities in the findings of the earlier investigations on the phenomenon
of shear-induced migration and the ongoing speculations on the mechanism for
migration provide motivation for further investigation on this subject, both
experimental and theoretical. There is also a need, from the perspective of macroscopic
modelling, to understand the mechanisms that cause microstructural changes in
suspensions and to help develop the proper conservation and constitutive equations.
The purpose of this study is two-fold. Firstly to verify earlier experimental findings on
inhomogeneous suspension flow and rationalize the differences between them. To this
end, we have performed dynamic simulations of the pressure-driven flow of a
suspension in a rectangular channel by Stokesian Dynamics (Brady
&
Bossis 1988), a
method which accurately and efficiently computes the many-body long-range
hydrodynamic interactions as well as the short-range lubrication interactions in Stokes
flow. Since our simulations are for a suspension of perfect hard spheres at zero
Reynolds number, factors such as inertial effects, surface roughness and non-
hydrodynamic forces can be controlled. This provides a ‘clean’ experiment with which
the results of earlier investigations can be compared. Secondly, it is hoped that this
work will provide further mechanistic insight into the phenomenon of shear-induced
particle migration and that the macroscopic transport equations we develop will
provide a rational framework for modelling suspension flows.
In
$2,
we derive the timescale for achieving steady state in the pressure-driven flow
of a viscous suspension, which is also valid in any other type of flow (e.g. wide-gap
Couette flow), and show that it is considerably longer than thcd in the laminar flow of
a pure Newtonian fluid. With this estimate of the timescale, we note that most
measurements of earlier investigations were made within the transition length, i.e.
before steady state was achieved, thus providing a probable explanation for the
differences in their observations. Next, in
$3,
we outline the simulation method and its
application to pressure-driven flow and discuss the importance of the suspension
temperature in the dynamics of concentrated suspensions. The results of the simulations
for various particle concentrations and channel widths are then presented in
$4,
where
we show substantial particle segregation and, concomitantly, blunting of the particle
velocity field for all cases studied. The results
of
the simulations are compared with the
experimental data of Karnis
et
al.
(1966) and Koh
et
al.
(1994) in
$5.
We observe that
most of the experiments of Koh
et
al.
were not at steady state, but for those that were
the concentration fields determined from our simulations are in qualitative (and
6-2
160
P.
R.
Nott
and
J.
F.
Brady
perhaps quantitative) agreement. However, the velocity measurements reported by
Koh et
al.
are in all cases considerably lower than the observations of this work. In
96,
we compare the simulation results with the predictions of the ‘diffusive flux’ model of
Leighton
&
Acrivos and our ‘suspension balance’ model. We discuss the relative merits
of the two models and show that the former can be derived as a special case of the
latter. Finally, we conclude with a discussion of how the self-diffusion of a marked
particle arises from the suspension balance equations, and present the criterion for a
homogeneous shear flow to be stable to small perturbations.
2.
Timescale for shear-induced migration
Karnis
et
al.
(1966) conducted one of the earliest studies of inhomogeneous
suspension flow under Stokes flow conditions. They measured velocity and
concentration profiles of neutrally buoyant spheres in a Newtonian fluid when the
suspension was pumped through a tube. For a range
of
the ratio
R/a (R
and
a
being
the radii of the tube and the particles, respectively) and particle volume fraction, they
reported substantial blunting
of
the velocity profiles. However, they did not detect any
inhomogeneity in the particle concentration
;
the particles were evenly dispersed in the
fluid, as they were at the entrance of the tube. More recently, Koh (1991) and Koh et
al.
(1994) conducted a systematic study of pressure-driven flow of a suspension through
a rectangular channel. Using the laser Doppler velocimetry (LDV) technique, they
measured velocities and concentrations of the suspended particles for a range
of
the
channel width and particle concentration. Their results also show blunting
of
the
velocity profile, but in contrast to the observations
of
Karnis et
al.,
they always
observed considerable inhomogeneity in the particle concentration due to migration of
particles towards the centre of the channel.
A number of other experimental studies have confirmed the migration of particles
from regions of high to low shear rate (Leighton
&
Acrivos 1987b; Abbott
et
al.
1991
;
Hookham 1986). However, the measurements differ in the extent of concentration
inhomogeneity and the degree of velocity blunting. The differences between the
observations may be attributed, in part, to inaccuracies in the experimental procedures
and to the different measurement techniques that were used. For instance, Leighton
&
Acrivos point out that the measurements
of
Karnis
et
al.
may be flawed owing to the
small number of tracer particles in their measurement volume. Sinton
&
Chow cite
flow-induced relaxation effects in their NMR imaging technique as a possible source
of error in their concentration measurements. In what follows, however, we consider
an aspect of the problem that has been overlooked in the majority
of
the earlier
investigations
-
the time taken to achieve steady state, or alternatively, the ‘transition
length’ in the channel (or tube) required for the flow to be fully developed.
A
simple
analysis, similar to that used by Leighton
&
Acrivos (1987b), is given below that yields
the time required to achieve steady state and the transition length.
Consider the flow of a suspension of particles of radii
a
suspended in a Newtonian
fluid driven by a pressure gradient (or by a uniform body force such as gravity) in a
channel or tube of half-width
H.
The particles, which are initially dispersed
homogeneously in the fluid, gradually migrate towards the centre of the channel (or
tube) until they reach a steady configuration. Using the shear-induced-diffusion
hypothesis of Leighton
&
Acrivos (1987b), the average distance travelled by the
particles perpendicular to the direction of flow,
y,
in time
t
is given by
y
=
2(Dt)l’Z,
Pressure-driven
$ow
of
suspensions 161
D
being the shear-induced diffusivity. If the mean distance the particles must travel is
taken to be of the order of the channel width, then the timescale for reaching steady
state is
f,,
-
fw4w
(1)
D
=
d($)ja2, (2)
Since the motion of the particles is driven purely by hydrodynamics,
where d($) is a non-dimensional function of the particle volume fraction
$
and
?i
is the
shear rate. We estimate
j
by its average value across the channel:
=
3(u)/H for
Poiseuille flow, where
(u)
is the average suspension velocity in the channel. Thus, (1)
becomes
The value of d($) can be estimated from measurements of the self-diffusivity in a
suspension of hard spheres (Leighton
&
Acrivos 1987a; Phung
&
Brady 1992; Phung
1993; Phan
&
Leighton 1994). (It might be more appropriate to use a collective as
opposed to self-diffusivity here, but for scaling purposes this estimate is sufficient.) For
dense suspensions
($
>
0.3), the value of 12d($) is approximately 1 and hence
t,,
-
(H/a)3a/(u). Equation
(3)
can be expressed equivalently as the length along the
channel required to reach steady state,
Note that this is just a characteristic length- (or time-) scale for the process and it may
in general require several transition lengths before steady state is finally achieved. In
contrast, the transition length in laminar flow of a homogeneous Newtonian fluid
(within which the boundary layer reaches the centreline) is given by
It should be clear that the time taken to achieve steady state can be much longer for
a viscous suspension than for a homogeneous Newtonian fluid. Moreover, the (H/LZ)~
scaling of the development length is rather stringent. In the experiments of Karnis et
al. the actual length was not given, but the distance
of
50cm was reported for the
translation of the microscope,
so
we use this as an estimate of the length. Using the
smallest tube diameter reported of
0.2
cm,
L/H
z
250, and from the values reported
in table 1 of Karnis et al. there is one experiment at (H/a)'
z
80,
one at 294 and the
remainder range from 320 to 1700. Thus, it is unlikely that any of the experiments of
Karnis et al. were at steady state. In the experiments of Sinton
&
Chow, the largest
L/H was 32.8, while the smallest (H/a)2 was 13520; clearly far from steady state. In
Koh's experiments the particle size and channel length were kept fixed at a
=
15 pm
and
L
=
12.7 cm, respectively, and the channel width varied to change
H/a.
Of the
four channel widths used (0.15, 0.0789, 0.047 and 0.026cm), only the last one is
potentially at steady state. This long development length is clearly a possible reason for
the disparity in the results of earlier investigations and is an important constraint to be
taken into account in future studies. We show in $4 that the above scaling for the
steady-state time is borne out in the results of this investigation. The above scaling