Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Collective and Individual Motions of Particulate Suspension
Shusaku Harada, Kodai Sato and Takashi Mitsui
Division of Field Engineering for Environment, Graduate School of Engineering,
Hokkaido University, N13W8, Sapporo, 0608628, JAPAN
haradaa eng hokudai ac jp. kodai@transer.eng.hokudai.ac.jp and mitsui@transer.eng.hokudai.ac.jp
Keywords: suspension, sedimentation, RayleighTaylor instability, linear stability analysis
Abstract
The sedimentation of solid particles partially suspended in fluid has been studied experimentally and theoretically. We observed the
settling of a stratified suspension which has both the upper and lower interfaces of concentration in quasitwo dimensional vessel.
We have examined whether collective or individual motion of particles is dominant, i.e., whether the particles settle as a continuous
suspension or they settle individually relative to surrounding fluid. The experimental results indicate that dense suspension
consisting of small particles behaves as a continuous fluid and the motion of constituent particles are governed by the
gravityinduced instability of suspensionfluid interface. On the other hand, individual motion of particles relative to fluid stands
out in dilute suspension with large particles. The existence of concentration interface plays a significant role with these extreme
behaviors. We propose a dimensionless parameter which expresses the transition from the collective to individual behaviors of
suspension quantitatively.
Introduction
The sedimentation of solid particles suspended in fluid is an
important phenomenon in engineering processes such as
coating, accumulation, separation and filtration of
particulate materials. It is known that hydrodynamic
interaction among particles plays a significant role on their
settling behaviors (Davis and Acrivos, 1985). Particularly in
case of fine particles in viscous fluid, not only does the
hydrodynamic interaction affect the mean settling velocity
but also it brings about the diffusive motion of suspended
particles which is called hydrodynamic diffusion. A lot of
experimental and theoretical studies concerned with the
hydrodynamic diffusion have been performed and have
been found that the diffusivity of particles has a strong
anisotropy and is influenced by many factors such as the
particle size, the vessel size and so on (Nicolai et al., 1995;
Martin et al., 1995; Davis, 1996; Mucha and Brenner,
2003).
The sedimenting behavior is more complicated when
particles are inhomogeneously suspended in fluid. A typical
example is a wellknown Boycott effect (Guyon et al, 2001).
When the suspended particles settle in an inclined tube, the
settling velocity is larger than that in a vertical tube. Such a
rapid settling is brought about the fluid convection caused
by the concentration gradient. The another example of
inhomogeneous suspension is falling clouds of particles in
liquid. Some researchers have found the interesting motion
of particle clouds such as leakage of particles, torus
formation and breakup of clouds (Adachi et al., 1978; Nitche
and Batchelor, 1997; Machu et al.,2001; Metzger et al.,
2007).
The horizontally stratified suspension is also a type of
inhomogeneous suspensions. When a homogeneous
suspension settles under the gravity, it is gradually stratified
and the variation of the concentration evolves vertically. The
settling velocity of constituent particle varies according to
local concentration, however it is the same order of
magnitude as the settling velocity of an isolated particle
(Kynch, 1952). On the other hand, when the stratified
suspension is put above pure fluid, lateral variation of the
concentration develops during the sedimentation (VOltz et al,
2001; Blanchette and Bush, 2005). This lateral variation of
concentration is similar to the gravityinduced instability
between pure fluids. As a result of such a collective motion
of suspension, the settling velocity of constituent particles
becomes far from that of an isolated particle.
The purpose of this study is to examine the collective
motion of particles in inhomogeneous suspensions. We
experimentally observed the settling of a stratified
suspension which has both the upper and lower interfaces of
concentration in quasitwo dimensional vessel. The observed
collective motion of suspension is compared with the
theoretical analysis of continuous fluid. We focus our attention
whether the particles settle as a continuous suspension or they
settle individually relative to surrounding fluid.
Experimental Setup
Figure 1 shows the schematic diagram of the experimental
apparatus. The test cell is quasitwo dimensional vessel with
a height L =240mm, a width W =100mm. The vessel depth
D is adjustable from 3mm to 12mm. In order to stratify the
suspension, we used two slits in back side of vessel so as to
put stainless steel blades into them. The thickness of the
blade is 0.5mm and the distance between two blades L, is
19.5mm. The blades divide the test cell into three parts in
vertical direction. The lower blade separates the lower pure
fluid and suspension, while the upper blade separates the
upper fluidsuspension. Therefore the distance between
blades L, corresponds to the initial height of the stratified
Paper No
suspension.
The experimental procedure is as follows. At first pure fluid
is filled into the lower part of the vessel and then the lower
part is closed by the lower blade. Next the suspension is
poured into the vessel above the lower blade until the
surface reaches the position of the upper blade. Finally the
upper blade is put into the vessel and the pure fluid is filled
above it. Consequently, the stratified suspension is held
between two blades. After these setups, the blades are
removed backward simultaneously. The settling behavior of
the suspended particles by gravity is recorded by a digital
video camera. We recorded images of settling behavior near
the center of the vessel where the effect of side walls can be
neglected.
The suspension contains silicone oil and glass or
polystyrene particles. The diameter of particle is d,=30pgm
or 100lm for glass and d,=550pm or 800m for
polystyrene. Mass density of glass and polystyrene are
pp=2500 kg/m3 and 1050 kg/m3 respectively. The
suspension is mixed by stirring for several hours and is
deaerated well in constant temperature (22 1C). The
silicone oil which has the different properties (density Oyf
=972 kg/m3, viscosity = 9711944 mPas) is used for
making suspension. The pure fluid which is used to fill the
upper and lower part of the vessel has the same properties as
constituent fluid of the suspension. Therefore the stratified
suspension can be interpreted as partially suspended
particles in a static pure fluid.
Stokes settling velocity Uo on our experimental condition is
the order of 0(104) to 0(102) mm/s. In order to verify
whether the uniformity of the suspension is kept during the
experimental setting, we estimate the settling distance of
particles during the setting. The period from the filling of
suspension to the removal of blades is a few minutes. We
can estimate the maximum settling distance of particle
during this period at a few particle size in all conditions.
Therefore we consider the uniformity of the suspension is
almost kept during the setting of the experiment.
suspension
lower fluid
digital video camera
:r
Stainless steel blades
y
 X
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Linear Stability Analysis
It is known that the interfacial instability is observed when
two immiscible fluids are separated by a horizontal
boundary with positive density gradient in vertical direction
(RayleighTaylor instability). RayleighTaylor instability
occurs by the balance between gravity force and interfacial
tension. In general, the growth of disturbance is determined
by the properties of two fluids (density, viscosity) and the
interfacial property between them.
RayleighTaylor instability has also observed at a
suspensionfluid interface even though there is no distinct
interface between them. V61tz et al. (2001) has studied
systematically the instability of the suspensionfluid
interface and it can be predicted from a similar analysis to
immiscible fluids on the assumption that the suspension has
apparent density p, and viscosity u, such as;
P = (1 )pf + pp, (1)
u, = (1+ 2.5)u (2)
where 0 is the concentration of suspended particles. Eq.(2)
is known as Einstein's apparent viscosity for dilute
concentration.
For understanding the collective motion of sedimenting
suspension, we performed linear stability analysis of
RayleighTaylor instability between two viscous fluids in
quasitwo dimensional vessel. The suspension is assumed as
the continuous fluid with apparent density and viscosity
given by Eqs.(1) and (2). The interfacial tension between
suspension and pure fluid is set to be zero. Since most part
of analytical procedure is similar to the wellknown linear
analysis of hydrodynamic stability (Chandrasekhar, 1961),
the different point from three dimensional analysis is
described here. On the assumption that the vessel has
narrow gap, the velocity in y direction is set to be zero and
the velocity and density fluctuation has parabolic Poiseuille
profiles (Huang and Edwards, 1996; V61tz et al, 2001).
u, (x, y, z,t) = (z) exp(ikx + nt)i(y), (3)
p(x,y,z,t)= 7(z) + p(z)exp(ikx + nt)i(y), (4)
where k is the wave number of disturbance in x direction,
n is the growth rate of disturbance, and '(y) indicates a
parabolic function in y direction defined by
(Y)= 6 D 2, (5)
where D is the vessel thickness. Substituting Eqs.(3) and (4)
to Stokes equation and continuity equation for viscous fluid
and averaging over y direction, we finally obtain the
relationship between the wave number k and the growth rate
n as follows;
g (, a2)+A +, 1[2 + ,lq2 k( + ,2
n
 4kl2 + 4k2 (
n
22 FX2qI1 Aq2 + k(l 2)]
Figure 1: Schematic diagram of experimental system
4k3
+ (ll 2)2F(q k)(q2 k)= 0, (6)
n
where F is kinematic viscosity (= 7/p ) and subscripts 1
and 2 indicate the lower and the upper fluid respectively, a, P
Paper No
and q are defined as follows;
a11
Oi1 p ,
P1 +a92
q1 = +k +17,
V17
a2 P2
+I P +2
2 a2 (1 F27
q2 =+k +r ,
\V2
where 77 = 92/9y2 = 12/D2 If the vessel thickness D
approaches infinity, 7 goes to zero and consequently Eq.(6)
is close to wellknown kn relation for infinite space which is
described in Chandrasekhar (1961).
For stratified suspension, we set 7,, v, (lower fluid) to be
properties of pure fluid p1 vf and set p F (upper fluid)
to be apparent properties of suspension p,, v, which are
defined by Eqs.(1) and (2). Calculating Eq.(6) by
NewtonRaphson method, we can obtain the fastestgrowing
wave number kmax and its growth rate ntheo.
Results and Discussion
Sedimentation Behavior of Stratified Suspension
Experiments were conducted by using various combinations
of particle and fluid. Figure 2 shows the examples of the
settling behavior of stratified suspension on the conditions
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
described in Table 1. We changed the particle density p,
diameter d, concentration 0, fluid viscosity /4 and vessel
thickness D. The Stokes settling velocity for an isolated
particle U0 = Pf )Sgd /18pu, is also shown in the
Table 1. The corresponding particle Reynolds number Rep =
p, Uodp/ l is less than 0.01 for all conditions.
Figure 2 indicates that a fingerlike perturbation of the lower
interface develops as time passes, while the suspension
keeps its upper interface almost flat. The flat upper
interfaces are due to a selfsharpening effect which results
from the dependency of the settling velocity on the local
particle concentration (Kynch, 1954). Dense suspension
with small particles (Figure 2a) behaves like immiscible
fluid and the perturbation can be found clearly. It should be
emphasized again that the system considered here is
partially suspended particles in a static pure fluid because
the suspension is made of the same fluid as the upper and
lower ones. Therefore there is no distinct border at
suspensionfluid interface. The instability at the lower
interface is obviously the gravityinduced instability
described in the previous section. Here we call such a
collective settling as fluidlike settling of suspension.
On the other hand, in case of dilute suspension with coarse
particles (Figure 2c), the suspended particles seem to settle
individually. It is known that the hindered settling velocity
of particle is expressed as f()Uo, whereft() is the hindered
Table 1: Physical properties of particle and fluid
p, (kg/m3) d, (gmn) 0 (mPas) D (mm) Uo (mm/s)
(a) glass particle 2500 30 0.03 1944 8 3.85 x 104
(b) polystyrene particle 1050 550 0.1 971 8 1.34 x 102
(c) polystyrene particle 1050 800 0.01 1944 12 1.40 102
t= Os
t= 50s
(a) d,=30g)m, i =0.03, D=8mm
(b) d=550gm,, =0.1,D=8mm
(c) d,=800g)m, =0.01,D=12mm
Figure 2: Settling behaviors of stratified suspension
t= 100s
Paper No
14
S8
I 6
2
0
o4 Uo
0 50 100 150
time t (s)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
14
, 12
~io
8
6
4
e
200 250
(a) d,=30pm, 0 =0.03, D=8mm
0 50 100 150
time t (s)
14
12
. 10
8
S6
4
200 250
(b) d,=550pm, =0.1, D=8mm
0 50 100 150 200 250
time I (s)
(c) d,=800pm, =0.01,D=12mm
Figure 3: Change in average length of fingers with time
settling function and is ( )"m with a constant m = 5 6
(Davis and Acrivos, 1985). Another factor of the settling
velocity change is the wall effect. The settling velocity of
particle is influenced by the presence of the wall and, for
our experimental condition, it decreases around ten percent
of the Stokes velocity (Happel and Brenner, 1973). At all
events, if we consider that the particle settles individually,
the settling velocity will be the same order of magnitude as
the Stokes settling velocity Uo. We call this kind of
individual settling as particlelike settling.
Figure 3 shows the change in the average length of fingers
with time. For all conditions, the fingers grow exponentially
at the beginning and then their growth speed becomes
constant. The exponential growth of fingers can be seen in
the RayleighTaylor instability of immiscible fluids and the
finger length z can be calculated by linear stability analysis
as the following form,
z = zo exp(nt), (7)
where n is the growth rate. We can define two experimental
values which express the settling speed of particles. As
shown in Figure 3, we obtained the growth rate of fingers
exp by exponential fitting of initial experimental data and
also obtained the growth velocity Uep by linear fitting of
succeeding data.
If the suspension behaves perfectly fluidlike, the
experimental growth rate nexp would be close to the growth
rate calculated by linear stability analysis ntheo. On the other
hand, if the suspension behavior is perfectly particlelike,
the change in the finger length with time will become
almost linear and the succeeding growth velocity Uep will
be the same order of magnitude as the Stokes settling
velocity Uo. As can be seen in Fig.3, it is found that the
settling velocity Uep on condition (a) is largest among three
conditions although it has the smallest Stokes velocity (see
Table 1). This indicates the macroscopic motion of
suspension enlarges the settling velocity of particles near
the lower interface. The quantitative discussion for the
settling velocity is given later.
0.08
0.07
S0.06
0.05
0.04
S0.03
S0.02
0.01
0 
0.0
(a) =18.4mm (D=8mm)
0.5 1.0 1.5
wave number k (mm 1)
(b)2 =11.6mm (D=5mm)
(c) =6.9mm (D=3mm)
Figure 4: Maximum wave number kma calculated by linear stability analysis and corresponding wave length = 27 A,,,, for
glass particle suspension (d,= 30gtm, 0= 0.03, 4f =ll44mPa s)i
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Comparison of Wave Length of Disturbance with Theory
The experimental results of dense suspension with small
particles are compared to the theoretical results for
immiscible fluids by linear stability analysis. Figure 4 shows
the theoretical results of the growth rate n for given wave
number k calculated by Eq.(6) under the conditions of d, =
30tm, 0= 0.03, if =1944mPas. We set p1, v, (lower
fluid) to be properties of pure fluid p ,vf and set p2,v2
(upper fluid) to be apparent properties of suspension pS, v,
which are defined by Eqs.(1) and (2). The theoretical results
indicate that the growth rate of disturbance is larger for larger
vessel thickness D and the fastestgrowing wave number kma
also depends on D.
Figure 4 also indicates the experimental pictures for same
conditions as theoretical analysis. It is found that the width of
finger is larger for larger vessel depth D. Yellow line in each
picture indicates the wave length of maximum growth of
disturbance 2 (=2/r/kma) predicted by linear stability
analysis. As can be seen, they are in quantitative agreement
with wave length of disturbance (finger width) observed in
experiment. Such perfectly fluidlike motion of suspension
have been studied by V61tz et al. (2001) Here we pay
attention to the transition from such a collective motion of
suspension to individual motion of constituent particles. In
the next section, we propose a dimensionless parameter
which expresses the transition from the collective to individual
motions quantitatively.
New Parameter Describing Transition from Collective to
Individual Motions of Suspension
Here we consider the parameter which describes the
transition from collective (fluidlike) to individual
(particlelike) motions of suspension as shown in Figure 2.
The parameter would be a function of particle density pp,
particle diameter dp, liquid density pf, liquid viscosity /,;
particle concentration 0 and vessel depth D. We propose a
new dimensionless parameter H = H (pp, d,, pf, /f, D) as
follows:
(a) / > 1
H= (8)
where 1 is average distance between particles in suspension
and is function of dp and 0. 2 =(pop, pf, uf, 0, D) is the wave
length of interfacial perturbation (finger) which can be
obtained from the linear stability analysis. Figure 5 indicates
the physical meaning of Eq.(8). It is found that the
parameter H expresses the border resolution of the density
interface. That is, when H is large, the particles in
suspension cannot form the finger clearly. By a simple
assumption, the average distance between particles is
calculated as 1 dp6 1/3 and consequently H is as follows.
dH
H = (9)
101/3
As explained above, if suspension behaves perfectly
fluidlike, the growth rate obtained experimentally nexp is
close to that by linear stability analysis ntheo. On the other
hand, if it behaves particlelike, the measured velocity Uexp
is close to the Stokes settling velocity Uo. Figure 6 shows
nexp/ntheo and Uexp/Uo obtained by 14 experiments with
variation in fluid and particle properties and also the vessel
thickness D. The horizontal axis is set to the parameter H
given by Eq.(9).
As can be seen in Fig.6, nexp/ltheo is around unity and
Uexp/Uo indicates larger value for H < 0.03. This results
indicate that the suspension behaves like fluid for smaller H.
On the contrary, Uexp/Uo is around unity and nexp/ltheo is
apart from unity for H > 0.2. This suggests the particlelike
settling of suspension for larger H. In conclusion, we can
classify the settling behaviors of suspension as fluidlike,
particlelike settling and their transition by new
dimensionless parameter H. Figure 6 also indicates some
pictures of the settling behavior of suspension. We can
understand visually that this parameter is reasonable to
classify the settling behavior.
(b) 2~ 1
(c) 2 <1
Figure 5: Relation between wave length of interfacial perturbation and average distance between particles
Paper No
H'
00000
0 0
0
0 0, 00
o @0. 00 @@ 0 o 00oo %o % %0%o~
so 0 o 0 000 o o o
00 0900 0. 0 0 oee OP 0 0004o00
2 2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1000
Figure 6: Transition from fluidlike to particlelike behaviors of suspension (nexp / ntheo and Uexp /
fluidlike and particlelike settling respectively)
Conclusions
The sedimentation of inhomogeneous suspensions in quasi
twodimensional vessel was investigated experimentally.
We have examined whether the particles behave as a
particle assembly (particlelike settling) or as a continuous
suspension (fluidlike settling). If the suspended particles
are adequately small, the settling behavior is completely
fluidlike and the settling velocity is dominated by
downstream flow due to the gravityinduced instability. The
experimental results showed that the settling velocity of
particles is influenced not only by the physical properties of
particle and fluid but the size of the vessel. The transition of
collective to individual motions can be predicted
quantitatively by a new parameter which expresses the
border resolution of the concentration interface.
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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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