Paper No
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A Cloud of Rigid Rods Sedimenting in a Viscous Fluid
Joontaek Park'*, Jason E. Butler2, Bloen Metzger3 and Elisabeth Guazzelli4
'University of Florida, Department of Chemical Engineering, P.O. Box 116005, Gainesville, FL 32611, USA
*Currently at the City College of the City University of New York, Levich Institute, 160 Convent Avenue, New York, NY
10031, USA
jpark@ccny.cuny.edu
2University of Florida, Department of Chemical Engineering, P.O. Box 116005, Gainesville, FL 32611, USA
butler@che.ufl.edu
3IUSTICNRS UMR 6596, PolytechMarseille, AixMarseille Universit'e (Ul), Technop^ole de
Ch^ateauGombert, 13453 Marseille cedex 13, France
Bloen.metzger@polytech.univmrs.fr
4IUSTICNRS UMR 6596, PolytechMarseille, AixMarseille Universit'e (Ul), Technop^ole de
Ch^ateauGombert, 13453 Marseille cedex 13, France
Elisabeth.guazzelli@polytech.univmrs.fr
Keywords: Particle Drop, Drop Breakup, Sedimentation, Rigid Rod, Stokes flow
Abstract
The deformation and the breakup of a cloud of rigid rodlike particles sedimenting in a viscous fluid have been studied
through experiments and numerical simulations. The timeevolution of the cloud of rods is observed to be qualitatively similar
to that of spherical particles: initially a spherical cloud deforms into a torus and breaks up into smaller cloudlets. However, the
mean breakup time of the rod cloud is found to be faster than that of the spherical particles. The simulation method developed
for slenderbody dynamics (Butler & Shaqfeh 2002) can correctly describe the timeevolution of the cloud. However, we made
a further approximation, which we named 'Fiblet', on the evaluation of the farfield hydrodynamic interactions in the
slenderbody dynamics simulation. The approximated formula demonstrates each contribution to the rod motions as the
selfmotion and the hydrodynamic dispersion. It also clearly shows that a single parameter which is related to the fluctuation
by the selfmotion of anisotropic particles controls the deformation and the breakup time. The results from experiments and
the two simulation methods are all in qualitatively and quantitatively good agreements, which shows that the approximated
farfield hydrodynamic interactions is enough for describing the dynamics of slenderbodies in the semidilute concentration
regime.
Introduction
The phenomenon whereby swarms of particles are dispersed
while sedimenting in a fluid can be found in nature as well
as industrial applications. Therefore, the deformation and
the breakup of the sedimenting cloud of particles have been
studied by many researchers.
An earlier study by Adachi et al. (1978) as well as many
subsequent studies, reviewed by Machu et al. (2001), have
described a general feature of shape evolution: starting from
a spherical shape, a particle cloud slowly deforms into a
torus structure and finally breaks up into smaller cloudlets
which repeat the same deformation procedure until they
shatter into the bulk fluid. The studies also have found that
the deformation and the breakup of the particle cloud are
due to either inertia or perturbation on the initial shape of
the cloud (Nitsche & Batchelor 1997; Schaflinger & Machu
1999; Machu et al. 2001; Bosse et al. 2005). However,
Bloen et al. (2007) later showed that the breakups can
happen without any inertia or initial perturbation.
As reviewed above, most of the studies so far have focused
only on spherical particles, although sedimentation of
anisotropic particles may happen more in nature and
industry. Hence, we studied the effect of the anisotropy of
particles on the cloud sedimentation. We focused on the
cloud of rigid rodlike particles or rigid fibers falling in a
quiescent Newtonian viscous fluid under the conditions of
the absence of inertia and surface tension. Both experiments
and numerical simulations were performed and the results
were compared. For simulations, two different levels of
approximations of the longrange hydrodynamic
interactions were tried in order to find a model which
Paper No
captures the minimal dynamics of the cloud. We found a
single parameter that controls the deformation and the
breakup of the cloud (Park et al. 2010).
Nomenclature
A Aspect ratio of a rod, (L/d)
c The selfterm parameter
d Length of the short axis or diameter of a rod
(cm)
e A unit vector in Cartesian coordinates
F Buoyancy force
I Identity tensor
L Length of the long axis of a rod (cm)
N The number of particles in a cloud
p Unit vector of rod orientation
p Rotational motion of a rod
R Horizontal radius of a cloud
r Centerofmass of a rod
i Translational motion of a centerofmess of a rod
s Coordinate along a rod long axis
t Time
u Flow disturbance
V Cloud (centerofmass) velocity
Greek letters
p viscosity (cP)
4) Volume fraction of rods
Subsripts
O Initial value
z The gravity direction
a Index of a rod
p Index of a rod
Superscripts
* Dimensionless value
Experimental Facility
Figure 1 illustrates the experimental setup. A glass vessel is
filled with a fluid mixture of Ucon oil and water (1:1 by
volume) which has a density of 1.0790.019 g cm3 and a
dynamic viscosity of p=217060 cP. A video camera is
installed on a sliding rail to record the timeevolution of the
cloud. A neon light tube and a side of a vessel wall covered
with a sheet of paper provide a homogeneous lighting. More
detailed information can be found in Metzger et al. 2007
and Park et al. 2010.
Copper rods, made by cutting a copper wire, are used as
rigid rodlike particles (Figure 2) which have a dimension of
L=0.1270.013 cm and d =0.01490.0020 cm and a density
of 8.970.03 g cm3.
Rigid rods are mixed with the suspending fluid to a desired
volume fraction ) (110%). Particle clouds are produced by
injecting the prepared suspension using a syringe. The
Reynolds number calculated from the fluid property, initial
radius and velocity of clouds are order of 10 2.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Experimental data are obtained by analysing recorded
movies using software (ImageJ). Breakup time is
determined by the naked eye when the toroidal cloud is
severely bent out of a round shape. Mean values are
obtained over several runs (typically 5).
Syringe
=s
Falling Cloud
Vessel filled
with
Ucon/water
mixture
Figure 1: Schematic diagram of the experimental setup.
Figure 2: A photo of copper rods used in the experiment.
Numerical Scheme
Figure 3 illustrates the sketch of the problem for the
formulation. A single particle cloud without surface tension
is falling in a Newtonian viscous fluid without inertia. Only
gravity is acting on the cloud in the positive zdirection.
Particles are hydrodynamically interacting with each other
and there is no other external flow, force field or boundary
effect. Particles are assumed to be rigid slender bodies with
high aspect ratio (Moran 63; Batchelor 70; Cox 70). Initially
No number of particles are randomly oriented in a spherical
cloud with Ro. The concentration inside the cloud in our
study is from a dilute to semidilute regime where the
longrange hydrodynamic interaction is dominant.
A simulation method developed for the dynamics of
slenderbody (Harlen et al. 1999; Butler & Shaqfeh 2002) is
used. Figure 4 illustrates how the motions of rods are
calculated. Each rod is moved directly by buoyancy force
(the selfmotion) and indirectly by flow disturbance induced
by motions of the other rods (the hydrodynamic dispersion).
Paper No
The evaluation of the flow disturbance acting on a rod
requires double integration of Stokeslets along the long axis
of all the rods as well as summation over all rods.
Equations (1) and (2) represent the rotation and the
centerofmass motions of a rod. The first term on the right
hand side of eq (2) corresponds to the selfmotion, derived
by the slenderbody theory, and the second term indicates
hydrodynamic dispersion induced by other particles.
No particles
A=L/d
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(1) and eq (2) has been further approximated to get a model
equation of minimal physics description. Moreover, because
the computational burden of the senderbody dynamics
makes simulations of a larger number of particles No>1000
difficult, the approximated simulation enables studies up to
No =4000.
Figure 5 illustrates the evaluation of flow disturbance in the
approximated simulation. Under the assumption that
separation distance between rods are far enough, the
longrange hydrodynamic interactions between rods are
approximated as those of point particles or Stokeslets
(named here as 'Fiblet'). Furthermore, flow disturbance on a
rod can be linearized around its centerofmass. Similar
approaches can be found in other references (Mackaplow et
al. 1998; Saintillan et al. 2006; Park et al. 2007; Park &
Butler 2009).
s, =L/2 "P
Pa V6 L/
Vo
Initial
sedimentation
rate
Figure 3: The sketch of
falling in a viscous fluid.
Sediment with
buoyancy force
the problem: a rigid rod cloud
Note that the actual simulation algorithm, similar to that of
described by Butler & Shaqfeh (2002), incorporates the
longrange hydrodynamic interactions until the stresslet
level by constructing a grand mobility tensor.
Hydrodynamic interactions between particles are calculated
using the OseenBurgers tensor. A fourthorder RungeKutta
method is used to integration of the equations of motions in
time.
s, =L2 =0
P ,
ral
2U
... ..t ..
/ .,
Figure 4: Schematic diagram of the slenderbody
dynamics simulation:
Pa = (Ipaa) f/2u(r + sa )sds, (1)
eL/2
r, l2A)F (I+pp ) f~ J 2u(r +aP.s (2)
Although the slenderbody dynamics simulation, described
above, can correctly describe the dynamics of rods in a
sedimenting cloud, the evaluation of flow disturbance in eq
ila Ua... (ra) ( +s'a'a
.* *
Figure 5: Schematic diagram of the Fiblet approximation.
The resulting approximated equations of motions are eq (3)
and eq (4). Note that the hydrodynamic term of the
translational motion of a rod is the same as that of the
spherical particles (Metzger et al. 2007). The selfterm in eq
(4) is derived using slenderbody theory. The difference in
the selfterm from spherical particles indicates that the
selfmotion of rods depends on each orientation.
Pa =(IPapa) (3)
Ia 2AF (I+ppa)e +ua(ra) (4)
Equations (3) and (4) are renormalized by the initial radius
of the cloud Ro and the velocity of a spherical cloud of
Stokeslets, NoF/5IKqRo (EkielJe'zewska et al. 2006). The
resulting dimensionless form of eq (4) is the following eq
(5).
.. 5c (i+ 5 N I r r*r*
8N0 8No, / r* r 3
Here, r is a distance between the centeroids of rods and a
new parameter c is defined as in eq (6). As a particle
becomes more anisotropic in shape, c increases. However, c
=0 corresponds to motions of a spherical particle on an
inertial frame since the selfterm which distinguishes the
motion of anisotropic particles vanishes.
2Ro ln(2A)
L
A BulirschStoer method (Press et al. 1994) is used to
perform the integration of eq (3) and eq (4). Detailed
Paper No
derivation and a complete form of renormalized eq (3) can
be found in Park et al. (2010).
For both simulation methods, shortrange interactions are
ignored for simplicity. Instead, pairs of rods within a relative
distance less than d are excluded from the calculation of
hydrodynamic interactions to avoid singularity. Mean values,
indicated as <...>, from simulation results are averaged
typically over 10.
Results and Discussion
Figure 6 shows sequential snapshots of a cloud of rigid rods
taken from the sedimentation experiment. It shows a general
evolution of the deformation of a particle cloud. Only
quantitative difference in time, depending on parameters (c
and No), is observed.
Figure 6: Snapshots of the time evolution of a cloud of rigid
rods (c =30, No =1000) falling in a viscous flow from the
experiment. Time evolution of the deformation is
represented by the sequence from the top images to the
bottom.
As a spherical drop of particles starts to sediment, vortices
inside of the drop, like the HadamardRybczynski solution,
occur and circulate the particles. However, the absence of
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
surface tension and the discrete nature of particles make the
drop flat, horizontal to the gravity direction, and lose some
particles at the top (Figure 6a). The expansion in the
horizontal direction and the particle leakage deform the drop
into a torus structure (Figure 6b). As the expansion
continues, the torus become unstable and breaks up into two
or more smaller droplets (Figure 6c). Each secondary
droplet repeats the same deformation procedure until it is
dispersed into a fluid.
Snapshots of Figure 7 are taken from the slenderbody
simulation of a cloud of rigid rods. It shows that each stage
of the deformation, (Figure 7a) particle circulation by the
inner vortex, (Figure 7b) a torus formation and (Figure 7c)
breakup into smaller droplets, simulated by the numerical
schemes is in qualitatively good agreement with the
experiment. Note also that the Fiblet simulation can also
generate similar sequential images (Park et al. 2010).
Figure 7: Snapshots of the time evolution of a cloud of rigid
rods (c =40, No =1000) falling in a viscous flow from the
slenderbody dynamics simulation. Images at onsets are
taken from the bottom view.
Qualitative agreement in the timeevolution between
experiments and the two simulation methods indicates that
the longrange hydrodynamic interactions incorporated in
the slenderbody dynamics simulation correctly capture the
dynamics of the rods in the sedimenting cloud, and the
approximation made in the Fiblet simulation is not severe.
In Figure 7c, the simulated breakup tends to produce more
numbers of droplets (3 or 4) than the experiment. Inclusion
of correct shortrange interactions in the simulation may
improve the agreement.
The effect of the anisotropy of the particle on the mean
breakup time has been studied in terms of the selfterm
parameter c. Simulations were performed with varying c
and No. The results are plotted in Figure 8. As c increases
with a fixed No, the anisotropy or the selfterm contribution
in eq (5) becomes larger to result in the reduction of the
breakup time. As c approaches 0, the break up time
becomes that of a cloud of spherical particles (Metzger et al.
Paper No
2007; Park et al. 2010). The increased breakup time with
increasing No at a fixed c also can be also explained by the
reduction of the selfterm.
Figure 8 also shows the comparison of data from the two
different simulation methods. Excellent agreement between
two simulations emphasizes that the Fiblet approximation is
enough to describe the dynamics of slenderbodies from a
dilute to semiconcentrated regime.
IFigure
Fiblet ai
function
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
anisotropic particles can be explained by investigating the
relation between the horizontal expansion rates of a cloud
and corresponding terms in eq (5) (Park et al. 2010). While
a drop of spherical particles is expanded only by the
hydrodynamic dispersion term in eq (5), a drop of rods has
the additional contribution of the selfterm in eq (5). The
flow topology study by Metzger et al. (2007) has shown that
the cavity inside of a torus becomes larger with the (slightly
exponential) expansion of the cloud, and finally
hydrodynamic pressure at the bottom penetrates the cavity
to break up the cloud when the ratio between the horizontal
radius and the vertical height of the cloud reaches a certain
critical value (around 1.7 for the spherical particles).
Therefore, the additional contribution of the selfterm
enhances the expansion to reach the critical value of the
breakup. However, the actual additional expansion rates of
the selfterm investigated by Park et al. (2010) seem to be
not large enough to explain the detailed mechanism of the
enhanced breakup. Some coupling effect of the selfmotion
and the hydrodynamic dispersion is conjectured to enhance
the fluctuation to break up the cloud faster.
Conclusions
0 20 40 60 80 100 Our experimental and numerical studies on a falling cloud
c of rigid rodlike particles in a viscous fluid have shown that
the qualitative picture of its deformation process is similar
8: Mean breakup time from (open symbols) the to that of the cloud of spherical particles, but the anisotropy
nd (filled symbols) the slenderbody simulations as a of particles results in quantitative difference in mean
of c with various No. Error bar indicates the breakup time.
standard deviations.
Figure 9 shows that the mean breakup time in Figure 8 can
be overlapped on a single master curve if rescaled with c /
No. Moreover, mean breakup times measured from the
experiments are also on top of the single trend obtained
from the simulation. This finding confirms that the
dynamics of anisotropic particles in a sedimenting cloud is
controlled by the magnitude of the selfterm, or a single
parameter c /No.
0.06 0.08 0.1 0.12 0.14
c/N0
Figure 9: Simulation and experimental results of mean
breakup time as a function of c / No
Decreased breakup time or faster deformation process for
The numerical study compared the approximations of the
farfield hydrodynamic interactions with two different levels
to find out that the Fiblet approximation is enough for
describing the physics of a cloud of rigid rods in a
semidilute concentration. The formulation of the
simulations clearly has shown that the coupling of the
longrange hydrodynamic interactions and the selfmotion
of anisotropic particles play an important role in the
deformation and the breakup. More importantly, a single
parameter which is closely related to fluctuation by the
selfmotion is found to control the cloud dynamics.
Further studies in higher concentrations are expected to
explore the limitations of the Fiblet approximation and the
importance of shortrange interactions. The coupling of the
selfmotion and the hydrodynamic dispersion is also worth
being explored more, such as, by modelling the expansion
and comparing the results with eq (5).
Acknowledgements
This work was supported by the National Science
Foundation through a CAREER Award (CTS0348205).
Visits were supported by the Partner University Fund on
'Particulate Flows' and by AixMarseille Universit'e (Ul)
visiting professorships.
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