Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 7.2.1 - A Cloud of Rigid Rods Sedimenting in a Viscous Fluid
ALL VOLUMES CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00102023/00175
 Material Information
Title: 7.2.1 - A Cloud of Rigid Rods Sedimenting in a Viscous Fluid Particle Bubble and Drop Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Park, J.
Butler, J.E
Metzger, B.
Guazzeli, E.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: particle drop
drop break-up
sedimentation
rigid rod
Stokes flow
 Notes
Abstract: The deformation and the break-up of a cloud of rigid rod-like particles sedimenting in a viscous fluid have been studied through experiments and numerical simulations. The time-evolution 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 break-up time of the rod cloud is found to be faster than that of the spherical particles. The simulation method developed for slender-body dynamics (Butler & Shaqfeh 2002) can correctly describe the time-evolution of the cloud. However, we made a further approximation, which we named ‘Fiblet’, on the evaluation of the far-field hydrodynamic interactions in the slender-body dynamics simulation. The approximated formula demonstrates each contribution to the rod motions as the self-motion and the hydrodynamic dispersion. It also clearly shows that a single parameter which is related to the fluctuation by the self-motion of anisotropic particles controls the deformation and the break-up time. The results from experiments and the two simulation methods are all in qualitatively and quantitatively good agreements, which shows that the approximated far-field hydrodynamic interactions is enough for describing the dynamics of slender-bodies in the semi-dilute concentration regime.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
 Record Information
Bibliographic ID: UF00102023
Volume ID: VID00175
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 721-Park-ICMF2010.pdf

Full Text

Paper No


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 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

3IUSTI-CNRS UMR 6596, Polytech-Marseille, Aix-Marseille Universit'e (Ul), Technop^ole de
Ch^ateau-Gombert, 13453 Marseille cedex 13, France
Bloen.metzger@polytech.univ-mrs.fr

4IUSTI-CNRS UMR 6596, Polytech-Marseille, Aix-Marseille Universit'e (Ul), Technop^ole de
Ch^ateau-Gombert, 13453 Marseille cedex 13, France
Elisabeth.guazzelli@polytech.univ-mrs.fr

Keywords: Particle Drop, Drop Break-up, Sedimentation, Rigid Rod, Stokes flow





Abstract

The deformation and the break-up of a cloud of rigid rod-like particles sedimenting in a viscous fluid have been studied
through experiments and numerical simulations. The time-evolution 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 break-up time of the rod cloud is found to be faster than that of the spherical particles. The simulation method developed
for slender-body dynamics (Butler & Shaqfeh 2002) can correctly describe the time-evolution of the cloud. However, we made
a further approximation, which we named 'Fiblet', on the evaluation of the far-field hydrodynamic interactions in the
slender-body dynamics simulation. The approximated formula demonstrates each contribution to the rod motions as the
self-motion and the hydrodynamic dispersion. It also clearly shows that a single parameter which is related to the fluctuation
by the self-motion of anisotropic particles controls the deformation and the break-up time. The results from experiments and
the two simulation methods are all in qualitatively and quantitatively good agreements, which shows that the approximated
far-field hydrodynamic interactions is enough for describing the dynamics of slender-bodies in the semi-dilute 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 break-up 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 break-up 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 break-ups 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 long-range 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
break-up of the cloud (Park et al. 2010).

Nomenclature

A Aspect ratio of a rod, (L/d)
c The self-term 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 Center-of-mass of a rod
i Translational motion of a center-of-mess of a rod
s Coordinate along a rod long axis
t Time
u Flow disturbance
V Cloud (center-of-mass) 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 set-up. 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 cm-3 and a
dynamic viscosity of p=217060 cP. A video camera is
installed on a sliding rail to record the time-evolution 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 ) (1-10%). 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 30-June 4, 2010

Experimental data are obtained by analysing recorded
movies using software (ImageJ). Break-up 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 set-up.










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 z-direction.
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 semi-dilute regime where the
long-range hydrodynamic interaction is dominant.

A simulation method developed for the dynamics of
slender-body (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 self-motion) 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
center-of-mass motions of a rod. The first term on the right
hand side of eq (2) corresponds to the self-motion, derived
by the slender-body 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 30-June 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 sender-body 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
long-range 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 center-of-mass. 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
long-range hydrodynamic interactions until the stresslet
level by constructing a grand mobility tensor.
Hydrodynamic interactions between particles are calculated
using the Oseen-Burgers tensor. A fourth-order Runge-Kutta
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 slender-body
dynamics simulation:


Pa = (I-paa) f/2u(r + sa )sds, (1)
eL/2
r, l2A)F (I+pp ) f~ J 2u(r- +aP.s (2)

Although the slender-body 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 self-term in eq
(4) is derived using slender-body theory. The difference in
the self-term from spherical particles indicates that the
self-motion of rods depends on each orientation.


Pa =(I-Papa) (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 (Ekiel-Je'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 self-term which distinguishes the
motion of anisotropic particles vanishes.


2Ro ln(2A)
L


A Bulirsch-Stoer 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, short-range 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 Hadamard-Rybczynski solution,
occur and circulate the particles. However, the absence of


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 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 6-a). The expansion in the
horizontal direction and the particle leakage deform the drop
into a torus structure (Figure 6-b). As the expansion
continues, the torus become unstable and breaks up into two
or more smaller droplets (Figure 6-c). Each secondary
droplet repeats the same deformation procedure until it is
dispersed into a fluid.

Snapshots of Figure 7 are taken from the slender-body
simulation of a cloud of rigid rods. It shows that each stage
of the deformation, (Figure 7-a) particle circulation by the
inner vortex, (Figure 7-b) a torus formation and (Figure 7-c)
break-up 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
slender-body dynamics simulation. Images at onsets are
taken from the bottom view.

Qualitative agreement in the time-evolution between
experiments and the two simulation methods indicates that
the long-range hydrodynamic interactions incorporated in
the slender-body 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 7-c, the simulated break-up tends to produce more
numbers of droplets (3 or 4) than the experiment. Inclusion
of correct short-range interactions in the simulation may
improve the agreement.

The effect of the anisotropy of the particle on the mean
break-up time has been studied in terms of the self-term
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 self-term contribution
in eq (5) becomes larger to result in the reduction of the
break-up 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 break-up time with
increasing No at a fixed c also can be also explained by the
reduction of the self-term.

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 slender-bodies from a
dilute to semi-concentrated regime.


IFigure
Fiblet ai
function


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 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 self-term 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 self-term
enhances the expansion to reach the critical value of the
break-up. However, the actual additional expansion rates of
the self-term investigated by Park et al. (2010) seem to be
not large enough to explain the detailed mechanism of the
enhanced break-up. Some coupling effect of the self-motion
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 break-up time from (open symbols) the to that of the cloud of spherical particles, but the anisotropy
nd (filled symbols) the slender-body simulations as a of particles results in quantitative difference in mean
of c with various No. Error bar indicates the break-up time.


standard deviations.

Figure 9 shows that the mean break-up time in Figure 8 can
be overlapped on a single master curve if rescaled with c /
No. Moreover, mean break-up 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 self-term, 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
break-up time as a function of c / No

Decreased break-up time or faster deformation process for


The numerical study compared the approximations of the
far-field 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
semi-dilute concentration. The formulation of the
simulations clearly has shown that the coupling of the
long-range hydrodynamic interactions and the self-motion
of anisotropic particles play an important role in the
deformation and the break-up. More importantly, a single
parameter which is closely related to fluctuation by the
self-motion 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 short-range interactions. The coupling of the
self-motion 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 (CTS-0348205).
Visits were supported by the Partner University Fund on
'Particulate Flows' and by Aix-Marseille Universit'e (Ul)
visiting professorships.


References

Adachi, K., Kiriyama, S. & Koshioka, N. The behaviour of


A No-4000
I 0 N=2000
Fiblet No=1000
-I No=1000

SO* No=500
'AsL Nr1000
Slender body N-=500


^ B
: I *


I






Paper No


a swarm of particles moving in a viscous fluid. Chem.
Engng. Sci. 33, 115-121 (1978).

Batchelor, G. K. Slender-body theory for particles of
arbitrary cross-section in Stokes flow. J. Fluid Mech. 44,
419-440 (1970).

Bosse, T., Kleiser, L., H"artel, C. & Meiburg, E. Numerical
simulation of finite Reynolds number suspension drops
settling under gravity. Phys. Fluids 17, 037101 (2005).

Butler, J. E. & Shaqfeh, E. S. G. Dynamic simulations of the
inhomogeneous sedimentation of rigid fibers. J. Fluid Mech.
468,205-237 (2002).

Cox, R. G. The motion of long slender bodies in a viscous
fluid. Part 1. General theory. J. Fluid Mech. 44, 791-810
(1970).

Ekiel-Je'zewska, M. L., Metzger, B. & Guazzelli, 'E.
Spherical cloud of point particles falling in a viscous fluid.
Phys. Fluids 18, 038104 (2006).

Harlen, O. G., Sundararajakumar, R. R. & Koch, D. L.
Numerical simulation of a sphere settling through a
suspension of neutrally buoyant fibers. J. Fluid Mech. 388,
355-388 (1999).

Machu, G., Meile,W., Nitsche, L. C. & Schaflinger, U.
Coalescence, torus formation and breakup of sedimenting
drops: Experiments and computer simulations. J. Fluid
Mech. 447, 299-336 (2001).

Mackaplow, M. B. & Shaqfeh, E. S. G. A numerical study
of the sedimentation of fibre suspensions. J. Fluid Mech.
376, 149-182 (1998).

Metzger, B., Nicolas, M. & Guazzelli, 'E. Falling clouds of
particles in viscous fluids. J. Fluid Mech. 580, 283-301
(2007).

Moran, J. P. Line source distributions and slender-body
theory. J. FluidMech. 17, 285-304 (1963).

Nitsche, J. M. & Batchelor, G. K. Break-up of a falling drop
containing dispersed particles. J. Fluid Mech. 340, 161-175
(1997).

Park, J., Bricker J. M. & Butler, J. E. Cross-stream
migration in dilute solutions of rigid polymers undergoing
rectilinear flow near a wall", Phys. Rev. E. 76, 040801
(2007).

Park, J. & Butler, J. E. Inhomogeneous distribution of a
rigid fibre undergoing rectilinear flow between parallel
walls at high Peclet numbers", J. Fluid Mech. 630, 267-298
(2009).

Park, J., Metzger, B, Guazzelli, 'E. & Butler, J. E. A cloud
of rigid fibres sedimenting in a viscous fluid. J. Fluid Mech.
In Press (2010).

Press, W.H., Teukolsky, S.A., Vetterling, W.T., and


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Flannery, B.P. Numerical Recipes in Fortran, The Art of
Scientific Computing. Cambridge: Cambridge University
Press 2nd ed., 719-725 (1994).

Saintillan, D., Shaqfeh, E. S. G. & Darve, E. The growth of
concentration fluctuations in dilute dispersions of orientable
and deformable particles under sedimentation. J. Fluid
Mech. 553, 347-388 (2006).

Schaflinger, U. & Machu, G. Interfacial phenomena in
suspensions. Chem. Engng. Technol. 22, 617-619 (1999).




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - Version 2.9.7 - mvs