Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.5.1 - Particle Focusing in a Corrugated Tube
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00260
 Material Information
Title: 10.5.1 - Particle Focusing in a Corrugated Tube Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Hewitt, G.F.
Marshall, J.S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: particle focusing
corrugated tube
 Notes
Abstract: A computational study is performed of the transport of a particulate suspension through a corrugated tube using a discrete-element method (Marshall, 2009). The tube is axisymmetric with a radius that varies sinusoidally, which in the presence of a mean suspension flow leads to periodic inward and outward acceleration of the advected particles. The oscillations in radial acceleration and straining rate lead to a net radial drift, with mean acceleration measuring an order of magnitude smaller than the instantaneous radial acceleration. Over time this radial drift leads to a focusing of small particles at the center of the tube. The foundations of particle focusing in this flow are examined analytically using lubrication theory together with a low Stokes-number approximation for the particle drift. This theory indicates that there are two competing mechanisms that give rise to the radial particle drift, one of which is related to the average of the fluid acceleration along lines parallel to the tube axis, and the other of which is related to the correlation between radial displacement of the particles and the radial gradient of the fluid acceleration field. Each of these effects gives rise to a particle drift velocity, which are often observed to be in competition with each other. The net radial particle drift, given by the sum of these two components, is found to vary linearly with Stokes number and as the square of the maximum slope of the tube wall.
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
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Volume ID: VID00260
Source Institution: University of Florida
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Resource Identifier: 1051-Hewitt-ICMF2010.pdf

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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010



Particle Focusing in a Corrugated Tube


Gregory F. Hewitt and Jeffrey S. Marshall
University of Vermont, School of Engineering
33 Colchester Ave, Burlington, Vermont 05405, U.S.A.
jeffm@cems.uvm.edu


Keywords: particle focusing, corrugated tube





Abstract

A computational study is performed of the transport of a particulate suspension through a corrugated tube using a discrete-element method
(Marshall, 2009). The tube is axisymmetric with a radius that varies sinusoidally, which in the presence of a mean suspension flow leads to
periodic inward and outward acceleration of the advected particles. The oscillations in radial acceleration and straining rate lead to a net radial
drift, with mean acceleration measuring an order of magnitude smaller than the instantaneous radial acceleration. Over time this radial drift
leads to a focusing of small particles at the center of the tube. The foundations of particle focusing in this flow are examined analytically using
lubrication theory together with a low Stokes-number approximation for the particle drift. This theory indicates that there are two competing
mechanisms that give rise to the radial particle drift, one of which is related to the average of the fluid acceleration along lines parallel to the
tube axis, and the other of which is related to the correlation between radial displacement of the particles and the radial gradient of the fluid
acceleration field. Each of these effects gives rise to a particle drift velocity, which are often observed to be in competition with each other. The
net radial particle drift, given by the sum of these two components, is found to vary linearly with Stokes number and as the square of the
maximum slope of the tube wall.


Introduction

Corrugated tube and channel flows are commonly used for
enhancement of fluid mixing and heat and mass transfer in
laminar flow fields. Studies have been performed to
examine flow regimes, fluid mixing, and heat transfer
enhancement for flow in both two-dimensional symmetric
corrugated channels and axisymmetric tubes (Guzman and
Amon, 1994, 1996; Rush et al., 1999; Nishimura et al.,
1990; NiCeno and Nobile, 2001; Mahmud et al., 2002;
Oviedo-Tolentino et al., 2008; Kim, 2001; Savvides and
Gerrard, 1984),. For low Reynolds numbers or sufficiently
small wall slope, the flow remains attached and the straining
rate on a fluid element oscillates as it is advected along the
channel (Vasudeviah and Balamurugan, 2001). At higher
Reynolds number or at higher wall slope, a recirculating
separated region forms within the crest of each corrugation
wave (Mahmud et al., 2002). These separated regions
display a rich array of flow regimes, starting with a steady
flow condition and passing through periodic, quasi-periodic,
and finally chaotic states as the Reynolds number is
increased (Guzman and Amon, 1994, 1996; Amon et al.,
1996). Conditions for instability of corrugated channel
flows have been reported in both theoretical and
experimental/computational studies (Selvarajan et al., 1999;
Cho et al., 1998; Cabal et al., 2002; Asai and Floryan, 2006).
Instability of a particle-laden suspension flow in a
wavy-walled channel is examined by Usha et al. (2005)
using a two-phase continuum theory; however, this work
does not consider particle migration in the channel.

Numerous applications of corrugated tube or channel flows
involve transport of a two-phase suspension through the


tube or channel. Examples include dust transport through a
radiator channel or collapsible plastic or metallic ducts used
for dryer ventilation. In biological systems, aqueous particle
or cell suspensions flow through corrugated tubes during in
transport of chyme through the colon, which is formed of
periodic "haustral pockets" positioned along its length (Putz
and Pabst, 2000).

Particle inertial focusing refers to the hydrodynamic-
induced drift of particles to certain preferential locations of
a microchannel flow in a manner that is dependent on the
particle inertia. Di Carlo et al. (2007) report the presence of
a continuous inertial focusing of 1-10 gm diameter particles
in a suspension flowing through in a stationary
anti-symmetric corrugated microchannel with repeated
S-shape curves to form a continuous undulating wave form.
Choi et al. (2008) and Choi and Park (2008) present a
hydrodynamic focusing method based on the flow response
and resulting particle drift induced by a series of obstacles
forming V-shaped patterns along different walls of the
microchannel. Other hydrodynamic focusing methods
based on the flow response to an array of obstacles placed in
the microchannel are proposed by Huang et al. (2004) and
Davis et al. (2006).

Recently, Marshall (2009a) showed that a particle clustering
phenomenon occurs when particles are exposed to an
oscillating straining field (in the particle frame), causing
particles to drift toward the nodal points of the straining
field. By using the low Stokes number approximation for
fluid drift velocity proposed by Ferry and Balachandar
(2003), this oscillatory clustering phenomenon can be
related to the non-zero time average of the fluid convective





Paper No


acceleration. A theoretical formulation for this phenomenon
was presented which reduces the particle motion in the
oscillating straining field to a damped Mathieu equation.
This theory predicts both the particle drift rate toward the
strain field nodal points and the stability limitations on
particle clustering. The method was applied by Marshall
(2009a) to examine particle drift in a peristaltic channel
flow, for which case particles drift toward the nodal points
of a standing peristaltic wave.

The current study applies this oscillatory clustering
phenomenon to suspension flow through an corrugated tube.
The tube corrugations induce an oscillatory straining on
particles advected through the tube. The oscillatory
clustering theory of Marshall (2009a) indicates that the
particles should drift toward the tube axis. The current paper
demonstrates that indeed particles do drift toward the tube
axis at low flow Reynolds numbers, but that at higher flow
Reynolds numbers the rate of oscillatory clustering within
the central part of the tube is greatly reduced, in part due to
the effects of flow separation within the furrow regions.


Nomenclature

a particle acceleration (ms-2)
A wave amplitude (m)
C particle drag coefficient (kg s 1)
d particle diameter (m)
E particle effective elastic modulus (Nm-2)
Ep particle elastic modulus (Nm-2)
f friction factor in drag force term
FA collision and adhesion force on particle (N)
Fd particle drag force (N)
FF fluid force on particle(N)
Fn normal force on particle (N)
Fnd dissipative part of particle normal force (N)
Fne elastic part of particle normal force (N)
Fs particle sliding resistance (N)
h tube radius (m)
H nominal tube radius (m)
I particle moment of inertia (kg m2)
k wavenumber (m 1)
kN particle normal force coefficient (Nm 1)
K particle nonlinear spring coefficient (Nm-3/2)
m particle mass (kg)
MA collision and adhesion torque on particle (Nm)
MF fluid torque on particle (Nm)
Mt twisting resistance torque (Nm)
n unit normal vector between two particles
p pressure (Nm-2)
r radial coordinate (m)
r, radius of ith particle (m)
q coefficient in Mathieu equation
R particle effective radius (m)
ReF fluid Reynolds number
s fluid straining rate (s 1)
St particle Stokes number
t time (s)
ts unit vector in sliding direction
u radial component of fluid velocity (ms-1)
u fluid velocity vector at particle location (ms-1)
UD particle drift velocity (ms-1)


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

v particle velocity vector (ms-1)
VR particle relative velocity at contact surface (ms-1)
vs particle slip velocity (ms-1)
w axial component of fluid velocity (ms-1)
W nominal axial velocity (ms-1)
We average fluid velocity (ms-1)
X, position vector of centroid of ith particle (m)
z axial coordinate (m)

Greek letters
a dimensionless wavenumber (= kH)
6N particle normal overlap (m)
At fluid time step (s)
Atc collision time step (s)
Atp particle time step (s)
s dimensionless particle diameter (= d/H)
a7 dimensionless wave amplitude (= A/H)
77N normal friction coefficient

2 wavelength (m)
u viscosity (kg mi's 1)
PF fluid density (kg m-3)
pp particle density (kg m-3)
co oscillation frequency of strain rate (s 1)
aO fluid vorticity vector (s 1)
2 particle rotation rate (s 1)
g damping coefficient in Mathieu equation

Symbols
prime Dimensionless
overbar furrow-averaged quantity


Numerical Scheme

The suspension flow is computed using an axisymmetric
finite-volume method for the fluid and a three-dimensional
discrete-element method for the particles. Computations are
performed with small particle concentrations (less than
0.5%) and small particle mass loading, so two-way phase
interactions can be neglected (Crowe et al., 1998).
Geometrical features of the tube corrugation waves are
illustrated in Figure 1. All variables are
non-dimensionalized using the nominal axial velocity W
and the nominal tube radius H for velocity and length scales,
respectively.
A


rH

W i z
W I



Figure 1: Schematic of flow in an axisymmetric
corrugated tube.


Fluid Flow Computational Method and Example Results
Computations of flow of an incompressible fluid in an
axisymmetric corrugated tube are performed using a






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


finite-volume method with a block-structured grid (Lai,
2000). The numerical method stores all dependent variables
at the cell centers of a structured grid in the r-z plane, and
uses a novel interpolation method to yield second-order
accurate spatial approximation of the diffusive and
convective fluxes on the cell boundaries for arbitrary
meshes. The PISO algorithm (Issa, 1985) is used to couple
momentum and continuity equations. Numerical stability is
enhanced by weighting the time derivative between
second-order and first-order upwind approximations, with
characteristically about 90-10 weighting ratio. The flow is
periodic over an axial distance often times the nominal tube
radius. The time step in all fluid flow computations is held
fixed at At = 0.01 .

Cases with corrugation wave amplitudes of 0.1 and 0.2 and
wavelengths of 1 and 2 were examined with flow Reynolds
numbers (ReF) of both 10 and 100. Each fluid run was
iterated until the pressure and velocity residuals reached
values of about 10-8. Large-amplitude cases with ReF =
100 exhibit detached, recirculating flows within the tube
corrugations, which subsequently reduces the magnitude of
straining flow oscillations to which the advected particles
are exposed. All of the ReF = 10 cases exhibit no
recirculation. Contours of the radial velocity and of the
negative of the radial acceleration are shown in Figure 2 for
a case with A=0.1 2=2 and ReF=100 This
particular flow does not exhibit a recirculation eddy in the
wave peaks, but instead the flow remains attached
everywhere within the tube. The flow is typified by
alternating regions of outward and inward flow along the
tube length. The transition between the outward and inward
flow results in a region of large outward-oriented
acceleration just under the wave trough. As discussed in the
next section, the particle drift relative to the streamlines is
proportional to the negative of the fluid acceleration, so the
contours of negative radial acceleration indicate regions of
inward and outward particle drift along the tube.
15. .


S225 23 235


24 245 25 255 26


S 006
004
0 02

-004


S02
015
01
0 05
0
-0 05
-0 1
-0 15
-02


Figure 2: Contours of (a) radial velocity and (b)
negative fluid acceleration for a case with A = 0.1,
2 =2 and Re =100 which exhibits no flow
recirculation. Dashed lines indicate negative contour
values and solid lines indicate positive contour values.


Figure 3: Grid sensitivity study showing the
axially-averaged radial acceleration across the
interrogation region as a function of radius for Grids
A-E. The coarsest grid (Grid A) is indicated using a
dashed line.


Grid independence of the fluid flow computations is
examined by comparing the radial acceleration averaged
across the tube length. Though quantities such as velocity
converge with a relatively coarse grid, quantities that are
averaged along the tube axis (such as the axially-averaged
acceleration, which is used in determining the particle drift
velocity) require a much finer grid to achieve convergence.
This difference is due to the fact that the average fluid
acceleration is about two orders of magnitude smaller than
the instantaneous acceleration. A representative grid
geometry of amplitude A = 0.1 and wavelength = 2 ,
with a Reynolds number of 100, was utilized for the grid
sensitivity study. The sensitivity study examined the effect
of grid density on the axially-averaged acceleration values
using five different grids, labeled Grid A (1200x 50
points), B (1800 x 100 points), C (2400 x 150 points), D
(3000 x 200 points) and E (4050 x 300 points), all over a
domain covering the region 0 r <1.1 and 0 < z <10 A
plot of the average radial acceleration for these five grids is
given in Figure 3. The peak value of the average
acceleration for the Grid D was found to be within 1.3% of
that for the most refined Grid E. All of the computations in
the paper are performed using grid resolution similar to that
in Grid D.

Discrete-Element Method for Particle Transport
A multiple-time scale discrete-element method (DEM)
(Marshall 2009b) is used to examine particle transport in
the corrugated tube at finite Stokes number. The
computational method assumed that the fluid time step
At = O(H IW), the particle time step At = O(d W), and

the collision time step At = O(d(p l/Eii)15) satisfy

At > At, > Ate. Here d is the particle diameter, p, is the

particle density, and E is the particle elastic modulus.
The method follows the motion of individual particles in
the three-dimensional fluid flow by solution of the particle
momentum and angular momentum equations


Paper No


i






Paper No


dv
m-= FFFA,
dt


dn
I = MF +MA, (1)
dt


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

where o, and o, are the Poisson ratios and E, and E,
are the elastic moduli of the individual particles. The
dissipation force F'd is given by


under forces and torques induced by the fluid flow (FF
and M ) and by the particle collision ( F and MA ). In
these equations, m is the particle mass and I is the moment
of inertia. The dominant fluid force is the drag force,
approximated by a modified form of the Stokes drag law

Fd = -3-dp(v u)f (2)

where v and u are the particle and local fluid velocities
and f is a friction factor that accounts for the effect of
local particle crowding using the correlation proposed by
Di Felice (1994). The viscous torque exerted by the fluid
arises from a difference in rotation rate of the particle and
the local fluid region and is given by


MF = -;-d3(,Q- ),
2


where 0 is the particle rotation rate and ca is the local fluid
vorticity vector. In addition to the drag force, the
fluid-induced forces also include the lift force (Saffman,
1965, 1968), the Magnus force (Rubinow & Keller, 1961),
and the added mass and pressure gradient forces. Particles
are assumed to be neutrally buoyant.

Particle collisions are simulated by employing a soft-sphere
collision model. Each collision involves a normal force F,
along the line passing through the particle centroids and
frictional resistances for sliding and twisting motions of the
particles. For a particle of radius r, the forces and torques
are given by

FA =Fn+Fts, MA =rFj(nxts)+An. (4)

The unit vector t vs/Ivsl indicates the direction of
sliding between the two particles, where vs is the slip
velocity at the contact point. There is no adhesive force
between the particles and no rolling resistance. The normal
force F is composed of the elastic force Fe and a
dissipative force Fd The Hertz (1882) expression gives
the elastic part of the normal force as

3/2
Fe = -kN6 = -KS2 (5)


where K = (4/3)Ej-R is an effective nonlinear spring

constant and 8A = r, + r, x, x is the particle overlap.
The particle effective radius and elastic modulus, R and E,
are defined by


1 1 1
--J+-
RrrJ


1- ,2 1- 0-2
F, J
E, EJ


Fnd =--N'7 VR "n,


where ar, is the normal friction coefficient. We use the
expression proposed by Tsuji et al. (1992) for 7N with the
restitution coefficient set equal to zero, which is consistent
with experimental results for small Stokes number particle
collisions (Joseph et al., 2001). The spring-dashpot-slider
model proposed by Cundall and Strack (1979) is used for
the sliding resistance. A rotational form of the
spring-dashpot model developed by Marshall (2009b) is
used for the twisting resistance.


Results and Discussion

Particle Clustering under Oscillatory Straining

Particle clustering under an oscillating straining flow was
examined by Marshall (2009a) using both analytical and
numerical methods. A simple theory explaining this
phenomenon can be developed based on one-dimensional
motion of an isolated particle in an oscillating straining flow,
with velocity components u1 =s(t)x and u, =-s(t)y
where the straining rate s(t) = A cos(ot) varies
harmonically in time. The equation of motion for a particle
of mass m immersed in the straining flow is


dv
m- = -C(v u)
dt


where C = 3n p and we assume (for purposes of
illustration) that the Stokes drag force is the only force
acting on the particle. Here u is the fluid velocity and v is the
particle velocity, both in the x-direction.

Non-dimensionalizing the terms in (8) using the inverse
forcing frequency c as the time scale and substituting
v= dx/dt yields the governing equation for the particle
motion in the form of a damped Mathieu equation:


d x' dx'
S+ 2 ,d 2qcos(2t')x' = 0,
dt'2 dt'


where =Cl/mo and q 2CA/mo2 Numerical
solutions of (9) are given in Figure 3 for the velocity and
position of both the particle and a passive marker. While the
passive marker just oscillates back and forth, the particle
drifts toward the center of the oscillating straining field due
to the velocity lag between the particle and the passive
marker. The particle inertia causes it to move downward a
bit more than the marker when it reaches the bottom of the
oscillation cycle, and as a result the particle lags the passive
marker during the outward-motion part of the cycle. The
particle consequently does not rise as high as the passive
marker at the top of the oscillation cycle before reversal of






Paper No


the straining flow. As the cycle continues, this difference in
phase between the passive marker and the particle results in
a net downward motion of the particle toward the position
of zero strain.


a 20* sol 2 0 0 0 40 ( O M INO
T r
Figure 3: Plots showing (a) particle velocity (solid line)
and the fluid velocity at the particle position (dashed-dotted
line) and (b) position of a passive particle (dashed line) and
a real particle (solid line). [Reproduced from Marshall,
2009a]

Lubrication Theory Approximation for Particle Drift in
a Corrugated Tube

We consider axisymmetric incompressible flow with
velocity u = u(r,z)er + w(r,z)ez in a corrugated tube with
radius h(z)=H +Acos(kz) At small flow Reynolds
number (ReF pWH /I <<1) and small slope ( kA <<1),
lubrication theory can be used to approximate the governing
equations for the flow as


--(ru)+-= 0,
r Or Cz


0= y(z)+ -(r ), (10)
r Or


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


where a =kH q =A/H and a prime denotes a
dimensionless variable.

A simplified form of the particle momentum equation valid
for low Stokes numbers is given by Ferry and Balachandar
(2003) as


Du'
v'= u'- St + O(St2),
Dt'


where v' and u' are the dimensionless particle velocity
and fluid velocity, respectively. The Stokes number can be
written in terms of the flow Reynolds number and the
dimensionless particle diameter e = d H as


St = 2 ReF/18 ,


where the density ratio X = 1 for the neutrally buoyant
particles.

The dimensionless particle drift velocity u' is given by
(14) to leading order in St as u' =-Sta' where
a' = Du'/Dt' is the dimensionless fluid acceleration at the
particle centroid location. We define F. as the average
radial position of particle n as it passes through a furrow, or


Fn(zn) =- rn(z)dz,
-A


where y(z) -Op /z is the negative pressure gradient and
p and p are the fluid density and viscosity, respectively.
Integration of (10) over r yields a solution for the velocity
components as


w(r,z) = [hz) r2], (lla)
4p

u(r,z)= (z)rh + (2h r 2) (Ilb)
l + +1 (2h2-r2) (llb)
4p dz 16p dz

Applying the no-slip condition at r =h(z) gives
h4y = C where C, is a constant of integration. The
average velocity over a cross-section of the tube is obtained
as


2 h(z)
Wc (.z) I w(r,z)rdr
h 0


C
8ph2(z)


where (rn,zn) denote the instantaneous radial and axial
coordinates of particle n. Variables averaged in accordance
with (16) are referred to as furrow-averaged quantities.
Since zn is a function of time, the furrow-averaged radial
position can also be written as a function of time (t) the
rate of change of which represents the radial particle drift.

In the case that the tube corrugation amplitude is small, the
fluid acceleration at the particle centroid can be expanded in
a Taylor series about the furrow-averaged position F((t) to
obtain
8a'
a'(r) = a'(') + (r )- +..., (17)

where the omitted terms are higher order in the small
parameter q than the two terms retained. The first and
second particle drift velocities, u' and uD2, are defined
as the furrow-averaged particle velocities generated by the
each of the terms on the right-hand side of (11), such that


A nominal axial velocity is defined by W -C /81uH2
Non-dimensionalizing u and w by W and
non-dimensionalizing r, h and z by H yields the
dimensionless velocity field as


w'=,2 [1-(r'/h')2], u'
h


2r- q [1 -(r'/h')2]sin(az'),
h


u1I =-St a'('), uD2


-St [(r'- F')8a'/Ir']. (18)


The first drift velocity is the average of the fluid
acceleration along the tube at the furrow-averaged particle
radial position. The second drift velocity is related to the
correlation between the radial oscillation of the particle
about the furrowed-averaged location and the variation of






Paper No


the radial acceleration gradient.

In a steady flow, the definition of a streamline can be
integrated to write r'-r'= [(u'lw')dz'. Substituting the
dimensionless velocity field (13) into the fluid material
derivative and averaging over the tube perturbation
wavelength yields the first and second fluid radial drift
velocities for the lubrication theory to leading order in 7 as


UD, = -2St(a)2r(1 r2)(3 72) ,
u2 = 2St(aq7)2(1 2)(1 -5r2) .


(19a)
(19b)


The first drift velocity is negative (inward drift) for particles
in the region 0 < r' < Ji37 and positive (outward drift)
for r3/7 < r' < 1 The second drift velocity is positive for
0 < r' < 1/ and negative for 1/ 1 < r' <1.

The net particle drift velocity u' is given by the sum of
u3I and '2 which is normalized by St(Ak)2 and
plotted in Figure 4 as a function of radius. Within the central
part of the tube (r'< 0.4), the magnitude of the first drift
velocity is significantly greater than that of the second drift
velocity, and the direction of the two velocities is opposite
to each other. The drift velocities each change sign within
the outer part of the tube, and the magnitudes of the two drift
velocities in this region are similar. The net effect is to keep
the total drift velocity negative throughout the tube,
implying that under the lubrication theory all particles will
drift toward the tube center at a velocity that is proportional
to St(Ak)2.



1 / \ \
15





-0.5






-20 02 04 06 08 '
r'

Figure 4: Plot showing first particle drift velocity uSI
(dashed line), second particle drift velocity u'2
(dashed-dotted line), and net particle drift velocity
uD = u3l +u'2 (solid line) for low Stokes number
particle transport in a tube given by the lubrication
theory. All velocities are normalized by St(aq)2.


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

DEM Simulations of Particle Focusing

The DEM calculations are initialized by placing particles in
an array in the tube with random initial velocities. A
preliminary calculation is then performed with no flow in
the tube and no fluid drag on the particles, during which
time the particles are allowed to bounce around within the
tube until they achieve a state in which the concentration
profile is essentially uniform. The particles are initialized
only within the inner 80% of the tube radius. The results of
this preliminary calculation are then used as the initial
condition for the DEM simulations with the corrugated
tube.

Results of a DEM simulation for a case with Re = 10 are
shown in Figure 5 for a case with e = 0.12, St = 0.008,
2 = 2 and A = 0.2 The particles are plotted in an end
view, looking down the end of the tube, at the initial time, at
an intermediate time (t = 25 ), and at a much later time
(t = 400 ) after which the particles have achieved a state of
statistical equilibrium. A solid circle and a dashed circle are
used in Figure 5a-c to indicate the innermost and outermost
values of the tube radii, respectively, corresponding to the
trough ( r = 0.8 ) and crest ( r = 1.2 ) radii of the wall
corrugations.

The particles are observed in Figure 5 to drift steadily
toward the center of the tube, with the drift rate highest near
the outer part of the tube. The slow contraction as the
particles become more dense near the center of the tube is
primarily caused by the radial variation in the particle drift
velocity (as seen in Figure 4), but more frequent particle
collisions also plays a role in mitigating particle inward drift
at large times. We note that the end-view figures in Figure 5
make the particles appear to be much more concentrated
than is actually the case. In a side view, as seen in Figure 5d,
it is apparent that the particles are spread out in a fairly
low-concentration suspension along the tube axis, so that
collisions are relatively infrequent.

The effect of flow Reynolds number was explored for a
fixed tube geometry with A= 0.1 and /=2 ,
corresponding to the non-recirculating flow case shown in
Figure 2. The furrow-averaged radial acceleration a,(r),
which is proportional to the negative of the first drift
velocity, is plotted in Figure 6 for cases with flow Reynolds
numbers of 1, 10 and 100, as well as the lubrication theory
(ReF = 0) solution (19a). Consistent with our theoretical
derivation, the lubrication theory result is truncated at r = 1
since the waves on the tube are assumed to be small. The
ar,(F) term dominates the inward drift velocity in the
center region of the tube, hence the plot in Figure 6 gives an
indication of the influence of flow Reynolds number on the
rate of inward particle drift. While all curves in Figure 6
have the same basic form, the magnitude of ,r(F) is
observed to decrease markedly with increase in flow
Reynolds number. For instance, the maximum positive
value of ar(r) is 0.2 for the lubrication theory but it is
only 0.02 for ReF =100.






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


I 0D.5 0 0.5 I


-1 -o.( D o. i

d)' . ... ':


1 0.5 0 0.5 I


(b) o o',o00 o o o



Figure 5: Time series showing particle drift toward the tube
center shown in (a) end view and (b) side view at final time
of series.


r
Figure 6: Plots showing the effect of Reynolds number
on the furrow-averaged radial acceleration (evaluated
at a fixed radial location) for cases with A = 0.1 and
2 = 2, and flow Reynolds numbers of 100 (solid line),
10 (dashed-dotted line), 1 (dashed line), and the
lubrication theory solution (dashed-double-dotted line).

Results of a DEM simulation for a case with ReF =100
and A = 0.2 are given in Figure 7, again showing an end
view looking down the tube. Dashed and solid lines
represent the maximum and minimum locations of the tube
surface waves. Particles are initialized with approximately
uniform concentration profile and with diameter e = 0.02.
The particles within the outer part of the tube quickly
contract inward toward the tube center. However, at a radius
of about r = 0.55 the inward motion of these particles
slows to such an extent that little further motion is
discernable in the computations. Particles at radial positions
between the tube center and a radius of r =0.55 are
observed to advect down the tube with little observable net
radial movement within the computational time period.


Figure 7: Time series showing formation of a particle
ring for a case with ReF = 100 at times (a) t = 0, (b)
t= 50, (c) t= 600 and (d) t =1900. The dashed and
solid lines represent the maximum and minimum
locations of the tube surface, corresponding to the wave
crest and trough, respectively.



Conclusions

Focusing of particles is investigated for suspension flow
down a corrugated, axisymmetric tube using both an
analysis based on lubrication theory and computations
based on the discrete-element method. At low flow
Reynolds numbers, the particles are observed to drift
toward the center of the tube at a rate that is proportional to
the Stokes number and the square of the tube maximum
corrugation wave slope. The particle drift is governed by the
sum of two competing drift velocities, one of which is
proportional to the axial average of the fluid radial
acceleration and the other of which is proportional to the
average of the particle radial deviation from its averaged
position times the gradient of the fluid radial acceleration.
The first of these drift velocities tends to move particles
inward within the inner part of the tube and outward within
the tube outer part, whereas the second drift velocity tends
to do the opposite. The rate of particle drift is observed to
decrease with increase in flow Reynolds number, due to the
decrease magnitude in the fluid acceleration oscillations at
higher Reynolds numbers.

The phenomenon reported in the current paper is shown to
be distinct from the well-known Segre-Silberberg effect or
other types of particle migration observed for straight tubes.
Specifically, the particle focusing discussed in the paper is
not observed in flows with straight tubes at the low values
of particle Reynolds number and particle concentration
used in the current study.


Paper No


ICC=L


(a)
..bs '
0





I Q'r 0 .
(c) -.






Paper No


Acknowledgements

This work was supported by the U.S. Department of
Transportation (grant number DTOS59-06-G-00048) and
by Vermont EPSCoR (grant number EPS 0701410).

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