Content uploaded by Phanindra Tallapragada
Author content
All content in this area was uploaded by Phanindra Tallapragada on Jan 12, 2016
Content may be subject to copyright.
Particle segregation by Stokes number for small neutrally buoyant spheres in a fluid
Phanindra Tallapragada*and Shane D. Ross†
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, (VPISU),
Blacksburg, VA 24061, USA
共Received 12 March 2008; revised manuscript received 14 June 2008; published 10 September 2008兲
It is a commonly observed phenomenon that spherical particles with inertia in an incompressible fluid do not
behave as ideal tracers. Due to the inertia of the particle, the planar dynamics are described in a four-
dimensional phase space and thus can differ considerably from the ideal tracer dynamics. Using finite-time
Lyapunov exponents, we compute the sensitivity of the final position of a particle with respect to its initial
velocity, relative to the fluid, and thus partition the relative velocity subspace at each point in configuration
space. The computations are done at every point in the relative velocity subspace, thus giving a sensitivity
field. The Stokes number, being a measure of the independence of the particle from the underlying fluid flow,
acts as a parameter in determining the variation in these partitions. We demonstrate how this partition frame-
work can be used to segregate particles by Stokes number in a fluid. The fluid model used for demonstration
is a two-dimensional cellular flow.
DOI: 10.1103/PhysRevE.78.036308 PACS number共s兲: 47.52.⫹j, 47.51.⫹a
I. INTRODUCTION
It has long been observed that particles with a finite size
and mass have different dynamics from the ambient fluid.
Because of their inertia the particles do not evolve as point-
like tracers in a fluid. This leads to preferential concentra-
tion, clustering, and separation of particles as observed in
numerous studies 关1–3兴. The inertial dynamics of solid par-
ticles can have important implications in natural phenomena,
e.g., the transport of pollutants and pathogenic spores in the
atmosphere 关4,5兴, formation of rain clouds 关6兴by coales-
cence around dust particles, and formation of plankton colo-
nies in oceans 关7兴. Similarly, the inertial dynamics of reactant
particles is important in the reaction kinetics and distribution
of reactants in solution for coalescence-type reactions 关8兴.
Mixing-sensitive reactions in the wake of bubbles have been
shown to be driven by buoyancy effects of reactants 关9兴.
Recently, a principle of asymmetric bifurcation of laminar
flows was applied to the separation of particles by size and
demonstrated the separation of flexible biological particles
and the fractional distillation of blood 关10,11兴. Innovative
channel geometries have been empirically designed to focus
randomly ordered inertial particles in microchannels 关12兴.
These phenomena and related applications rely on the non-
trivial dynamics of inertial particles in a fluid.
Many theoretical and numerical studies have been done
on the dynamics of inertial particles in some model flows,
using the Maxey-Riley equation 关13兴. Maxey 关14兴studied the
settling properties and retention zones of nonbuoyant inertial
particles in vertical cellular flows, under the influence of
gravity. The sensitive dependence on the initial conditions of
inertial particle trajectories and clustering was further studied
by 关15兴using a two-dimensional cellular flow for neutrally
buoyant particles.
Several studies have been concerned with characterizing
the surface in physical space to which inertial particles clus-
ter, using, for instance, fractal dimension and rates of con-
vergence to this surface. Among these, 关16–18兴studied clus-
tering of inertial particles in two-dimensional flow past a
cylinder. Similar studies 关19–22兴considered inertial particle
clustering in turbulent flows. In the turbulent flow studies,
the spectrum of 共long-time兲Lyapunov exponents was com-
puted to calculate the fractal dimension of the surfaces of
particle clustering. While we also use Lyapunov exponents,
we use them in a different way, as described below.
A method to segregate inertial particles from an initial
mixture by different sizes 共i.e., Stokes numbers兲was numeri-
cally demonstrated in 关23兴, but no physical description was
given of how segregation arose. We provide a description by
partitioning phase space into zones of initial phase space
locations where inertial particles will evolve to different final
locations, according to Stokes number. To achieve this, we
calculate the phase space distribution of short-time Lyapunov
exponents, and use topological features of this distribution to
find the partition boundaries. This partition is parametrized
by the Stokes number. We demonstrate this method using a
simple test model of two-dimensional flow, namely, cellular
flow, demonstrating that this partitioning scheme can explain
the sensitive dependence of trajectories on initial conditions
and the consequent clustering effects. We argue that this
methodology can be used as a systematic tool to achieve
segregation of inertial particles in a fluid.
We employ a simplified form of the Maxey-Riley equa-
tion 关13兴as the governing equation for the motion of inertial
particles in a fluid. The dynamics of a single particle occur in
a four-dimensional phase space. The sensitive dependence of
the particle motion on initial conditions is quantified using
the finite-time Lyapunov exponents 共FTLEs兲. It has been
shown previously 关24,25兴, that the ridges in the FTLE field
act as separatrices. These are in general time dependent and
go by the name of Lagrangian coherent structures 共LCSs兲.
We chose to do a simplified sensitivity analysis by perturbing
the initial conditions in only two dimensions, in the initial
relative velocity subspace. We obtain a sensitivity field akin
to a FTLE field but restricted to the relative velocity sub-
space, and demonstrate numerically that the ridges in this
*tssap@vt.edu
†sdross@vt.edu
PHYSICAL REVIEW E 78, 036308 共2008兲
1539-3755/2008/78共3兲/036308共9兲©2008 The American Physical Society036308-1
field act as separatrices. The partitions in the relative velocity
subspace created by these separatrices determine the even-
tual spatial distribution of particles in the fluid. Using this
partitioning scheme we show how the Stokes number acts as
a parameter in the separation of particles of different inertia
or size.
The paper is organized as follows. In Sec. II we review
the equation governing the inertial particle dynamics in a
fluid and its simplified form. In Sec. III we briefly review the
background theory of phase space distributions of finite-time
Lyapunov exponents, which we use to quantify the sensitiv-
ity of the physical location of inertial particles with respect
to perturbations in the initial relative velocity. We also de-
scribe our computational scheme to obtain the sensitivity
field in the relative velocity subspace. In Sec. IV we present
results for the sensitivity field of the inertial particles in a
cellular flow. In Sec. V we demonstrate our procedure for the
segregation of particles by their Stokes number using the
results from Sec. IV In Sec. VI we give numerical justifica-
tion for the robustness of the sensitivity field to perturbations
in the velocity field of the fluid. In Sec. VII we discuss the
results and give conclusions.
II. GOVERNING EQUATIONS
Our starting point is Maxey and Riley’s equation of mo-
tion of a rigid spherical particle in a fluid 关13兴:
p
dv
dt =
f
Du
Dt +共
p−
f兲g−9
f
2a2
冉
v−u−a2
6
ⵜ2u
冊
−
f
2
冋
dv
dt −D
Dt
冉
u−a2
10
ⵜ2u
冊
册
−9
f
2a冑
冕
0
t1
冑t−
d
d
冉
v−u−a2
6
ⵜ2u
冊
d
,共1兲
where vis the velocity of the solid spherical particle, uthe
velocity field of the fluid,
pthe density of the particle,
fthe
density of the fluid,
the kinematic of the viscosity of the
fluid, athe radius of the particle, and gthe acceleration due
to gravity. The terms on the right-hand side are the force
exerted by the undisturbed flow on the particle, the force of
buoyancy, the Stokes drag, the added mass correction, and
the Basset-Boussinesq history force, respectively. Equation
共1兲is valid under the following restrictions:
a共v−u兲/
Ⰶ1,
a/LⰆ1,
冉
a2
冊冉
U
L
冊
Ⰶ1, 共2兲
where Land U/Lare the length scale and velocity gradient
scale for the undisturbed fluid flow. The derivative
Du
Dt =
u
t+共u·兲u共3兲
is the acceleration of a fluid particle along the fluid trajec-
tory, whereas the derivative
dv
dt =
v
t+共v·兲v共4兲
is the acceleration of a solid particle along the solid particle
trajectory.
Equation 共1兲can be simplified by neglecting the Faxen
correction and the Basset-Boussinesq terms 关15兴. We restrict
our study to the case of neutrally buoyant particles, i.e.,
p
=
f. Writing W=共v−u兲, the relative velocity of the particle
and the surrounding fluid, the evolution of Wbecomes
dW
dt =−共J+
I兲W,共5兲
and the change in the particle position is given by
dr
dt =W+u,共6兲
where Jis the gradient of the undisturbed velocity field of
the fluid, r=共x,y兲is the position of the solid particle, and
=2
3St−1 is a constant for a particle with a given Stokes
number St. Equations 共5兲and 共6兲can be rewritten as the
vector field
d
dt =F共
兲,共7兲
with
=共r,W兲=共x,y,Wx,Wy兲苸R4. Equation 共7兲defines a
dissipative system with constant divergence −4
3
. It has been
shown by Haller 关17兴that an exponentially attracting slow
manifold exists for general unsteady inertial particle motion
as long as the particle Stokes number is small enough. For
neutrally buoyant particles this attractor is W=0共the xy
plane兲. Despite the global attractiveness of the slow mani-
fold, domains of instability exist in which particle trajecto-
ries diverge 关15,16,18兴.
III. SENSITIVITY ANALYSIS
The Lyapunov characteristic exponent is widely used to
quantify the sensitivity to initial conditions. A positive
Lyapunov exponent is a good indicator of chaotic behavior.
We have used the finite-time version of the Lyapunov expo-
nents, the FTLEs, as a measure of the maximum stretching
for a pair of phase points.
We review some important background regarding the
FTLEs below, following 关24–26兴. The solution to Eq. 共7兲can
be given by a flow map
t0
t, which maps an initial point
共t0兲
at time t0to
共t兲at time t,
PHANINDRA TALLAPRAGADA AND SHANE D. ROSS PHYSICAL REVIEW E 78, 036308 共2008兲
036308-2
共t兲=
t0
t„
共t0兲….共8兲
The evolution over a time Tof the displacement between two
initially close phase points
共t0兲and
共t0兲+
␦
共t0兲is given by
␦
共t0+T兲=
d
t0
t0+T共
兲
d
␦
共t0兲+O共储
␦
储2兲.共9兲
Neglecting the higher-order terms, the magnitude of the per-
turbation is
储
␦
共t0+T兲储 =冑
冓
␦
共t0兲,
d
t0
t0+T共
兲*
d
d
t0
t0+T共
兲
d
␦
共t0兲
冔
.
共10兲
The matrix
C=
d
t0
t0+T共
兲*
d
d
t0
t0+T共
兲
d
共11兲
is the right Cauchy-Green deformation tensor. Maximum
stretching occurs when the perturbation
␦
is along the ei-
genvector nmax corresponding to the maximum eigenvalue
max of C. The growth ratio is given by
储
␦
共t0+T兲储/储
␦
共t0兲储 =e
1„
共t0兲…兩T兩,共12兲
where
1„
共t0兲…=1
兩T兩ln 冑max共C兲共13兲
is the maximal finite-time Lyapunov exponent. One can as-
sociate an entire spectrum of finite-time Lyapunov exponents
with
共t0兲, ordering them as
1„
共t0兲…⬎
2„
共t0兲…⬎
3„
共t0兲…⬎
4„
共t0兲….共14兲
The entire spectrum of the Lyapunov exponents can be com-
puted from the state transition ⌽共t,t0兲=d
t0
t共
兲/d
matrix
using singular value decomposition, where t=t0+T,
⌽共t,t0兲=B共t,t0兲⌳共t,t0兲1/2R共t,t0兲.共15兲
The diagonal matrix ⌳gives all the Lyapunov exponents
while
⌺共tf,t0兲=ln关⌳共tf,t0兲1/2兩T兩兴,共16兲
where T=tf−t0and ⌺共tf,t0兲=diag共
1,...,
4兲. An arbitrary
perturbation in the fixed basis can be transformed using a
time-dependent transformation 关26兴
␦
⬘共t兲=A共t,t0,tf兲
␦
共t兲,共17兲
such that in the new basis 共the primed frame兲, the variational
equations become
␦
˙⬘共t兲=⌺共tf,t0兲
␦
⬘共t兲.共18兲
Since ⌺共tf,t0兲is a constant diagonal matrix, we have
␦
⬘共t兲=e共t−t0兲⌺共tf,t0兲
␦
⬘共t0兲.共19兲
The first coordinate in the new frame grows as
␦
1
⬘共t兲
=e共t−t0兲
1
␦
1
⬘共t0兲. The time-dependent transformation A共t兲is
given by 关26兴
A共t,t0,tf兲=e共t−t0兲⌺共tf,t0兲R共tf,t0兲*R共t,t0兲⌺共t,t0兲−1/2B共t,t0兲.
共20兲
A. Sensitivity to initial relative velocity
Since the dynamics of the inertial particle is in a four-
dimensional phase space, the separatrices, that is, LCSs de-
fined by ridges in the field of the maximal FTLEs, are three-
dimensional surfaces 共see 关25兴兲. However, because the
system is dissipative and the global attractor is the xy sub-
space, we can obtain meaningful information by restricting
the computations to a lower-dimensional subdomain of the
phase space. This we do by considering an initial perturba-
tion only in the relative velocity subspace and study how this
perturbation grows in the xy plane, the configuration space,
i.e.,
␦
共t0兲=关0,0, ⌬Wx,⌬Wy兴*,共21兲
where ⌬Wxand ⌬Wyare the perturbations in the relative
velocity subspace. It is to be noted that when perturbations
are applied to the initial velocity of the solid particle, an
initial drag is experienced by the particle, but this effect is
negligible 关14兴.
Using the time-dependent transformation A共t,t0,tf兲the
evolution of the perturbation is given by
␦
共t兲=A−1共t兲e共t−t0兲⌺共tf,t0兲A共t0兲
␦
共t0兲.共22兲
The growth of perturbations in the xy plane is given by the
first two components of the above vector. The last two com-
ponents are the evolution of the perturbations in the relative
velocity subspace. Since the xy plane is a global attracting
set, these tend to zero. One can choose a finite time Tsuch
that the evolution of the initial perturbation comes arbitrarily
close to the xy plane. In this way the sensitivity of the final
spatial location of the particles with respect to initial relative
velocity can be computed.
B. Numerical computation of the sensitivity field
The evolution of a perturbation is along the four basis
vectors. For an arbitrarily oriented initial perturbation the
growth may not be dominated in the direction of greatest
expansion for short integration times. This can be overcome
by sampling multiple perturbations in the different direc-
tions. A reference point and its neighbors are identified and
after a finite time their positions in configuration space are
PARTICLE SEGREGATION BY STOKES NUMBER FOR…PHYSICAL REVIEW E 78, 036308 共2008兲
036308-3
computed. The state transition matrix can then be computed
at each point in the xy plane, by using a central finite-
difference method. For initial perturbations restricted to
WxWysubspace, this gives
⌽,rW =
冢
xi,j,k+1,l共t0+T兲−xi,j,k−1,l共t0+T兲
⌬Wx共t0兲
xi,j,k,l+1共t0+T兲−xi,j,k,l−1共t0+T兲
⌬Wy共t0兲
yi,j,k+1,l共t0+T兲−yi,j,k−1,l共t0+T兲
⌬Wx共t0兲
yi,j,k,l+1共t0+T兲−yi,j,k,l−1共t0+T兲
⌬Wy共t0兲
冣
.共23兲
In the four-dimensional finite-difference grid 共indexed by
i,j,k,l兲, each reference point has eight neighboring points,
one along each of the positive and negative directions in
each phase space direction. Since we are looking at the initial
perturbations in relative velocity only, we fix the initial spa-
tial position of the particle 共i,j兲but vary its relative velocity
共k,l兲. In the relative velocity, points are separated by con-
stant amounts ⌬Wx,⌬Wyin the x,yrelative velocity direc-
tions, respectively.
The relative velocity sensitivity field
共Wx,Wy兲is given
by
共Wx,Wy兲=1
兩T兩ln 冑max共⌽,rW
*⌽,rW兲.共24兲
Ridges on this sensitivity surface are one dimensional struc-
tures similar to LCSs. The ridges in the maximal sensitivity
field
共Wx,Wy兲partition the relative velocity subspace. We
applied the above procedure to a cellular flow.
We make a note on the terminology used here. The field
measuring the sensitivity of the final location of particles in
configuration space with respect to perturbations in initial
relative velocity is analogous to the FTLE field, but not iden-
tical. To obtain the true FTLE field, one would have to com-
pute the 4⫻4 state transition matrix ⌽. Using the notation of
Eq. 共23兲,
⌽=
冉
⌽,rr ⌽,rW
⌽,Wr ⌽,WW
冊
.共25兲
The FTLE field is then given by
共x,y,Wx,Wy兲
=共1/兩T兩兲ln冑max共⌽*⌽兲. Ridges in this field are three-
dimensional structures and represent the true LCSs. Ridges
in the relative velocity sensitivity field
共Wx,Wy兲are one-
dimensional structures which can be considered “slices” of
the full three-dimensional structure, where the slices are pa-
rametrized by the two-dimensional location of the initial spa-
tial point 共x,y兲.
IV. EXAMPLE FLUID MODEL: CELLULAR FLOW
We demonstrate the computation of the relative sensitivity
field using a simple test model of two-dimensional flow. We
choose cellular flow, as it has been used in previous studies
关14,15兴, and is a simple example of a fluid with separatrices.
This flow is described by the stream function
共x,y,t兲=acos xcos y.共26兲
The velocity field is given by
u=−acos xsin y,共27兲
v=asin xcos y共28兲
There are heteroclinic connections from the stable and
unstable manifolds of the fixed points 共2n+1兲共
/2兲, shown
by the arrows in Fig. 1, which are also the boundaries of the
cells. These coincide with LCSf, the LCSs for fluid particles,
which have no relative velocity 共Wx=Wy=0兲and evolve ac-
cording to the fluid velocity field, Eqs. 共27兲and 共28兲. The
LCSfis to be distinguished from the LCS of the inertial
particle in the full four-dimensional phase space. By choos-
ing initial perturbations of the form given by Eq. 共21兲at
different points along a streamline, we follow how these per-
turbations grow in the xy plane by integrating the particle
FIG. 1. Streamlines of
=acos xcos yform an array of cells.
The arrows indicate the heteroclinic fluid trajectories connecting the
fixed points of the velocity field formed by
. For this velocity
field, the heteroclinic trajectories coincide with the LCSf, i.e., the
separatrices or transport barriers, for fluid particles.
PHANINDRA TALLAPRAGADA AND SHANE D. ROSS PHYSICAL REVIEW E 78, 036308 共2008兲
036308-4
trajectories numerically from which the sensitivity field is
computed. Figure 2shows the sensitivity field computed for
initial perturbations in the relative velocity subspace, at dif-
ferent points on the streamline
共x,y,t兲=0. The ridges in
this field have high values of sensitivity. It can be seen that
there is a continuous variation in the ridges of the sensitivity
(
a)
(
b)
(
c
)
(
d)
(
e)
(
f)
(g
)
(
h)
(
i)
(
j)
(
k
)
(
l
)
FIG. 2. Ridges in the sensitivity field for
=acos xcos y. Initial spatial position varies from 共a兲共x0,y0兲=共
/2,
/2兲to 共k兲共0,
/2兲,at
the points shown in 共l兲, along
=0, at intervals of 0.05
. The plots show a smooth variation in the structure of the ridges in the sensitivity
field. Parameters: a=100, St=0.2, T=0.24.
PARTICLE SEGREGATION BY STOKES NUMBER FOR…PHYSICAL REVIEW E 78, 036308 共2008兲
036308-5
field with respect to the initial 共x,y兲coordinates. In each case
the sensitivity field at a given point depends on the underly-
ing LCSfof the fluid flow.
The ridges in the sensitivity field have meaningful infor-
mation about the dynamics of inertial particles even when
computed at points far from the saddle points of the fluid
flow. This is shown in Fig. 3共a兲which is the sensitivity field
computed at 共x,y兲=共3
/8,3
/8兲. The ridges in the sensitiv-
ity field partition the relative velocity subspace according to
the final location of particles. In Fig. 3共b兲the ridges in the
sensitivity field are used to identify regions in the relative
velocity subspace, which produce qualitatively different tra-
jectories. Particles that start at the same physical location,
but are in different regions of the relative velocity subspace,
are neatly separated from particles that started in other re-
gions, as shown in Fig. 3共c兲. Thus the ridges in the sensitivity
field have the property of a separatrix.
V. SEGREGATION OF PARTICLES BY STOKES NUMBER
Equation 共5兲can be diagonalized as
dWd
dt =
冉
−−
0
0−
冊
Wd,共29兲
where are the eigenvalues of the Jacobian of the fluid
velocity field. If
=2
3St−1 is very large, then both the com-
ponents of Adwill decay. For low values of
, one compo-
nent of Adwill grow. Therefore the dynamics of an inertial
particle depend on the value of
, that is, on the Stokes
number. It is reasonable to expect that the computations of
the sensitivity of the particle location to the initial relative
velocity also would depend on the Stokes number. That this
is indeed the case is shown by the computations of the sen-
sitivity field for a particle with Stokes number 0.1 for the
time-independent flow, as shown in Fig. 4共a兲. The thick lines
are the ridges in the sensitivity field for particles with St
=0.1 and the hatched lines are those of St=0.2. It can be seen
that, though the structure of the sensitivity field is similar, the
ridges are present at different locations in the relative veloc-
ity subspace. This fact can be exploited to design a process to
separate particles by their Stokes number. In this section we
illustrate a simple procedure for doing this.
The ridges of the sensitivity fields computed for the two
different particles of Stokes number 0.1 and 0.2, respectively,
are superimposed on the same plot, as shown in Fig. 4. The
subdomain of the relative velocity subspace sandwiched be-
tween the ridges of the sensitivity fields of the two types of
particles forms a zone of segregation. One such sample zone
is shown in gray in Fig. 4共a兲. Two particles, with St=0.1 and
0.2, respectively, with common initial coordinates 共x,y兲
=共3
/8,3
/8兲and initial relative velocities belonging to the
gray region, have trajectories that separate in physical space.
To illustrate this, the trajectories of 500 particles of each
Stokes number, starting at the same initial physical point
共x,y兲=共3
/8,3
/8兲and with initial relative velocities val-
ues belonging to the gray region were computed. Figures
4共b兲–4共j兲show snapshots of the particle positions as a func-
tion of time. The particles are completely segregated into two
(
a)
(b)
(
c)
FIG. 3. 共a兲Ridges in the sensitivity field partition the velocity
subspace into regions of distinct qualitative dynamics. Three such
partitioned regions are shown. 共b兲Particles starting with relative
initial velocities belonging to distinct partitions in the relative ve-
locity subspace are segregated into different cells in the xy plane.
Same parameters as in Fig. 2:a=100, St=0.2, T=0.24. The initial
position of all particles is 共x0,y0兲=共3
/8,3
/8兲, shown by the ⫻
marker in 共c兲.
PHANINDRA TALLAPRAGADA AND SHANE D. ROSS PHYSICAL REVIEW E 78, 036308 共2008兲
036308-6
(a) (b) (c)
(d)
(
e
)
(f)
(g) (h) (i)
(j)
FIG. 4. 共a兲Ridges in the sensitivity field for particles with St= 0.2 共hatched兲and St = 0.1 共thick兲. The initial position of all particles is
共x0,y0兲=共3
/8,3
/8兲. The gray patch is a sample region sandwiched between the ridges of the two Stokes numbers. 共b兲–共j兲A mixture of
St=0.1 and St= 0.2 particles starting at 共x0,y0兲=共3
/8,3
/8兲, with initial relative velocity in the gray patch in 共a兲, are separated into
different cells in the xy plane after a short time. t=共b兲0.005, 共c兲0.030, 共d兲0.060, 共e兲0.085, 共f兲0.110, 共g兲0.135, 共h兲0.160, 共i兲0.185, and
共j兲0.210.
PARTICLE SEGREGATION BY STOKES NUMBER FOR…PHYSICAL REVIEW E 78, 036308 共2008兲
036308-7
different cells after a short time 共t⬅0.060兲. Notice that the
particles separate as they approach a portion of the LCSf, the
boundary of the two cells. The above procedure can be ap-
plied to any other region sandwiched between the two types
of ridges and can be extended to more than two particle
sizes.
VI. ROBUSTNESS OF THE SENSITIVITY FIELD TO
PERTURBATIONS IN THE STREAM FUNCTION
The time-independent flow given by the stream function
in Eq. 共26兲is perturbed by making it weakly time dependent.
The modified fluid flow is given by the stream function 关15兴
共x,y,t兲=acos共x+bsin
t兲cos y.共30兲
The velocity field is given by
u=−acos共x+bsin
t兲sin y,共31兲
v=asin共x+bsin
t兲cos y.共32兲
For time-varying vector fields the location of the LCSf
and LCSs depends on the choice of initial time. For the com-
putation of the sensitivity field, the locations of ridges in the
relative velocity subspace depend on the initial spatial coor-
dinates of the particle as well as the initial time. However,
our computations show that the dependence of the ridge
structure on the initial time is weak. Figure 5shows the
ridges in the sensitivity field. As the initial time is increased,
it is seen that there is a “squeezing” of the sensitivity field in
some regions of the relative velocity subspace. A comparison
of Fig. 4with Fig. 3shows that the ridge locations in the
sensitivity field remain qualitatively the same for the three
cases in Fig. 5where the initial time is small. This offers
numerical evidence that the sensitivity field is robust to small
perturbations in the fluid velocity.
VII. CONCLUSION
The dynamics of inertial particles in a fluid flow can ex-
hibit sensitivity to initial conditions. We demonstrated that
ridges in the relative velocity sensitivity field at each spatial
point effectively partition phase space into zones of different
particle fates, i.e., inertial particles initially located on either
side of a ridge will evolve to different spatial locations after
a short time. The phase space location of these ridges de-
pends on the Stokes number, and by implication the size of
the inertial particles of interest. This dependence can be ex-
ploited to make particles of different sizes cluster in different
regions of the fluid and thus separate and segregate them.
We used this method to achieve segregation using a
simple test model of two-dimensional flow: cellular flow. By
“injecting” a mixture of inertial particles of different sizes
into the fluid at a common relative velocity range that is
sandwiched between the ridges of different Stokes number,
the particles are segregated by size in a short time. Though
we have based our results on only cellular flow, the method-
ology presented only requires that the underlying flow 共1兲
has a spatial partition, i.e., separatrices in the fluid itself and
共2兲is of low Reynolds number. These requirements ensure
(
a)
(
b)
(
c)
FIG. 5. 共a兲Ridges in the sensitivity field for the time-dependent
stream function
共x,y,t兲=acos共x+bsin
t兲cos yfor 共x0,y0兲
=共3
/8,3
/8兲. The hatched and thick lines are the ridges corre-
sponding to St=0.2 and 0.1, respectively. Parameters: a= 100, b
=0.25,
=1, T= 0.24. Initial times t0=共a兲0; 共b兲0.25; 共c兲0.5.
PHANINDRA TALLAPRAGADA AND SHANE D. ROSS PHYSICAL REVIEW E 78, 036308 共2008兲
036308-8
that segregated particles do not remix. The method does not
rely on any other flow characteristic or specific stream func-
tion. Extending this to turbulent flows may present difficul-
ties, since the ridges in the sensitivity field may not persist
for long enough times to achieve a clean separation of the
inertial particles. This aspect requires further investigation.
In future work, the approach employed here can be adapted
to segregate non-neutrally-buoyant particles, and to segre-
gate particles by other characteristics, e.g., density and
shape, with a goal of designing flows that can fractionally
separate particles for a range of inertial parameters.
关1兴G. Segre and A. Silberberg, Nature 共London兲189, 209 共1961兲.
关2兴M. Tirumkudulu, A. Tripathi, and A. Acrivos, Phys. Fluids 11,
507 共1999兲.
关3兴T. Shinbrot, M. M. Alvarez, J. M. Zalc, and F. J. Muzzio, Phys.
Rev. Lett. 86, 1207 共2001兲.
关4兴S. A. Isard, S. H. Gage, P. Comtois, and J. H. Russo, Bio-
Science 55, 851 共2005兲.
关5兴D. G. Schmale, A. S. Shah, and G. C. Bergstro, Phytopathol-
ogy 95, 472 共2006兲.
关6兴M. C. Facchini, M. Mircea, S. Fuzzi, and R. J. Charlson, Na-
ture 共London兲401, 257 共1999兲.
关7兴E. R. Abraham, Nature 共London兲391, 577 共1998兲.
关8兴T. Nishikawa, Z. Toroczkai, and C. Grebogi, Phys. Rev. Lett.
87, 038301 共2001兲.
关9兴A. Athanas and G. K. Johannes, Chem. Eng. Technol. 29,13
共2006兲.
关10兴L. R. Huang, C. E. Cox, R. H. Austin, and J. C. Sturm, Science
304, 987 共2004兲.
关11兴J. A. Davis, D. W. Inglis, K. J. Morton, D. A. Lawrence, L. R.
Huang, S. Y. Chou, J. C. Sturm, and R. H. Austin, Proc. Natl.
Acad. Sci. U.S.A. 103, 14779 共2006兲.
关12兴D. DiCarlo, D. Irimia, R. G. Tompkins, and M. Toner, Proc.
Natl. Acad. Sci. U.S.A. 104,48共2007兲.
关13兴M. R. Maxey and J. J. Riley, Phys. Fluids 26, 883 共1983兲.
关14兴M. R. Maxey, Phys. Fluids 30, 1915 共1987兲.
关15兴A. Babiano, J. H. E. Cartwright, O. Piro, and A. Provenzale,
Phys. Rev. Lett. 84, 5764 共2000兲.
关16兴I. J. Benczik, Z. Toroczkai, and T. Tel, Phys. Rev. Lett. 89,
164501 共2002兲.
关17兴G. Haller and T. Sapsis, Physica D 237, 573 共2008兲.
关18兴G. Haller and T. Sapsis, Phys. Fluids 20, 017102 共2008兲.
关19兴J. Bec, Phys. Fluids 15, L81 共2003兲.
关20兴E. Balkovsky, G. Falkovich, and A. Fouxon, Phys. Rev. Lett.
86, 2790 共2001兲.
关21兴K. Duncan, B. Mehlig, S. Ostlund, and M. Wilkinson, Phys.
Rev. Lett. 95, 240602 共2005兲.
关22兴J. Bec, Phys. Fluids 18, 091702 共2006兲.
关23兴I. J. Benczik, Z. Toroczkai, and T. Tel, Phys. Rev. E 67,
036303 共2003兲.
关24兴S. C. Shadden, F. Lekien, and J. Marsden, Physica D 212, 271
共2005兲.
关25兴F. Lekien, S. C. Shadden, and J. E. Marsden, J. Math. Phys.
48, 065404 共2007兲.
关26兴K. Ide, D. Small, and S. Wiggins, Nonlinear Processes Geo-
phys. 9, 237 共2002兲.
PARTICLE SEGREGATION BY STOKES NUMBER FOR…PHYSICAL REVIEW E 78, 036308 共2008兲
036308-9
