# Anisotropic Particles Focusing Effect in Complex Flows

Séverine Atis  Massachusetts Institute of Technology, Department of Mechanical
Engineering, Cambridge, Massachusetts 02139.  University of Chicago,
Department of Physics, Chicago, Illinois 60637.  Matthieu Leclair
Massachusetts Institute of Technology, Department of Mechanical Engineering,
Cambridge, Massachusetts 02139.  Themistoklis P. Sapsis  Massachusetts
Institute of Technology, Department of Mechanical Engineering, Cambridge,
Massachusetts 02139.  Thomas Peacock  Massachusetts Institute of Technology,
Department of Mechanical Engineering, Cambridge, Massachusetts 02139.

###### Abstract

Finite size effects can lead neutrally buoyant particles to exhibit different
dynamics than tracer particles, and can alter their transport properties in
fluid flows. Here we investigate the effect of the particle’s shape on their
dispersion in 2-dimensional complex flows. Combining numerical simulations
with laboratory experiments, we show that particles with isotropic and
anisotropic shapes exhibit different Lagrangian coherent structures, resulting
in distinct dispersion phenomena within a given flow field. Experiments with
rod shaped particles show a focusing effect in the vicinity of vortex cores.
We present a simple model that describes the dynamics of neutrally buoyant
ellipsoidal particles in two-dimensional flow and show that particle aspect
ratio and orientation dependent forces can generate clustering phenomena in
vortices.

##  I Introduction

The dispersion of a tracer in a fluid flow is influenced by the Lagrangian
motion of fluid elements. Even in laminar regimes, the irregular chaotic
behavior of a fluid flow can lead to effective stirring that rapidly
redistributes tracers throughout the domain. In this context, small finite-
size objects can behave considerably different from infinitesimal fluid
particles, and lead to altered clustering and dispersion phenomena. Ranging
from pollutant transport [1] and plankton population evolution on the surface
of the ocean [2], to interstellar dust dynamics [3, 4], quantifying the impact
of finite-size effects is therefor essential to describe material transport in
natural and industrial processes.

However, depending on their size and buoyancy properties, inertial particles
can exhibit a variety of dynamics, resulting in markedly different
trajectories [5, 6, 7, 8]. The shape of a particle can also affect its
response to fluid flow [9, 10, 11, 12]. For instance, in cloud formation the
anisotropic shapes of ice particles influence their transport [10, 13] and
aggregation [14, 15] in turbulent atmospheric flows. Fiber transport in
cellular flows, on the other hand, have been shown to exhibit hydrodynamic
dispersion phenomena controlled by the buckling properties of the material
[16, 17]. Under marine conditions, preferential alignment of elongated
microorganisms in the local flow direction can affect light backscattering and
change global carbon fixation rates within phytoplankton blooms [18]. In
recent studies, predictive models suggested that the irregular shape of
microplastic contaminants could also have an impact on their long term
transport properties on the ocean surface [19, 20, 21].

In this paper, we investigate the effect of anistropic shape on the particles
dynamics and long term behavior in chaotic flows. We experimentally show that
when advected in two-dimensional cellular flows for an extended period of
time, finite-size particles trajectories with an ellipsoidal shape start
diverging from the spherical ones. The ellipsoidal particles exhibit a strong
clustering in the vicinity of vortices with an aggregation rate that depends
on the particle’s aspect ratio. We present a simple model that combines
finite-size effects with an orientational-dependent dynamics, and perform in
parallel, numerical simulations under identical conditions to the experiments.
Our results suggest that neutrally buoyant ellipsoidal particles present
stronger inertial effects, and are attracted to the coherent regions of the
flow at a higher rate than spherical particles.

##  II Finite-size anisotropic particles dynamics

###  II.1 Isotropic inertial particles

Inertial effects arise when the advected particle and fluid flow accelerations
become different; namely, the particle is responding to changes in the
underlying flow field over a finite time and additional forces are generated.
For instance when the particle density is different from the carrying fluid
density, even for a small difference, these effects can accumulate over time
and eventually lead to a drastic deviation between the fluid flow and particle
trajectory. Experiments conducted in 3-dimensional turbulent flows [22, 7, 23,
24, 25], and in 2-dimensional chaotic flows [26, 6], have shown that the
particles behavior can be partly quantified by the particle’s Stokes number
$\mathrm{St}$:

$\mathrm{St}=\frac{2}{9}\frac{\rho_{p}}{\rho_{f}}\left(\frac{a}{L}\right)^{2}\mathrm{Re}\text{,}$
(1)

where $\rho_{p}$ is the particle density, $\rho_{f}$ the fluid density, $a$
the particle radius, $L$ the flow characteristic length scale, and
$\mathrm{Re}=UL/\nu$ is the flow Reynolds number with $\nu$ the fluid
kinematic viscosity. Assuming $a\ll L$, the dynamics of small but finite-size
inertial particles can be described with a simplified Maxey-Riley equation
[27], neglecting the Faxén correction and memory terms [28, 26, 5, 29, 30].
After rescaling space by $L$, and time by $T=L/U$, with $U$ the characteristic
flow velocity, the momentum equation becomes [29]

$\frac{d\mathbf{v}}{dt}=\frac{3R}{2}\frac{D\mathbf{u}}{Dt}-\frac{R}{\mathrm{St}}(\mathbf{v}-\mathbf{u})+\left(1-\frac{3R}{2}\right)\mathbf{g}\text{.}$
(2)

The particle’s velocity $\mathbf{v}$ depends on three forces in the right hand
side of the equation: the first term corresponds to advection and added mass,
where $\mathbf{u}$ is the flow velocity, the second term is the Stokes drag
force, and the third term is the buoyancy force, where the density ratio is
given by $R=2\rho_{f}/(\rho_{f}+2\rho_{p})$, and $\mathbf{g}=-g\
\mathbf{e_{z}}$ is gravity. Note that
$d\mathbf{.}/dt=\frac{\partial.}{\partial t}+(\mathbf{v}\cdot\nabla).$ and
$D\mathbf{.}/Dt=\frac{\partial.}{\partial t}+(\mathbf{u}\cdot\nabla).$ are
different here [26].

When the particles are neutrally buoyant, such that $\rho_{p}=\rho_{f}$,
$R=2/3$ and Equation (2) reduces to [29]

$\frac{d\mathbf{v}}{dt}=\frac{D\mathbf{u}}{Dt}-\frac{2}{3}\mathrm{St}^{-1}(\mathbf{v}-\mathbf{u})\text{.}$
(3)

This shows that particles with finite $\mathrm{St}$ number can still exhibit a
dynamics with a finite relaxation time due to the flow interactions with the
particle’s surface [26, 5, 6, 7]. Note that equation (3) converges toward the
fluid elements motion when $\mathrm{St}\rightarrow 0$ for infinitesimal
particles.

###  II.2 Anisotropic particles angular dynamics

In comparison with spherical particles, anisotropic particles can exhibit
additional torques and forces due to the coupling between their rotational and
translational behavior. The motion of small, neutrally buoyant ellipsoidal
particles in laminar flow has been first studied by G. B. Jeffery in 1922
[31]. His theoretical work on the torque experienced by non-inertial
ellipsoidal particles combined with more recent numerical simulations with
orientation dependent relaxation times [9, 32, 21], suggest that the dynamics
of even very small objects, i. e. $\mathrm{St}\ll 1$, could be affected by
their anisotropic shape. Experiments in turbulent flow, have also indicated
that the turbulent kinetic energy of a suspension is altered accordingly with
the suspended particles’ shape [33]. Finally, numerical models with small
inertial effects coupled with orientation dependent forces, showed that the
transport of anisotropic particles under a wave can also lead to distinct
behaviors over long period of times [21].

Neutrally buoyant ellipsoidal particles experience a torque that tends to
align their principal axis parallel to the surrounding fluid principal axis of
distortion. For non-inertial particles, the particle’s velocity is identical
to the surrounding fluid $\mathbf{v}=\mathbf{u}$ at all time. In this
situation, there is no resultant force and the torque acting on the particle
vanishes at first order [31]. The particle angular velocity $\dot{\theta}$ is
then simply expressed as a function of the flow vorticity $\omega$ and strain-
rate: [34, 35, 30]

$\dot{\theta}=\frac{\omega}{2}+\frac{1}{2}\left(\frac{1-\mathrm{\gamma}^{2}}{1+\mathrm{\gamma}^{2}}\right)\left[\sin
2\theta\left(\frac{\partial u_{x}}{\partial x}-\frac{\partial u_{y}}{\partial
y}\right)-\cos 2\theta\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial
u_{x}}{\partial y}\right)\right]$ (4)

where $\mathrm{\gamma}=a_{2}/a_{1}$ is the particle aspect ratio, with $a_{1}$
and $a_{2}$ the semi-minor and semi-major axes respectively, and $u_{x}$ and
$u_{y}$ are the horizontal and vertical components of the flow velocity in the
fixed frame. The particle’s orientational dynamics is described using 3
coordinate systems: a fixed frame $\mathbf{x}=(x,y,z)$, a particle frame
$\hat{\mathbf{x}}=(\hat{x},\hat{y},\hat{z})$, and a co-moving frame
$\hat{\hat{\mathbf{x}}}=(\hat{\hat{x}},\hat{\hat{y}},\hat{\hat{z}})$, see
supplemental Fig. C.1 for a schematic [30]. In this configuration, $\theta$
corresponds to the angle between the particle’s major axis and $\mathbf{x}$ or
$\mathbf{\hat{\hat{x}}}$.

###  II.3 Anisotropic inertial particles

To describe the combined effects of finite-size and anisotropic shape with a
minimal model, we present here an equation of motion that couples the
particle’s orientation with the drag force only. In order to consider a
finite-size particle, we need to take into account inertial forces that will
tend to separate the particle’s trajectory from the fluid trajectory, such
that now $\mathbf{v}\neq\mathbf{u}$. The dynamics of anisotropic particles is
not well approximated by equation (3) anymore, and need to be modified to
account for the orientation specific dynamics [36, 37]. For instance, the
added mass and drag force terms can become dependent to the particle’s
instantaneous orientation [32, 9, 21]. Here, we use a linear momentum equation
with a the hydrodynamic drag force term coupled with the ellipsoidal
particle’s orientation [38, 39, 40, 41]

$\mathbf{F}=-6\pi
a_{s}\mu\;\mathbf{\hat{\hat{K}}}(\mathbf{v}-\mathbf{u})\text{,}$ (5)

where $a_{s}=a_{1}\mathrm{\gamma}^{1/3}$ is the radius of the equivalent
sphere with the same volume as the ellipsoid, $\mu$ is the dynamic viscosity
of the fluid and $\mathbf{\hat{\hat{K}}}$ represents the resistance tensor of
an ellipsoidal particle in the co-moving frame [40, 9]; see Supplemental
Material for more details [30]. The resistance tensor $\mathbf{\hat{K}}$ is
initially determined in the particle frame, and relates to
$\mathbf{\hat{\hat{K}}}$ in the co-moving frame through the particle’s
rotation matrix $\mathbf{A}$, such that

$\mathbf{\hat{\hat{K}}}=\mathbf{A^{-1}}\mathbf{\hat{K}}\mathbf{A}\text{.}$ (6)

In the case of a prolate ellipsoidal particle, e.g. invariant by rotation
along its major axis, the resistance tensor reduces to a diagonal matrix, and
for a 2-dimensional flow, we have

$\displaystyle\mathbf{\hat{K}}=\frac{8}{3}\mathrm{\gamma}^{-1/3}\begin{pmatrix}\frac{\mathrm{\gamma}^{2}-1}{\frac{2(2\mathrm{\gamma}^{2}-1)}{\sqrt{\mathrm{\gamma}^{2}-1}}\ln(\mathrm{\gamma}+\sqrt{\mathrm{\gamma}^{2}-1})-\mathrm{\gamma}}&0\\\
0&\frac{\mathrm{\gamma}^{2}-1}{\frac{2\mathrm{\gamma}^{2}-3}{\sqrt{\mathrm{\gamma}^{2}-1}}\ln(\mathrm{\gamma}+\sqrt{\mathrm{\gamma}^{2}-1})+\mathrm{\gamma}}\end{pmatrix}$

Substituting the drag force term applying to finite-size spheres in equation
(3), and noting
$\mathbf{\hat{\hat{K}}}=8/3\mathrm{\gamma}^{-1/3}\mathbf{\hat{\hat{k}}}$, we
obtain

$\frac{d\mathbf{v}}{dt}=\frac{D\mathbf{u}}{Dt}-\frac{16}{9}(\mathrm{\gamma}\mathrm{St})^{-1}\;\mathbf{\hat{\hat{k}}}(\mathbf{v}-\mathbf{u})\;\text{,}$
(7)

such that now, the equation of motion becomes coupled to the orientation of
the particle given by equation (4). The drag force term’s prefactor is
proportional to $(\mathrm{\gamma}\mathrm{St})^{-1}$, indicating that the
relaxation time of the particle will now also depend on its aspect ratio; i.e.
the more elongated the particle the longer the response time. Similarly to
Eq.(3), drag forces can introduce perturbations and also deviate elongated
particles from fluid particle trajectories. Note that although the particles
are passive, these perturbations allow the particles to cross flow streamlines
and explore larger portions of the flow field, akin to active self-propelled
particles [26, 42, 43, 44, 43]. However the dynamics exhibited by motile and
inertial particles differ as they experience distinct forces. While prolate
particles that can swim possess an intrinsic velocity aligned with the
particle’s long axis at all times, passive particles on the other hand exhibit
a distinct velocity from the fluid only when inertial forces are non
negligible. These forces depend on both the particle’s orientation and the
local flow geometry, such that, as opposed to motile particles with a fixed
velocity, inertial particles intrinsic velocity can vary over time and point
in different directions relative to the particle’s major axis.

##  III Materials and Methods

Figure 1: (a) Experimental setup of the rotor-flow. One rotor attached to a
traverse are respectively ensuring the rotation and translation of the rod. A
mirror at 45 degrees is reflecting the recorded image of the interface from
the bottom of the tank. The particles are positioned at a fixed height in a
glycerol and salt density stratification. (b) Experimental flow velocity field
determined from PIV measurements in the steady regime with a rotor speed at
$200$ rpm and a fixed transversal position. The color code corresponds to the
norm of the flow velocity. (c) Corresponding FTLE field computed for a time
window $t=[0;60]$ $\tau$ where $\tau=0.3$ s is the rotor period.

The experimental setup consists in a rectangular container of dimensions
$L\times W\times h=100\times 400\times 200$ mm. To circumvent any surface
tension effect [45], the particles are trapped at the interface between two
miscible fluids of different density. The tank is filled with two stratified
layers of pure glycerol with density $\rho_{1}=1.26\pm 0.001\text{ g.cm}^{-3}$
and a solution of glycerol and salt with $\rho_{2}=1.30\pm 0.01\text{
g.cm}^{-3}$, as represented in Fig. 1(a). To prevent viscous instabilities at
the interface, the viscosity of the layers are matched by adding $4\%$ water
to the glycerol and salt solution. The particles have a density
$\rho_{p}=1.28\pm 0.005$ and are stabilized by buoyancy forces on a
quasi-2-dimensional plane, see Supplemental Material for pictures [30]. The
bottom of the tank is filled with a 1 cm layer of low viscosity fluorinated
oil to reduce friction on the lower wall, and a layer of vegetable oil is
added on top to prevent water absorption by the glycerol over time. The
particles motion is recorded with a 45 degree tilted mirror at the bottom the
setup, and particle tracking velocimetry (PTV) is performed from digitized
images with a 14 bit LaVision camera.

We use 3D-printing ABS filament of $1.75$ mm diameter with variable length to
create anisotropic particles of the desired aspect ratio. To minimize
particle-particle interactions, a small number of particles (32) is initially
uniformly deposited and spread across the domain. For each particle’s aspect
ratio, five independent realizations were prepared manually by randomly
redistributing the initial positions of the particles at the
glycerol/glycerol-and-salt interface. Because of the stable stratification
configuration and the high viscosity of the glycerol, the interface remained
well define for several consecutive realizations of the experiment. The
stratification is then renewed by changing the fluid mixture in the tank, and
different particles are deposited at the interface.

The flow is created by stirring the fluid with a rotor attached to a
longitudinal translation stage; see [46] for a similar experimental system
description. The rotation is controlled with a Lexium NEMA 34 motion control
motor with a $2$ cm diameter Plexiglas rod, and the translation of the rotor
with a Parker lead-screw traverse attached on top of the tank. The resulting
flow is quasi-2-dimensional in the stratified fluid layer, and similar to the
rotor-oscillator flow topology described by Hackborn [47, 48, 30, 49]. In the
steady regime, when the position of the rotor is fixed, the rotation of the
rod close to the wall creates a hyperbolic stagnation point in the center and
two Moffatt eddies [50] on each side of the rotor. Particle image velocimetry
(PIV) is performed by seeding the upper glycerol layer with $30$ $\mu$m
diameter hollow glass beads. A YAG laser $532$ nm is used to illuminate a
sheet in the horizontal plane and visualize the flow near the plane containing
the inertial particles. A typical flow velocity snapshot is displayed in Fig.
1(b). The inner region in the direct proximity of the rotor exhibits the
highest velocities $\|u\|\gtrsim 10\text{ mm.s}^{-1}$, and rapidly decays
within a rod radius away from the rotating boundary. The two co-rotating
Moffatt eddies are also visible, with an average flow speed in the outer
region of $u\simeq 4\pm 1\text{ mm.s}^{-1}$ in their vicinity.

We perform in parallel numerical simulations solving the governing equations
(3), (4) and (7) under identical conditions with the experiments. The size and
the aspect ratio of the particles are set by choosing identical Stokes number
and $\mathrm{\gamma}$ with the experimental particles. The particles initial
positions are on a grid that matches our PIV field coverage, the initial
orientation of the anisotropic particles $\theta_{0}$ are randomly chosen and
the initial velocity is set slightly smaller than the fluid velocity at the
particle’s location $\mathbf{v}(t=0)=0.9\mathbf{u}$.

##  IV Numerical results

###  IV.1 Focusing effect in the steady double gyre flow

Figure 2: Numerical particles positions in the steady regime for $t>500$ in
the steady vortex flow with $A=100$ and $\omega=2\pi/10$, for particles with
$\mathrm{St}=5\ 10^{-4}$ and initially at rest, (a) spheres and (b) ellipsoids
with $\mathrm{\gamma}=5$. Corresponding radial component of the drag force
experienced by the particles in the transient regime. Each of the color plots
are normalized by the respective drag force prefactor $\mathrm{St}^{-1}$ and
$(\gamma\mathrm{St})^{-1}$ in Eq. (3) and Eq. (7), (c) spheres and (d)
ellipsoids with $\mathrm{\gamma}=5$. (e) Particle velocity $v_{x}$ and fluid
velocity $u_{x}$ difference versus time at the particle’s position for a
spherical (orange) and ellipsoidal (purple) particle selected at the same
initial position and with $\mathrm{St}=10^{-2}$. (f) Particles spatial
distribution around the vortex center in the steady regime for 3 different St
number, for spherical particles (orange circles), and ellipsoids with
$\mathrm{\gamma}=5$ (purple triangles). The radial density profile is
azimuthally averaged and normalized by the total number of particles $N_{0}$.
(g) Radial position of density profile maximum $r_{p}$ as a function of St.

We first numerically investigate the effect of particle size and shape on
their trajectories over a large range of Stokes number
$10^{-4}\leqslant\mathrm{St}\leqslant 10^{-1}$, and aspect ratio
$1\leqslant\mathrm{\gamma}\leqslant 10$. We consider a double gyre flow with
periodic boundary conditions defined by the streamfunction

$\psi(x,y,t)=A\sin[(\pi f(x,t)]\sin(\pi y)\text{,}$ (8)

where $A$ sets the flow amplitude, $f(x,t)=x+B(x^{2}-2)\sin\omega t$, and $B$
sets the oscillation amplitude. In the steady regime $B=0$, and the
streamfunction reduces to $\psi(x,y)=A\sin(\pi x)\sin(\pi y)$, leading to a
simple periodic array of counter-rotating vortices.

The resulting spatial distributions of neutrally buoyant particles with
$\mathrm{St}=5\ 10^{-4}$ and different shapes around a single vortex are shown
in Figs. 2(a) and 2(b). The streamfunction amplitude is set to $A=100$, such
that all the particles converge rapidly to a steady spatial distribution after
a short transient regime and ellipsoidal particles orientation aligns with the
local flow streamlines. Figures 2(c) and 2(d) show the corresponding radial
component of the drag forces during the early transient dynamics. For both
types of particles, a negative radial component is displayed near the center
of the vortex, and indicates attractive forces toward this region of the flow.
Drag forces are systematically larger for the ellipsoidal particles, leading
to stronger deviations in average than for spherical particles. As can be seen
in Fig. 2(e), even for the largest simulated Stokes number
$\mathrm{St}=10^{-2}$, spherical particles velocity rapidly relaxes toward the
fluid flow velocity while elongated particles velocity doesn’t converge to the
flow velocity. Anisotropic particles systematically present a stronger
focusing effect, converging toward smaller orbit radii than the spherical
particles at equal $\mathrm{St}$ number. To illustrate this effect, we repeat
the same simulation and vary the St number by two order of magnitude. Figure
2(f) shows the resulting radial distribution of particles in the steady
regime. As the St number increases, the particles spatial distribution becomes
more peaked and moves closer to the vortex center. As can be seen in Fig.
2(g), the distance of the particle density peak to the vortex center $r_{p}$
decreases faster for ellipsoidal particles as a function of the St number
before becoming similar for both types of particles when $\mathrm{St}\gtrsim
3\ 10^{-3}$.

Even at small $\mathrm{St}$ number $\mathrm{St}\approx 10^{-4}$, when inertial
effects are expected to be negligible for spherical particles, anisotropic
particles exhibit a more complex dynamics with a stronger attraction to vortex
cores.

###  IV.2 Particles dispersion in time-varying flows

Figure 3: Inertial particles positions and corresponding backward FTLE fields
in simulations for spherical particles, left column, and ellipsoidal particles
with $\mathrm{\gamma}=5$, right column, after $t=20$ time steps in a double
gyre flow with $A=100$, $e=0.25$, and $\omega=2\pi A$. (a)-(d)
$\mathrm{St}=10^{-3}$ and particles initial velocity
$\mathbf{v}_{0}=0.9\mathbf{u}$, (e)-(h) $\mathrm{St}=5\ 10^{-4}$ and
$\mathbf{v}_{0}=0$.

We now use the full time-dependent double gyre flow, and set $e=0.25$. In time
varying flows, attractive and repulsive structures are only defined over a
finite-time period. In this context, Lagrangian coherent structures (LCS)
provide a useful framework to understand the transport of material in complex,
unsteady flow fields [1]. To identify these regions, we first compute the
finite time Lyapunov exponent (FTLE) field corresponding to our flow and the
inertial particles trajectories for a given time period [8, 30, 51, 52].
Regions with a low FTLE value are indicative of attractive properties; i.e.
initially neighboring particles remain close to each other over a given time
window, while regions with a high value correspond to repulsive regions of the
flow, where the spatial separation over the same time window between
neighboring fluid particles is maximum [53].

Figure 3 displays spatial distributions and corresponding FTLE fields for
inertial particles with two different initial velocity and $\mathrm{St}$
number. In all configurations, after $t=2/A$, where $A$ is the gyre rotation
frequency, the ellipsoidal particles distribution systematically present a
more inhomogeneous spacial density variation than the spherical ones. At
$\mathrm{St}=5\ 10^{-4}$, the corresponding spherical particles FTLE field is
similar to the flow FTLE field, see Supplemental Material [30]. The
ellipsoidal particles FTLE fields are altered and exhibit additional
structures located in regions with stronger shear. This suggests that in time
varying flows, ellipsoidal particles more complex dynamics can lead to a
departure from passive transport even at very small St numbers.

##  V Experimental results

###  V.1 Focusing effect in the rotor-flow

To test our findings, we perform a series of experiments and compare the
resulting trajectories with our model by computing the numerical particles
dynamics in the same, experimentally measured, flow field. The flow has the
geometry of a rotor-flow in the steady regime, and a snapshot of the
experimental velocity field with the corresponding FTLE field is shown in
Figs. 1(b) and (c). We impose a constant rotor speed of $200$ rpm for all
three different particle aspect ratio $\mathrm{\gamma}=$ 1, 2 and 5. In this
configuration, the Reynolds number typically ranges between
$0.1\leq\mathrm{Re}\leq 10$ in the flow central region, away from the direct
vicinity of the rotor indicated by the gray disk in Figs. 1 and 4. This leads
to relatively small particle Stokes number $\mathrm{St}\approx
10^{-4}-10^{-3}$, such that, in the absence of anisotropic effects, we expect
the particles with $\mathrm{\gamma}=1$ to behave closer to tracer particles in
our experiments.

The particles are initially distributed uniformly across the experimental
domain with no preferential alignment across the quasi-2-dimensional plane
defined by the two-layer stratification, as described in the Method section
[30]. The particles remain in the same plane for the entire duration of the
experiments, and the principal axis of the elongated particles doesn’t display
off-plane motions except near the rotor wall. Immediately after the flow is
started, particles with $\mathrm{\gamma}>1$ change their orientation and
exhibit a preferential alignment of their principal axis with the local flow
direction, as shown in the supplemental movie 1 [30]. This suggests that the
orientational behavior has a negligible response time to the changes in the
flow field, in good agreement with equation (4), and previously reported
findings for flows at higher Reynolds number [35, 10].

Figure 4: (a)-(c) experimental particles final positions at $t=12000$ $\tau$,
(d)-(f) particles positions over a time window $t=[5000;9200]$ $\tau$ in the
corresponding experiments, (g)-(i) numerical particles positions advected in
the experimental flow field shown in Fig. 1(b) for the entire duration of the
simulation equivalent to $t=6700$ $\tau$. Spherical (orange) and ellipsoidal
(purple) particles, color intensity indicates time.

After a time $t\gtrsim 4000$ $\tau$, where $\tau=0.3$ s corresponds to the
rotor rotation time period, the particles spatial distribution approaches a
stationary regime which strongly depends on their aspect ratio, as shown in
Figs. 4(a)-(c) and the supplemental movies 2-4 [30]. The final position of the
particles with aspect ratio $\mathrm{\gamma}=1$ do not exhibit any
preferential concentration, and remain uniformly spread across the domain,
while particles with $\mathrm{\gamma}=2$ and $\mathrm{\gamma}=5$ display a
migration from the outer regions of the flow into the vortex cores and their
vicinity. Remarkably, for $\mathrm{\gamma}=5$, all the particles are focused
in the central region. Note that for $\mathrm{\gamma}=2$ and
$\mathrm{\gamma}=5$, some particles can be attracted to the secondary vortices
located near the lateral edges of the tank not covered in our field of views
in Figs. 1 and 4.

In order to quantify this behavior, we determine the individual particles
trajectories using particle tracking velocimetry (PTV), and monitor their
spatial distribution over time. Figures 4(d)-(f) display all the particles
positions as a function of time for the entire duration of the experiment
shown in Figs 4(a)-(c) respectively. As can be seen in Fig. 4(e) and 4(f),
some of the elongated particles’ final trajectories form closed orbits inside
the vortices. While for $\mathrm{\gamma}=2$, the particles remain more evenly
distributed outside the vortices, particles with $\mathrm{\gamma}=5$ exhibit a
strong concentration in the direct vicinity of the vortices.

Figure 5 displays the measured particle density variation between $0\leq t\leq
12000$ $\tau$ in the central region of the flow delimited by the white dashed
lines in Fig. 4(c). During a first transient regime, the spatial density of
the elongated particles increases linearly over time and exhibit an aspect
ratio dependent migration rate, with the largest rate for $\mathrm{\gamma}=5$.
After $t=t^{*}$, the migration slows down and the density of particles in the
central region of the flow converges toward a constant density over time.
Interestingly, while both the migration rate and the final particle
concentration depend on $\gamma$, the saturation time scale $t^{*}\simeq 4000$
$\tau$ seems similar for particles with aspect ratio $\mathrm{\gamma}>1$.
Particles with $\mathrm{\gamma}=1$ present a negligible focusing effect, and
their spatial distribution remains nearly unchanged over time.

###  V.2 Comparison with the numerical model

We perform in parallel numerical simulations using the experimentally measured
flow field shown in Fig. 1(b). The particles $\mathrm{St}$ number and aspect
ratio are now chosen identical to the experimental particles. The flow field
is first nondimensionalized as described in Supplemental Material [30]. We set
$x^{*}=x/L$ with $L=100$ mm corresponding to the width of the tank, and
$u^{*}=u/U$, where $U=L/T$ is the characteristic flow velocity. To estimate
$U$ from the experiments, we select the velocity that gives the best match
between the numerical and experimental particles velocities located in
identical regions of the flow. The best agreement with the experimental
particles velocities is obtained for $U=5\pm 2$ mm.s-1, in good agreement with
the largest flow structures average velocity in Fig. 1(b).

Figure 5: Spatial particle density variation as a function of time in the
central region of the flow indicated by the white dashed lines in Fig. 4c).
Lines with shaded region, experimental average value and corresponding error.
Circles, numerical average value with $\mathrm{St}=5\ 10^{-4}$ in the
experimentally measured flow field. Blue : $\mathrm{\gamma}=1$, orange:
$\mathrm{\gamma}=2$, and green: $\mathrm{\gamma}=5$. Dashed lines, linear fit
of the experimental data in the transient regime for $t<t^{*}$.

The resulting numerical trajectories are shown in Figures 4(g)-(i), and
qualitatively agree with the experimental trajectories spatial features. They
display a similar convergence toward the vicinity of the vortices for
$\mathrm{\gamma}>1$, although a larger asymmetry between the left and right
attractive regions within the vortices is visible, and only the right vortex
traps the most elongated particles on a distinct inner orbit. The migration
rate of the numerical particles are shown in Fig. 5, and display a similar
trend with the experimental data in the transient regime. After $t\gtrsim
3000$ $\tau$, the predicted aggregation rates start differing from the
experimental curves and lead to a less contrasted particle spatial
distribution as a function of $\mathrm{\gamma}$.

Comparing the shape of high particle concentration regions in our experiments
and simulations with the rotor-flow FTLE field shown in Fig. 1(c), we can
identify which flow structures attract the most the elongated particles. The
flow FTLE field displays several ridges: two minima within the vortex cores
corresponding to the two ellipsoidal regions where elongated particles are
trapped in the experiments, and a third structure formed by the minima
encircling the two vortices. Both experimental and numerical particles with
the largest aspect ratio $\mathrm{\gamma}=5$ are specifically attracted to
this third region and display the large closed orbit visible in Figs. 4(f) and
4(i).

##  VI Discussion

In our experiments, the observed migration toward vortices for elongated
particles with a small Stokes number $10^{-4}\lesssim\mathrm{St}\lesssim
10^{-3}$, suggests that even weak deviations from passive trajectories can
build up over time, leading to distinct spatial organization over long time
periods. These perturbations are not sufficient to significantly alter
particles trajectories with an aspect ratio $\mathrm{\gamma}=1$, and particles
with $\mathrm{\gamma}=2$ exhibit a considerably weaker focusing effect than
those with $\mathrm{\gamma}=5$. A direct comparison of these observations with
our numerical simulations shows that our minimal model qualitatively captures
this behavior, although the particles focusing effect exhibits a smaller
dependence to the particle aspect ratio $\mathrm{\gamma}$ than in the
experiments. This difference could be attributed to second order orientation
dependent terms, such as the added mass [36, 37, 32, 9, 21], that were
neglected here.

These results suggest that anisotropic particles respond to the flow with a
larger lag. In addition to the orientation dependent nature of the drag forces
in equation (7), $\mathrm{St}^{\prime}=\mathrm{\gamma}\mathrm{St}$ could be
interpreted as an “effective” Stokes number that is proportional to
$\mathrm{\gamma}$. For weak inertial effects $\mathrm{St}\ll 1$, the
difference between $\mathrm{St}$ and $\mathrm{St}^{\prime}$ on the particles
trajectories becomes only visible at large time scales, when small deviations
have accumulated over longer time periods.

This effect could potentially enhance the migration rate of irregularly shaped
objects such as microplastic in the ocean toward gyres of marine debris [20,
21], and influence the garbage patches formation dynamics [19, 54]. On the
opposite length and time scales, in biology this interplay between the shape
of the particle and the flow field geometry could influence the selective
capture of microorganisms by larger animals. For instance, flows created by
the beating of cilia can generate a local flow that attract only
microorganisms of an optimal size [55]. Identical mechanisms could potentially
select for an optimum shape, such as the aspect ratio of a given bacteria.

In conclusion, even neutrally buoyant particles with small particle Stokes
number $\mathrm{St}\simeq 10^{-4}$ exhibit trajectories that deviate from the
fluid flow when they possess an anisotropic shape. Both experimental and
numerical particles exhibit a migration toward regions in the vicinity of
vortices. The orientational dependence of the drag force term in our minimal
model describes qualitatively the particles dynamics, suggesting that this
mechanisms could be the dominant effect for the separation of isotropic and
anisotropic particle trajectories in cellular flows. In addition, the effect
of additional orientation dependent terms and experiments with unsteady flows
should be investigated in future work.

##  VII Acknowledgments

The authors would like to thank A.W. Murray and E. Climent for stimulating
discussions, G. Haller, T. Witten and E. Kanso for their thought provoking
comments. S. A. was supported by the French government DGA/DS - Mission pour
la Recherche et l’Innovation Scientifique fellowship during her visit at MIT.
T. P. acknowledges funding from ONR Grant N000141812762.

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