# Enhancing Transport Barriers with Swimming Microorganisms in Chaotic Flows

Ranjiangshang Ran1,2  Paulo E. Arratia1<EMAIL_ADDRESS>1 Department
of Mechanical Engineering and Applied Mechanics, University of Pennsylvania,
Philadelphia, PA 19104, USA 2 Department of Physics, Emory University,
Atlanta, GA 30322, USA

###### Abstract

We investigate the effects of bacterial activity on the mixing and transport
properties of two-dimensional, time-periodic flows in experiments and in a
simple model. We focus on the interactions between swimming E. coli and the
flow Lagrangian Coherent Structure (LCS), which are computed from
experimentally measured velocity fields. Experiments show that such
interactions are non-trivial and can lead to transport barriers through which
the tracer flux is significantly reduced. Using the Poincaré map, we show that
these transport barriers coincide with the outermost members of elliptic LCSs
known as Lagrangian vortex boundaries. Numerical simulations further show that
elliptic LCSs can repel elongated swimmers and lead to swimmer depletion
within Lagrangian coherent vortices. A simple mechanism shows that such
depletion is due to the preferential alignment of elongated swimmers with the
tangents of elliptic LCSs. Our results provide insights into understanding the
transport of microorganisms in complex flows with dynamical topological
features from a Lagrangian viewpoint.

###### keywords:

##  1 Introduction

Many microorganisms live in environments characterized by currents (e.g.,
oceans, rivers, human intestines), and flow mediates many important microbial
processes such as infection (Costerton et al., 1999; Yawata et al., 2016),
nutrients uptake (Taylor & Stocker, 2012; Musielak et al., 2009), and
reproduction (Riffell & Zimmer, 2007; Zimmer & Riffell, 2011). Flow exerts
forces and torques on microorganisms that affect their motility dynamics and
spatial distribution (Guasto et al., 2012; Wheeler et al., 2019). It also
controls the transport of essential molecules including nutrients, oxygen, and
signals of mates and predators (Kim et al., 2016). Microorganisms, in turn,
can adapt their swimming behavior to these physical and chemical gradients
(Stocker et al., 2008). Such interactions can lead to non-trivial phenomena
such as rheotaxis (Hill et al., 2007; Marcos et al., 2012; Mathijssen et al.,
2019), gyrotaxis (De Lillo et al., 2014; Borgnino et al., 2018), and
chemotaxis (Locsei & Pedley, 2009; Stocker & Seymour, 2012), to name a few
examples.

Even simple shear flows, when coupled to cell motility and morphology, can
give rise to complex transport and cell motility behavior (Rusconi et al.,
2014; Durham et al., 2009; Ezhilan & Saintillan, 2015; Gustavsson et al.,
2016). For instance, fluid shear can cause a torque that constantly rotates
elongated cell body and can result in accumulation of motile bacteria and
phytoplankton in high-shear-rate regions of the flow (Rusconi et al., 2014;
Barry et al., 2015; Ezhilan & Saintillan, 2015). Near surfaces, shear flows
can orient flagellated cells against the flow, causing bacteria and
spermatozoa to swimming upstream (Tung et al., 2015; Zaferani et al., 2018;
Mathijssen et al., 2019). Away from surfaces, shear gradients can trap bottom-
heavy gyrotactic swimmers at certain depth of water column, causing the
formation of intense cell assemblages called “thin layers” (Durham et al.,
2009, 2013; Gustavsson et al., 2016). In unsteady and/or complex flows,
transport of microorganisms shows intriguing phenomena such as aggregation,
dispersion, and pattern formation (Torney & Neufeld, 2007; Khurana et al.,
2011; Khurana & Ouellette, 2012; Zhan et al., 2014; Qin & Arratia, 2022), but
are less understood. In time-periodic flows, simulations show that elongated
swimmers can be trapped or repelled by elliptic islands depending on their
shape and swimming speed (Torney & Neufeld, 2007); such trapping effects can
lead to a reduction in long-term swimmer transport (Khurana et al., 2011).
Recent experiments and simulations show that microswimmers can be trapped,
repelled, or dispersed by vortices depending on the dimensionless path length
and swimming speed (Qin & Arratia, 2022). In isotropic turbulence, simulations
show that elongated swimmers, while remaining rather uniformly distributed,
exhibit preferential alignment with instantaneous Eulerian fields such as
local velocity (Borgnino et al., 2019), vorticity (Zhan et al., 2014) and
velocity gradient (Pujara et al., 2018).

Recently, it has been shown that Lagrangian Coherent Structure (LCS) can be a
useful concept to understand the transport properties of swimming
microorganisms in complex flows in both numerical simulations and experiments
(Khurana & Ouellette, 2012; Dehkharghani et al., 2019; Ran et al., 2021; Si &
Fang, 2021, 2022; Yoest et al., 2022). Simulations in chaotic flows show that
elongated swimmers align with repelling LCSs of hyperbolic fixed points
(Khurana & Ouellette, 2012), while later numerical studies show that elongated
active particles have a much stronger alignment with attracting LCSs (Si &
Fang, 2021, 2022), similar to passive elongated particles (Parsa et al.,
2011). Experimental investigations that examine the interactions of swimming
organisms and flow LCSs are few but show some intriguing phenomena.
Experiments in model porous media show that bacteria align and accumulate near
attracting LCSs and induce filamentous density patters (Dehkharghani et al.,
2019). In time-periodic flows, experiments show that the accumulation of
bacteria near attracting LCSs can attenuate stretching and hinder large-scale
transport, although small-scale mixing is locally enhanced (Ran et al., 2021).
Most, if not all, previous studies focus on attracting and repelling LCSs
associated with the flow hyperbolic fixed points. That is not surprising since
one expects large levels of strain near or around those fixed points. Less
understood are swimmer interactions with elliptic LCSs, i.e., vortex-like flow
dynamical features (Haller, 2015; Haller et al., 2016; Farazmand & Haller,
2016). That is the focus of this manuscript.

Here, we experimentally investigate the effects of bacterial activity on the
mixing and transport properties of a passive scalar in a time-periodic flow in
experiments and simulations. We focus on the interaction of swimming bacteria
(E. coli) with the elliptic LCSs of the flow. Results show that such
interaction leads to transport barriers through which the fluxes of the
passive tracer are significantly reduced. By constructing the Poincaré map
from velocimetry data, we show that these transport barriers coincide with the
outermost member of elliptic LCSs, or namely, Lagrangian vortex boundaries
(LVBs). We further test these results in numerical simulations and find that
elliptic LCSs repel elongated swimmers and lead to swimmer accumulation
outside (or swimmer depletion inside) the Lagrangian vortices. A simple
mechanism shows that the repulsion of swimmers is due to the preferential
alignment of elongated swimmers with the tangents of elliptic LCSs. Overall,
these results can be useful in understanding the transport of microorganisms
in chaotic flows with elliptic dynamical features or nontrivial vortex
structures.

Figure 1: Photographs of the dye concentration field for chaotic mixing in (a)
buffer solution ($\phi_{b}=0$) and (b) active suspension ($\phi_{b}=0.5\%$).
Images are taken at $N=300$ periods after the start of the experiments. The
imaged region is $60\times 60$ mm2; the scale bar represents 6 mm. The
Reynolds number and path length of the flow are $Re=7.2$ and $p=1.0$,
respectively. (c) Vorticity (color code) and velocity (arrows) fields of the
flow, corresponding the the first peak of a time period. (d) Poincaré map of
the flow in the same imaged region as the dye field. The map is colored by the
trajectory rotation average (TRA) calculated over 20 time periods.

##  2 Methods

Experiments are performed in the flow cell setup where a thin conductive fluid
layer is placed above an array of permanent magnets (Voth et al., 2002; Ran et
al., 2021, 2022). As a sinusoidal forcing (electrical current, 0.2 Hz
frequency) is imposed on the fluid layer, the magnetic field induces Lorentz
force and creates spatially disordered vortex patterns [see flow field in Fig.
1(c) and Supplementary Materials]. The resulting flow is characterized by two
parameters: the Reynolds number and the path-length. The Reynolds number is
defined as $Re=U\\!L/\nu$, where $L=6.0$ mm and $U=1.2$ mm/s are the
characteristic length and the average flow speed, and $\nu=1.0$ mm2/s is the
fluid kinematic viscosity (water-like). The path-length is the normalized mean
displacement of a typical fluid parcel in one period of forcing, defined as
$p=UT/L$, where $T=5$ s is the forcing period. Here, the Reynolds number and
path-length are $Re\approx 7.2$, $p\approx 1.0$; these conditions are known to
lead to chaotic advection in this system (Ran et al., 2021; Voth et al.,
2003).

Two main types of experiments are performed: dye mixing and particle tracking
velocimetry (PTV). Dye mixing experiments are performed by labelling half of
the fluid layer with a passive dye tracer ($6.25\times 10^{-5}~{}$M sodium
fluorescein). Initially, the labelled and unlabelled portions are separated by
a physical barrier. As the flow starts, the barrier is lifted and the labelled
fluid progressively penetrates to the unlabelled portion with time. Dye
concentration field is recorded by a CMOS camera (Flare 4M180) operating at 5
frames/s and 20002-pixel resolution. Particle tracking experiments are
performed by seeding the fluid with 100-m large fluorescent polystyrene
particles. Particle positions are recorded by the aforementioned camera
operating at 30 frames/s and 12002-pixel resolution. Active suspensions are
prepared by adding a strain (K12 MG1655) of Escherichia coli to an aqueous
buffer solution of 2% KCl and 1% NaCl by weight. The strain of E. coli has a
swimming speed of 10–20 m/s and a rod-shaped body of on average 2 m long and
0.8 m in diameter. The bacterial volume fraction ($\phi_{b}$) is adjusted
between 0% (pure buffer) and 0.5%; this volume fraction range is considered
dilute (Kasyap et al., 2014; Ran et al., 2021), without introducing large-
scale collective behaviors.

##  3 Results and Discussion

###  3.1 Experimental Results

Figures 1(a) and 1(b) show sample snapshots of dye mixing in the buffer
solution ($\phi_{b}=0\%$) and active suspension ($\phi_{b}=0.5\%$),
respectively. Both snapshots are taken after $N=300$ periods of forcing, with
dye initially confined on the right half of the images at $N=0$. The dye
patterns reveal complex patterns and the underlying vortex structures, which
are similar in the buffer and the active suspension. However, a main
difference is the existence of undyed (void-like) regions near the center of
vortex structures in the active suspension [Fig. 1(b)], while similar regions
are dyed in the buffer case [Fig. 1(a)]. The undyed void-like regions near the
vortex center appear primarily in the left half of the image that is initially
not covered by dye. This suggests that the interplay of microbial activity and
vortex structures leads to (enhanced) transport barriers through which the dye
fails to penetrate or penetrates much more slowly. A video of the dye mixing
processes in the buffer and the active suspension are shown in Movie 1.

To further understand the interaction between activity and vortex structures,
we plot in Fig. 1(d) the flow Poincaré map obtained from PTV experiments.
Lines in the map are stroboscopic particle trajectories that connect a
particle’s initial position to its next position after a period ($T=5$ s).
These trajectories are obtained by numerical integration of fluid particles in
the experimentally measured velocity fields. The Poincaré map reveals several
nested families of closed trajectories, whose behaviors mimic
Kolmogorov–Arnold–Moser (KAM) tori in Hamiltonian dynamical systems (Ottino,
1989). These nested torus families define Lagrangian coherent vortices known
as the elliptic LCSs, with each torus acting as a barrier that blocks tracer
(dye) transport within the vortex (Katsanoulis et al., 2020; Haller et al.,
2020; Aksamit & Haller, 2022). Unlike hyperbolic LCSs, elliptic LCSs are
rotation-dominated regions that do not experience substantial (fluid)
stretching. We note that Lagrangian coherent vortices are fundamentally
different from the vortices defined by Eulerian fields such as vorticity.
While we find a striking correspondence between the dye mixing patterns [Figs.
1(a) and 1(b)] and the Lagrangian coherent vortices in the Poincaré map [Fig.
1(d)], such similarity is absent for the Eulerian vorticity field in Fig.
1(c).

To quantitatively locate the elliptic LCSs, we calculate the trajectory
rotation average (TRA) in the Poincaré map, as shown by the color code in Fig.
1(d). The TRA characterizes the average angular velocity of a particle
trajectory and is defined as:

$\overline{\mathrm{TRA}}_{0}^{N}(\mathbf{x}_{0})=\frac{1}{NT}\sum_{i=0}^{N-1}\cos^{-1}\frac{\langle\dot{\mathbf{x}}_{i},\dot{\mathbf{x}}_{i+1}\rangle}{\left|\dot{\mathbf{x}}_{i}\right|\left|\dot{\mathbf{x}}_{i+1}\right|}.$
(1)

Here, $\dot{\mathbf{x}}_{i}$ is a stroboscopic velocity related to the rate of
change of a particle’s position in the Poincaré map, and $N$ is the number of
periods over which the quantity is calculated. Although the TRA is originally
defined for actual particle trajectories (Haller et al., 2021), we use it here
for the stroboscopic trajectories in the extended phase space of the Poincaré
map. We interpolate the values of the TRA onto a uniform spatial grid to
obtain the TRA field (see Movie 2). In two-dimensional flows, elliptic LCSs
can be located from the closed convex contours of the TRA field (Haller et
al., 2021; Aksamit & Haller, 2022). Since elliptic LCSs are nested families,
we identify the outermost convex TRA contours as the boundaries of Lagrangian
coherent vortices, and the centroid of the innermost convex TRA contours as
elliptic fixed points. For concision, we refer to the outermost members of
elliptic LCSs as Lagrangian vortex boundaries (LVBs).

Figure 2: Enlarged photographs of the dye concentration field for (a) buffer
and (b) active suspension, in the dashed square regions shown in Fig. 1. Green
contours are Lagrangian vortex boundaries (LVBs) identified from the TRA field
and red dots are elliptic fixed points. (c) Dye flux as a function of time
$N$, for the vortex with the label “4”. Inset: average dye flux as a function
of vortex Reynolds number $Re_{\Gamma}$, for the buffer (blue) and the active
suspension (red). Different markers represent different vortices in the flow
field. Error bars are standard deviations.

Figures 2(a) and 2(b) show Lagrangian vortex boundaries (LVBs) (green
contours) and elliptic fixed point (red dots) identified from the TRA field,
superimposed on the dye mixing patterns in the buffer and the active
suspension, respectively. Although LVBs in chaotic flows are known to
correspond to transport barriers (Katsanoulis et al., 2020; Haller et al.,
2020; Aksamit & Haller, 2022), the LVBs in the buffer case [Fig. 2(a)] contain
dye inside. On the other hand, the LVBs in the active case [Fig. 2(b)] enclose
void-like undyed regions around the elliptic fixed points. This suggests that
bacterial activity enhances the strength of LVBs as transport barriers in
chaotic flows. We also notice some differences in the size and shape of LVBs
for the buffer and active cases. Since the LVBs are calculated from
experimentally measured velocity fields, this indicates that microbial
activity also modifies the underlying velocity field.

To quantify the strength of the transport barriers, or the reduction in dye
transport, we calculate dye flux into the Lagrangian vortices using the
following mass balance equation:

$\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{S(t)}C\,\mathrm{d}A=\oint\limits_{B(t)}D(\mathbf{\nabla}C\cdot\mathbf{n})\,\mathrm{d}l-\oint\limits_{B(t)}C(\mathbf{v}-\mathbf{v}_{B})\cdot\mathbf{n}\,\mathrm{d}l.$
(2)

Here, $S(t)$ and $B(t)$ denotes the enclosed area and the contour of the LVBs,
$C$ is the dye concentration field, $D$ is the dye diffusivity, $\mathbf{n}$
is the normal vector of LVBs, $\mathbf{v}$ is fluid velocity, and
$\mathbf{v}_{B}$ is the velocity induced by the motion and deformation of
LVBs. The first and second terms on the right hand side of Eq. 2 are the
contributions to the dye flux by diffusion and convection, respectively. The
left hand side of Eq. 2 gives a simple way to calculate the dye flux as the
rate of change of the total dye enclosed by the LVBs. Figure 2(c) shows the
dye flux as a function of time, for vortex with label “4” in the buffer and
the active suspension. We find a significant reduction of dye flux into the
LVBs with bacterial activity, especially at earlier times ($N<150$). At later
times ($N>200$), the dye flux in the active case is larger than that in the
buffer, as the result of a larger diffusive flux due to large dye
concentration gradients.

We now explore the generality of the reduction of dye flux in active
suspensions. We start by calculating the circulation of a Lagrangian vortex,
defined as:

$\Gamma(t)=\oint\limits_{B(t)}\mathbf{v}\cdot\,\mathrm{d}\mathbf{l}=\int\limits_{S(t)}\omega_{z}\,\mathrm{d}A.$
(3)

Here, $\omega_{z}$ denotes the vorticity component in the out-of-plane
direction. We then define the the vortex Reynolds number as:
$Re_{\Gamma}=\left|\overline{\Gamma}\right|/\nu$, where $\overline{\Gamma}$
denotes the time-averaged circulation over a period, and the absolute value is
included since $\overline{\Gamma}$ can be positive or negative depending on
the sign of the vorticity. We calculate $Re_{\Gamma}$ for each Lagrangian
vortex in Figs. 2(a) and 2(b); the labels correspond to the magnitude of their
$Re_{\Gamma}$. Inset of Fig. 2(c) shows the averaged dye flux as a function of
the vortex Reynolds number $Re_{\Gamma}$, for the buffer and active cases. We
find that in both cases the average dye fluxes scale linearly with
$Re_{\Gamma}$. This is because both the average dye flux and $Re_{\Gamma}$ are
linearly proportional to the area of the Lagrangian vortex. More importantly,
however, we find that the slope of the scaling is larger in the buffer case
compared to the active case. This result shows quantitatively that bacterial
activity reduces the dye flux into the Lagrangian vortices and therefore
enhances the strength of transport barriers.

###  3.2 Numerical Simulations

To understand how swimming microorganisms interact with an imposed time-
periodic flow, we perform numerical simulations of swimming particles using
the experimentally measured velocity field. Microorganisms are modelled as
axisymmetric ellipsoids with a constant swimming speed $v_{s}$, in the
direction $\mathbf{q}$ along their symmetry axis. The swimmer’s position
$\mathbf{x}$ is modelled as:

$\dot{\mathbf{x}}=\mathbf{v}_{f}(\mathbf{x},t)+v_{s}\mathbf{q},$ (4)

where $\mathbf{v}_{f}$ is experimentally measured fluid velocity in the active
suspension, and $v_{s}=20~{}\umu$m/s is the swimming speed of E. coli. As a
control, simulations of elongated passive particles are also performed with
$v_{s}=0$ in the velocity field measured in the buffer. The swimmer’s
orientation is described by the Jeffery’s equation (Jeffery & Filon, 1922):

$\dot{\mathbf{q}}=[\mathbf{\Omega}(\mathbf{x},t)+\Lambda\mathbf{D}(\mathbf{x},t)]\mathbf{q}-\Lambda[\mathbf{q}\cdot\mathbf{D}(\mathbf{x},t)\mathbf{q}]\mathbf{q},$
(5)

where $\mathbf{D}$ and $\mathbf{\Omega}$ are the symmetric and skew-symmetric
parts of the velocity gradient tensor, $\nabla\mathbf{v}_{f}$. Here,
$\Lambda=(1-\alpha^{2})/(1+\alpha^{2})$ is a shape factor, with $\alpha$ being
the swimmer aspect ratio; the factor is roughly $\Lambda\approx 0.88$ for E.
coli.

Initially, both passive and active particles are uniformly distributed in the
flow field with random orientations. As the simulations begin, passive and
active particles start to develop complex patterns following the morphology of
Lagrangian vortices (see Movie 3). Figures 3(a) and 3(b) show the spatial
distribution of passive and active particles at $N=150$, respectively. The
color code in the plots represents the local particle number density,
$\rho_{N}$, normalized by the initial number density, $\rho_{0}$. We find that
active particles are mostly depleted within the LVBs (green contours),
suggesting that Lagrangian vortices repel elongated swimmers. This is in
contrast to the passive case in which LVBs trap passive particles and lead to
particle accumulation within the Lagrangian vortices. This results suggests
the repulsion of active particles by the Lagrangian vortices is not a generic
feature of the velocity field itself. Rather, it is the result of the
interaction between Lagrangian vortices and elongated swimmers.

The repulsion of elongated swimmers by the LVBs is quantified by the radial
distribution function: $g(r)=\langle\rho_{N}(r)\rangle/\rho_{0}$, where the
angle bracket denotes a radial average in all directions calculated from the
elliptic fixed points. The function $g(r)$ represents the likelihood of
finding a particle at a radial distance $r$ from the elliptic fixed point of a
Lagrangian vortex. Figure 3(c) shows $g(r)$ for both passive and active
particles in the Lagrangian vortex with the label “4” in Fig. 2. The dashed
line in Fig. 3(c) is a nominal radius of Lagrangian vortex, defined as:
$r_{n}=\sqrt{A_{S}/\pi}$, where $A_{S}$ is the area enclosed by the LVBs. We
find that the likelihood of finding an active particle within the nominal
radius of a Lagrangian vortex is smaller than that of a passive particle, and
vice versa for outside the nominal radius. This result further corroborates
the repulsion of elongated active particles by Lagrangian vortices.

Figure 3: Spatial distributions of (a) passive particles, and (b) active
particles, at time $N=150$. Particles are colored by their normalized local
number density, $\rho_{N}/\rho_{0}$. Passive particles accumulate inside and
active particles accumulate outside the LVBs (green contours). (c) Radial
distribution function $g(r)$ calculated from the elliptic fixed points of the
Lagrangian vortex with the label “4”; the dashed line is the nominal radius of
the vortex. (d) Spatially averaged normalized number density within the LVBs,
$\langle\rho_{N}\rangle_{S}/\rho_{0}$, as a function of time $N$ for the same
vortex as in (c).

To quantify the time evolution of the repulsion of elongated swimmers, we
calculate the average number density within the LVBs,
$\langle\rho_{N}\rangle_{S}$, normalized by the initial number density
$\rho_{0}$. Figure 3(d) shows $\langle\rho_{N}\rangle_{S}/\rho_{0}$ as a
function of time $N$. Results show that $\langle\rho_{N}\rangle_{S}$ decreases
non-monotonically with time for active particles, and drops to only 20% of the
initial number density $\rho_{0}$ at $N=200$. This suggests that most active
particles escape and accumulate outside the Lagrangian vortex as mixing
progresses. In the control case, $\langle\rho_{N}\rangle_{S}$ for passive
particles increases with time and reaches four times of the initial number
density at $N=200$, which suggests the repulsion of elongated particles by
LVBs is not an intrinsic feature of the velocity field. Although it has been
observed that symmetric and circular Eulerian vortices can repel elongated
swimmers (Torney & Neufeld, 2007; Ran et al., 2021; Qin & Arratia, 2022),
here, we extend these results in a more general case of asymmetric and non-
circular vortices from Lagrangian criteria.

Figure 4: (a) Stroboscopic trajectories of passive (blue) and active (red)
particles that are initially inside the Lagrangian vortices illustrated by the
TRA field. While sharing the same initial condition, passive particles remain
trapped in the vortices and active particles spiral outward and escape. The
trajectories of active particles are colored by their time, which appear more
reddish as time progresses. (b) The PDFs of the inner product of the particle
orientation vector, $\mathbf{q}$, and the tangent vector of members of
elliptic LCSs, $\mathbf{t}$, as defined in the Inset. The initial condition
($N=0$) is approximately a uniform distribution for particles with random
initial orientations. The PDFs at a later time ($N=50$) show that both passive
and active particles preferentially align with members of elliptic LCSs at
$\mathbf{q}\cdot\mathbf{t}=\pm 1$.

To gain further insights into how swimming particles are repelled by
Lagrangian vortices, we plot the stroboscopic trajectories and orientations of
passive and active particles as a function of time, as shown in Fig. 4(a). The
Lagrangian vortices are illustrated by the TRA field (color map); passive and
active particles are shown as blue and red bars, respectively. The
trajectories of passive particles are blue, while active particles’
trajectories are colored by elapsed time that become more reddish with time.
Despite sharing the same initial conditions for position and orientation,
passive particles remain trapped in the Lagrangian vortices while active
particles spiral outward with time and escape the vortices. We note that the
swimming number of the active particles, $\Phi=v_{s}/U$, is at the order of
$10^{-2}$ in the simulations, which suggests that the swimming speed is fairly
negligible compared to the flow speed. However, even such (relatively) low
swimming speed can drive active particles out of equilibrium and escape the
Lagrangian vortices. That is, even small levels of swimming activity is enough
to initiate the expulsion process. In addition, we find that the orientations
of both passive and active particles tend to align with the level curves of
the TRA field. Since the convex TRA contours locate the nested families of
elliptic LCSs, this result indicates elongated particles—whether self-
propelled or not—preferentially align with the members of elliptic LCSs. The
difference is that the self-propulsion of active particles causes them to move
outward and leave the vortices in a spiral manner.

Next, we test the alignment of the elongated particles, we calculate the inner
product of the particle orientation vector, $\mathbf{q}$, and the tangent
vector of the elliptic LCSs, $\mathbf{t}$. The inset of Fig. 4(b) shows a
schematic of the two vectors. Since elliptic LCSs are nested families, we
partition the elliptic LCSs into 10 members using the convex TRA contours. The
inner product is calculated between the orientation vector of an elongated
particle and the tangent of the nearest member of elliptic LCSs. Figure 4(b)
shows probability density functions (PDF) of $\mathbf{q}\cdot\mathbf{t}$ for
both passive and active particles at two different times. At $N=0$, the
initial condition of the PDFs shared by both passive and active particles is
approximately a uniform distribution due to the random initial orientations.
At a later time of $N=50$, the PDFs become bimodal at
$\mathbf{q}\cdot\mathbf{t}=\pm 1$, suggesting a parallel or anti-parallel
alignment between $\mathbf{q}$ and $\mathbf{t}$; we do not find a substantial
difference between the PDFs of passive and active particles, indicating self-
propulsion does not play a significant role in affecting the alignment of
elongated particles with elliptic LCSs. However, self-propulsion causes active
particles to travel toward the outer members of the elliptic LCSs in both
parallel and anti-parallel alignment configurations (see Movie 4), resulting
in the spiral escaping trajectories of the active particles in Fig. 4(a).
Consequently, the interplay between self-propulsion and the alignment of
elongated particles with elliptic LCSs leads to the depletion of active
particles inside the elliptic LCSs, that is, the accumulation of active
particle outside the Lagrangian vortex boundaries (LVBs). This accumulation of
microswimmers outside the LVBs can further obstruct dye transport into the
Lagrangian vortices, which is responsible for the observed transport barriers.

##  4 Conclusion

In this manuscript, we investigate the interaction between swimming
microorganisms and Lagrangian coherent vortices known as the elliptic LCSs in
time-periodic flows in experiments and in simulations. Our results show that
even small amounts of swimming activity can affect (i) the dynamics of active
particles in the flow (Fig. 4) and consequently (ii) the mixing and transport
of passive scalars (Fig 1) in chaotic flows. Experiments show that the
interaction between organisms and elliptic LCSs leads to transport barriers
through which the tracer flux is significantly reduced. Using the Poincaré map
and the TRA field, we show that these transport barriers coincide with
outermost member of elliptic LCSs, or Lagrangian vortex boundaries (LVBs). To
further understand the formation of the transport barriers, we perform
numerical simulations of elongated microswimmers in experimentally measured
velocity fields. Results show that elliptic LCSs can repel elongated swimmer
and lead to swimmer accumulation outside LVBs. This accumulation of
microswimmers effectively reduces the transport into elliptic LCSs. We further
show that the interplay between self-propulsion and the preferential alignment
of elongated particle with the tangents of elliptic LCSs leads swimmers to
escape the Lagrangian vortices. Overall, our results allow quantitative
prediction of the Lagrangian transport of microorganisms and passive tracer
quantities (e.g., temperature, oxygen, and nutrients) in chaotic flows with
nontrivial vortex structures. Although there have been previous studies on the
interaction between microorganisms and LCSs (Khurana & Ouellette, 2012;
Dehkharghani et al., 2019; Ran et al., 2021; Qin & Arratia, 2022; Si & Fang,
2021, 2022; Yoest et al., 2022), our work extend study of swimmer-LCS
interaction to elliptic LCSs (i.e., vortex-like dynamical structures). From a
practical perspective, our results may be useful in understanding the organic
matter flux in the oceans, algal blooms in lakes, and harmful bacterial
infections.

## Acknowledgement

We thank Tom Solomon, Kevin Mitchell, Simon Berman, and Albane Théry for
insightful discussions, and Brendan Blackwell for help with early work. This
work was supported by NSF-DMR-1709763.

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