# Jeffery’s Orbits and Microswimmers in Flows: A Theoretical Review

Kenta Ishimoto<EMAIL_ADDRESS>Research Institute for
Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

(August 28, 2024)

###### Abstract

In this review, we provide a theoretical introduction to Jeffery’s equations
for the orientation dynamics of an axisymmetric object in a flow at low
Reynolds number, and review recent theoretical extensions and applications to
the motions of self-propelled particles, so-called microswimmers, in external
flows. Bacteria colonize human organs and medical devices even with flowing
fluid, microalgae occasionally cause huge harmful toxic blooms in lakes and
oceans, and recent artificial microrobots can migrate in flows generated in
well-designed microfluidic chambers. The Jeffery equations, a simple set of
ordinary differential equation, provide a useful building block in modeling,
analyzing, and understanding these microswimmer dynamics in a flow current, in
particular when incorporating the impact of the swimmer shape since the
equations contain a shape parameter as a single scalar, known as the
Bretherton parameter. The particle orientation forms a closed orbit when
situated in a simple shear, and this non-uniform periodic rotational motion,
referred to as Jeffery’s orbits, is due to a constant of motion in the non-
linear equation. After providing a theoretical introduction to microswimmer
hydrodynamics and a derivation of the Jeffery equations, we discuss possible
extensions to more general shapes, including those with rapid deformation. In
the latter part of this review, simple mathematical models of microswimmers in
different types of flow fields are described, with a focus on constants of
motion and their relation to periodic motions in phase space, together with a
breakdown of the degenerate orbits, to discuss the stable, unstable, and
chaotic dynamics of the system. The discussion in this paper will provide a
comprehensive theoretical foundation for Jeffery’s orbits and will be useful
to understand the motions of microswimmers under various flows and to analyze
and design computer and laboratory experiments, as well as in the active
matter and artificial smart swimmer studies.

##  I Introduction

A century ago, the British physicist George B. Jeffery published a paper [1]
describing the slow motion of an ellipsoidal particle in a viscous fluid.
According to Google Scholar metrics, as of the date of this writing, Jeffery’s
article has been cited more than 5000 times, and half of these citations were
made after 2010. This dramatic revival of this classical study is related to
the recent progress in fluid dynamics on the cellular scale, microfluidics,
and their applications to biological and medical problems, as well as
microrobots and active soft material sciences [2, 3, 4, 5, 6, 7].

For a spheroidal body (an ellipsoidal body-of-rotation), the equation of the
angular evolution is called Jeffery’s equation, and later Bretherton [8] found
that the equation is also applicable to an arbitrary body of revolution.

In a simple shear, the orientation of the particle obeys periodic rotation
orbits, known as Jeffery’s orbits. These simple dynamics are obtained as an
exact solution to the Stokes equations and are thus useful when modeling a
particle in more complex flow environments. These background flows are
ubiquitous in nature, as in marine turbulence and flows in saturated soils, as
well as in microfluidic devices in the laboratory [9, 10, 11, 12]. The
rotational dynamics of a particle become more important when the particle is
active, or in other words self-propelled, because a tiny difference in the
orientation eventually yields a large deviation of the position after a while.

In this article, we provide a comprehensive introduction and review of the
theoretical basis of self-propelled swimming particles at a small Reynolds
number, focusing on the swimming dynamics in flows in connection with the
Jeffery equations. Self-propelled particles in fluids are called
microswimmers, and examples include swimming microorganisms, swimming cells
such as spermatozoa in broad taxa [13], and microscopic synthetic and
artificial self-propelled particles [14, 15].

A sphere is the simplest model of a swimming particle, and thus a spheroid is
the next-order approximation of the shape of the swimmer. Indeed, the shapes
of biological microswimmers are very diverse, from the spherical algae of
Volvox to elongated bacterium and from the symmetrical shape of diatoms to
largely deformed shapes, such as vigorous movement of sperm [16, 9, 17, 18].
To elucidate the effects of shape on the motions of microswimmers in flows,
Jeffery’s equations have been popularly and widely used as a theoretical
building block in modeling and analyzing their complex behaviors. A notable
feature is the close, periodic orbit emerging as a constant of motion in non-
linear differential equations. In this review, we examine the microswimmer
dynamics through constants of motion, focusing on how they are derived and
when the constancy is violated.

The response of organisms to a background flow is generally called rheotaxis,
and some rheotactic behaviors in biological and artificial microswimmers are
understood as hydrodynamical interactions between the swimmer shape and its
surrounding flow, enabling us to understand the mechanisms and design sorting
and navigating systems. From a simple shear and quadratic Poiseuille flow in
pipes to more complex swirling vortical flow and mixing flow with hyperbolic
stagnation points, we discuss their theoretical developments and experimental
realizations.

We start with the fluid dynamics of a self-deforming particle in Sec. II and
provide a detailed derivation for the Jeffery equations and Jeffery’s orbits
in Sec. III. Theoretical extensions for the Jeffery equations are discussed in
Sec. IV, which contains the factors that violate periodic Jeffery orbits. We
then move on to swimming behaviors in simple to complex background flows in
Sec. V, where we review how the Jeffery equations are used in the modeling of
swimming dynamics in flows. Future perspectives and topics that are not fully
covered in this review are briefly discussed in Sec. VI, after which the
conclusion is provided.

##  II Microswimmer Hydrodynamics

###  II.1 Governing Equations

We start with the governing equations for the hydrodynamics of microswimmers.
The fluid motion obeys the incompressible Navier-Stokes equations. The
Reynolds number for the flow around a swimmer is given by $Re=\mu UL/\rho$,
where $\mu$ is the viscosity and $\rho$ is the fluid mass density. The
characteristic length $L$ and $U$ are typically the size and swimming speed,
respectively, of the microswimmer. Many swimming organisms have a Reynolds
number much less than unity, for example, $Re\sim 10^{-5}$ for bacteria such
as Escherichia coli, $Re\sim 10^{-3}$ for sperm cells, and $Re\sim 10^{-1}$
for a ciliate Paramecium [19], where the Stokes approximation well represents
the flow dynamics. Hence, neglecting the inertia terms in the Navier-Stokes
equation reduces our fluid equations to the steady Stokes equation given by

$\nabla p=\mu\nabla^{2}\bm{u},$ (1)

where $p$ is the pressure and $\bm{u}$ is the fluid velocity that satisfies
the incompressible condition $\nabla\cdot\bm{u}=0$. The Stokes equations may
be rewritten as $\partial_{j}\sigma_{ij}=0$ by using the Newtonian stress
tensor, $\sigma_{ij}=-p\delta_{ij}+2\mu E_{ij}$. Here, we use the Einstein
summation convention for the repeated spatial indices $i,j=\\{1,2,3\\}$,
$\partial_{i}=\partial/\partial x_{i}$, $\delta_{ij}$ is the Kronecker delta,
and $E_{ij}=(\partial_{i}u_{j}+\partial_{j}u_{i})/2$ is the rate-of-strain
tensor.

We now consider the equation of motion of a swimmer immersed in fluid. To
specify the motion of a deforming object, we introduce two reference frames,
the laboratory-fixed frame, $\\{\bm{e}_{i}\\}$, and the swimmer-fixed frame,
$\\{\hat{\bm{e}}_{i}\\}$ (Fig. 1). The linear and angular momentum
conservation should hold. Again, at low Reynolds numbers, all the inertia
terms are negligible and thus these momentum equations are reduced to the
force and torque balance relation. The total force $\bm{F}$ and the torque
$\bm{M}$ acting on the object should vanish as

$F_{i}=\int_{S}\sigma_{ij}n_{j}\,dS{\color[rgb]{0,0,0}=0},\leavevmode\nobreak\
M_{i}=\int_{S}\epsilon_{ijk}(x_{j}-X_{j})\sigma_{k\ell}n_{\ell}\,dS=0,$ (2)

if there is no other external forces or torques. Here, $S$ indicates the
surface of the swimmer, the unit normal $\bm{n}$ is taken outwards to the
surface, and $\epsilon_{ijk}$ is the Levi-Civita permutation symbol. The
vector $\bm{X}$ indicates the origin of the body-fixed coordinates. At the
limit of negligible inertia, replacements of the body-fixed coordinates and
their origin do not affect the equations of motion (2). This corresponds to
the gauge freedom of the system, and the swimmer dynamics are determined
essentially by the shape gait [20]. To solve the swimmer dynamics, we may
practically fix the body-fixed coordinates, or fix the gauge arbitrarily,
although a theoretically reasonable choice is to set the origin as the center
of mass [21].

We also consider the background flow $\bm{u}^{\infty}$ and assume this given
function also satisfies the Stokes equations. The total fluid velocity is a
sum of the background flow and the disturbed flow due to the presence of the
swimmer $\bm{u}^{d}$, that is, $\bm{u}=\bm{u}^{\infty}+\bm{u}^{d}$. In many
situations in nature and the laboratory, the length scale for modulation of
the background flow is larger than the size of the swimmer. Even in the
turbulent ocean, the smallest length scale of the vortices is estimated at
around 10–30 mm [9], which is sufficiently large to approximate the local
background flow to be linear in position. The background $\bm{u}^{\infty}$ may
be regarded as the flow induced by other swimmers or external boundaries, and
these hydrodynamic interactions are discussed in later sections.

Now, let us consider the dynamics in the absence of a flow
($\bm{u}^{\infty}=\bm{0})$ and write down the velocity of the swimmer surface
$\bm{v}$ to describe the no-slip boundary conditions of the fluid equation
(1):

$\bm{u}^{d}=\bm{v}\leavevmode\nobreak\ \textrm{on}\leavevmode\nobreak\ S.$ (3)

Let $\bm{a}$ be the Lagrangian label for the material point on the surface
$S$. We can specify the point on the surface by using the body-fixed
coordinate as $\bm{x}(\bm{a},t)=\bm{X}+x^{s}_{i}(\bm{a},t)\hat{\bm{e}}_{i}$.
Its time derivative gives

$\bm{v}=\frac{d\bm{x}}{dt}=\frac{d\bm{X}}{dt}+x^{s}_{i}\frac{d\hat{\bm{e}_{i}}}{dt}+\frac{\partial
x^{s}_{i}}{\partial t}\hat{\bm{e}_{i}}$ (4)

when the time derivative in the final term is taken with a fixed Lagrangian
label. We then define the translational velocity $\bm{U}$ and rotational
velocity $\boldsymbol{{\it\Omega}}$ as the difference between the two frames
as $\bm{U}=d\bm{X}/dt$ and
$d\hat{\bm{e}}_{i}/dt=\boldsymbol{{\it\Omega}}\times\hat{\bm{e}}_{i}$.
Introducing the deformation velocity as $\bm{u}^{s}=(\partial
x^{s}_{i}/\partial t)\hat{\bm{e}}_{i}$, we may obtain the surface velocity as

$\bm{v}=\bm{U}+\boldsymbol{{\it\Omega}}\times(\bm{x}-\bm{X})+\bm{u}^{s}.$ (5)

In this article, we particularly focus on the kinematic problem, in which one
solves the swimming velocity, $\bm{U}$ and $\boldsymbol{{\it\Omega}}$, and
thus the swimming trajectory under a given shape gait $x^{s}_{i}(\bm{a},t)$.
If we need to solve the shape gait as an unknown function by solving a
coupling problem with the material elasticity, this is called
elastohydrodynamics [22, 23, 24, 25], and the constitutive relation of the
swimmer, $\sigma^{\textrm{int}}_{ij}(t)$, is required to complete the model
equations. At the zero inertia limit, the Cauchy momentum equation inside the
swimmer is $\partial_{j}\sigma^{\textrm{int}}_{ij}=0$ and by the Gauss theorem
we have

$F_{i}=-\int_{S}\sigma^{\textrm{int}}_{ij}n_{j}\,dS=0,$ (6)

which indicates that no force is generated from the interior of the swimmer
and also for the torque, regardless of the material property. Hence, the
system is closed by the Stokes equations and the force- and torque-free
conditions.

In summary, the governing equations in the kinematic problem of microswimmer
hydrodynamics are therefore Eqs. (1) and (2) with the no-slip boundary
condition $\bm{u}^{d}=\bm{v}$ on the surface $S$ given by Eq. (5). Depending
on the confinement that encloses the fluid, we also need further boundary
conditions on the surface of the container. If the swimmer is located in a
free space, we impose $\bm{u}^{d}\rightarrow\bm{0}$ as
$|\bm{x}-\bm{X}|\rightarrow\infty$.

FIG. 1:  (Color online) Schematic of a microswimmer in a fluid. Two reference
systems are considered to describe the swimming dynamics: The laboratory-fixed
coordinates $\\{\bm{e}_{i}\\}$ and the body-fixed coordinates
$\\{\hat{\bm{e}}_{i}\\}$, whose origin is denoted by $X_{i}$. The orientation
of the swimmer is represented by a rotational matrix that corresponds to the
two coordinate systems. Figure modified from Ishimoto [26] under the creative
commons license, http://creativecommons.org/licenses/by/4.0. (copyright 2020
by The Author(s))

###  II.2 Swimming Formula

We now consider the hydrodynamic forces and torques acting on a microswimmer
and derive the swimming formula that provides us with the swimming velocities
for a given shape gait.

To do so, we first recall the Lorentz reciprocal theorem, which is a relation
between the solutions to the Stokes equations with the same boundary geometry
$S$ but with different boundary values [27]. Let $(u_{i},\sigma_{ij})$ and
$(\tilde{u}_{i},\tilde{\sigma}_{ij})$ be the two distinct Stokes solutions,
and from the Lorentz reciprocal theorem, these satisfy

$\int_{S}u_{i}\tilde{\sigma}_{ij}n_{j}dS=\int_{S}\tilde{u}_{i}\sigma_{ij}n_{j}dS,$
(7)

where $\bm{n}$ indicates the unit normal on the surface.

Now we assume that the former solution $(u_{i},\sigma_{ij})$ corresponds to
solutions around a deforming swimmer whose surface velocity is given by Eq.
(5) in a background flow, and that the latter with the hat symbol
$(\tilde{u}_{i},\tilde{\sigma}_{ij})$ is chosen for a flow around a rigid body
with arbitrary translational and rotational velocities, $\tilde{\bm{U}}$ and
$\tilde{\boldsymbol{{\it\Omega}}}$, respectively, in the absence of a
background flow.

With the linearity of the flow equation, we may decompose the surface force on
the rigid body (from the surrounding fluid)
$\tilde{f}_{i}=-\tilde{\sigma}_{ij}n_{j}$, with $\bm{n}$ being the outward
unit normal to the fluid region, in the form

$\tilde{f}_{i}=-{\it\Sigma}^{U}_{ij}\tilde{U}_{j}-{\it\Sigma}^{\Omega}_{ij}\tilde{{\it\Omega}}_{j}.$
(8)

The second-rank tensors, ${\it\Sigma}^{U}_{ij}$ and
${\it\Sigma}^{\it\Omega}_{ij}$, are called the translational and rotational
surface force resistance tensors, respectively [28]. As seen below, these
tensors are fundamental quantities to solve the swimmer dynamics from a
theoretical point of view, even though standard direct numerical computations
of the Stokes flow bypass these by using the boundary integral formulation
[28]. Note, however, that these tensors are functions depending only on the
instantaneous shape of the swimmer when represented in the body-fixed
coordinates once the fluid viscosity is fixed.

After integrating these variables over the surface $S$, we obtain the familiar
resistance tensors by introducing four $3\times 3$ tensors as
$K^{FU}_{ij}=\int_{S}{\it\Sigma}^{U}_{ij}dS$,
$K^{F{\it\Omega}}_{ij}=\int_{S}{\it\Sigma}^{\it\Omega}_{ij}dS$,
$K^{MU}_{ij}=\int_{S}\epsilon_{ik\ell}r_{k}{\it\Sigma}^{U}_{\ell j}dS$, and
$K^{M{\it\Omega}}_{ij}=\int_{S}\epsilon_{ik\ell}r_{k}{\it\Sigma}^{\it\Omega}_{\ell
j}dS$, where we have introduced $r_{k}=x_{k}-X_{k}$. These are integrated into
a single $6\times 6$ tensor,

$\mathcal{K}=\begin{pmatrix}{\bf K}^{FU}&{\bf K}^{F{\it\Omega}},\\\ {\bf
K}^{MU}&{\bf K}^{M{\it\Omega}}\end{pmatrix},$ (9)

called the grand resistance tensor, which is symmetric and positive-definite
[29]. Hence, one can define its inverse, $\mathcal{M}=\mathcal{K}^{-1}$,
called the grand mobility tensor. We then introduce 6-dimensional velocity and
force vectors as

$\mathcal{U}=\begin{pmatrix}\bm{U}\\\
\boldsymbol{{\it\Omega}}\end{pmatrix},\leavevmode\nobreak\
\hat{\mathcal{U}}=\begin{pmatrix}\tilde{\bm{U}}\\\
\tilde{\boldsymbol{{\it\Omega}}}\end{pmatrix},\leavevmode\nobreak\
\mathcal{F}=\begin{pmatrix}\bm{F}\\\ \bm{M}\end{pmatrix},\leavevmode\nobreak\
\hat{\mathcal{F}}=\begin{pmatrix}\tilde{\bm{F}}\\\
\tilde{\bm{M}}\end{pmatrix}.$ (10)

Plugging this representation into Eq. (7), we may rewrite it in the form [30,
21, 31]

$\mathcal{U}_{a}\hat{\mathcal{F}}_{a}+\hat{\mathcal{U}}_{a}\mathcal{F}^{\textrm{prop}}_{a}+\hat{\mathcal{U}}_{a}\mathcal{F}^{\infty}_{a}=\hat{\mathcal{U}}_{a}\mathcal{F}_{a},$
(11)

where we use the indices $a,b,\cdots,\in\\{1,2,\cdots,6\\}$ to distinguish
from the indices $i,j,k,\cdots,\in\\{1,2,3\\}$ for the spatial labels. Here,
we have written

$\mathcal{F}^{\textrm{prop}}=\begin{pmatrix}\bm{F}^{\textrm{prop}}\\\
\bm{M}^{\textrm{prop}}\end{pmatrix},\leavevmode\nobreak\
\mathcal{F}^{\infty}=\begin{pmatrix}\bm{F}^{\infty}\\\
\bm{M}^{\infty}\end{pmatrix},$ (12)

and these are explicitly given by

$F^{\textrm{prop}}_{j}=-\int_{S}u^{\prime}_{i}{\it\Sigma}^{U}_{ij}dS,\leavevmode\nobreak\
M^{\textrm{prop}}_{j}=-\int_{S}u^{\prime}_{i}{\it\Sigma}^{{\it\Omega}}_{ij}dS,$
(13)

and

$F^{\infty}_{j}=\int_{S}u^{\infty}_{i}{\it\Sigma}^{U}_{ij}dS,\leavevmode\nobreak\
M^{\infty}_{j}=\int_{S}u^{\infty}_{i}{\it\Sigma}^{{\it\Omega}}_{ij}dS.$ (14)

Eq. (11) is an integral formula first derived by Stone & Samuel [32]. Here,
the force and torque for a rigid body are simply written as
$\hat{\mathcal{F}}_{a}=-\mathcal{K}_{ab}\hat{\mathcal{U}}_{b}$, and Eq. (11)
may be further reduced to

$\mathcal{F}_{a}=-\mathcal{K}_{ab}\mathcal{U}_{b}+\mathcal{F}^{\textrm{prop}}_{a}+\mathcal{F}^{\infty}_{a}$
(15)

by noting that $\hat{\mathcal{U}}_{a}$ is taken arbitrarily. For a swimmer
under the force- and torque-free conditions (2), that is, $\mathcal{F}_{i}=0$,
the swimming formula is then simply given by

$\mathcal{U}_{a}=\mathcal{M}_{ab}\,(\mathcal{F}^{\textrm{prop}}_{b}+\mathcal{F}^{\infty}_{b}),$
(16)

from which one can calculate the swimming velocities for a given shape gait
and background flow. With the swimming formula (16), one may be able to
explicitly demonstrate Purcell’s scallop theorem[33], which states that a
microswimmer cannot generate net locomotion by reciprocal motion. The
reciprocal motion should be defined as the time-reversal deformation using the
body-fixed surface position. Note, however, that the surface velocity
$\bm{u}^{s}(\bm{a},t)=(\partial x^{s}_{i}/\partial t)\hat{\bm{e}}_{i}$ depends
on the instantaneous orientation of the swimmer, and we thus need
representations of the resistance tensors in the body-fixed coordinates to
provide a formal proof for the scallop theorem [21].

###  II.3 Squirmer

In this subsection, we consider a specific swimmer model, known as the
squirmer (Fig. 2. The squirmer was first proposed by Lighthill [34] as a
slightly deforming sphere and later corrected and applied by Blake to analyze
a swimming ciliate [35]. Presently, however, the squirmer refers to a
mathematical model of a rigid body with a surface slip velocity instead of
actual deformation [36]. Due to the model’s simplicity, the squirmer has been
extensively studied in the past few decades to elucidate universal features of
microswimmers and their hydrodynamic interactions, as well as to model the
behaviors of specific microorganisms, such as ciliates and volvox and
artificial swimming colloids [37, 38, 39, 40, 41, 42, 43, 36, 44, 45].

\begin{overpic}[width=241.84842pt]{fig_squirm.eps} \put(1.0,45.0){\large(a)}
\put(55.0,45.0){\large(b)} \end{overpic} FIG. 2:  (Color online) Schematic of
a slightly deforming microswimmer. (a) Sphere with slight deformation. The
instantaneous shape shown with a thick red curve is denoted by $S(t)$, whereas
the original unperturbed sphere is shown as a dotted black circle denoted by
$S_{0}$. The point on the surface with the Lagrangian label $\bm{a}$ moves
from $\bm{x}_{0}(\bm{a},t)$ to $\bm{x}(\bm{a},t)$. (b) Schematic of a swimming
ciliate whose shape is well described by the envelope shown with a broken red
curve.

Let us consider a general deforming object, express a position vector on its
surface as $\bm{x}(\bm{a},t)=\bm{X}+x^{s}_{i}(\bm{a},t)\hat{\bm{e}}_{i}$, and
write its deformation as $x^{s}_{i}(\bm{a},t)=\epsilon\alpha_{i}(\bm{a},t)$,
where $\epsilon$ denotes the magnitude of the deformation and $\alpha_{i}$ is
taken as $|\alpha_{i}|=O(1)$. We analyze the swimming dynamics for a small
deformation, specifically $O(\epsilon)\ll 1$.

Our primary aim is to calculate the surface force resistance tensors (8) to
solve the hydrodynamic force/torque or to apply the swimming formula (16).
Because it is impractical to obtain these quantities in an analytical form for
a general deforming object, we instead construct an equivalent but simpler
boundary condition in which the Stokes solutions remain the same. To do so, we
introduce the reference surface $S_{0}(t)$, which denotes the surface in the
absence of deformation, and this is obtained by setting $\epsilon=0$ in the
expression for $x^{s}_{i}$. By Eq. (4), the surface deformation velocity is
given by $\bm{u}^{\prime}=\epsilon\dot{\alpha}_{i}\hat{\bm{e}}_{i}$, where the
dot indicates the Lagrangian time derivative.

Instead of the no-slip boundary conditions on $S(t)$, we consider a slip
surface velocity $\bm{u}^{\prime}_{0}=A_{i}(\bm{a},t)\hat{\bm{e}}_{i}$ on
$S_{0}(t)$, and determine the coefficient $A_{i}$ to ensure that the fluid
velocity behaves properly on $S(t)$. We expand the swimming velocities and the
coefficient $A_{i}(\bm{a},t)$ as
$\mathcal{U}_{i}=\epsilon\mathcal{U}^{(1)}_{i}+\epsilon^{2}\mathcal{U}^{(2)}_{i}+\cdots$
and $A_{i}=\epsilon A^{(1)}_{i}+\epsilon^{2}A^{(2)}_{i}+\cdots$. The velocity
fields are then also expanded around the point $\bm{x}_{0}$ so that

$\bm{u}(\bm{x},t)=\bm{u}(\bm{x}_{0},t)+\bm{s}\cdot\nabla\bm{u}+O(\epsilon^{2}),$
(17)

where $\bm{s}=\bm{x}-\bm{x}_{0}=O(\epsilon)$. We substitute the two
expressions of the boundary conditions on $S(t)$ and $S_{0}(t)$ into Eq. (17),
and equating the $O(\epsilon)$ contributions in both cases yields the simple
relation $A^{(1)}_{i}=\dot{\alpha}_{i}$. By setting
$\bm{u}^{\prime}_{0}=\epsilon A^{(1)}_{i}\hat{\bm{e}}_{i}+O(\epsilon^{2})$,
the leading-order contribution to the propulsive force is therefore obtained
as

$F^{\textrm{prop}}_{j}=-\int_{S_{0}}u^{\prime}_{0i}{\it\Sigma}^{U}_{0ij}dS+O(\epsilon^{2}),$
(18)

where $\Sigma^{U}_{0ij}$ indicates the surface force tensor evaluated at the
reference shape $S_{0}$. The propulsive torque is also written only in terms
of the surface force tensor at $S_{0}$. From the swimming formula, it is
therefore found that the leading-order velocities $\bm{U}^{(1)}$ and
$\boldsymbol{{\it\Omega}}^{(1)}$ are the same as those of a swimmer with a
slip velocity on its reference surface $S_{0}$. In other words, for a small
deformation, a deforming swimmer is well approximated by a rigid body with
surface slip velocity, that is, a simple squirmer.

The squirmer model has been extensively studied, in particular to study the
hydrodynamic interactions between the external boundaries and between the
cells. In particular, the symmetric, traceless part of the first force moment
$S_{ij}$, called the stresslet, determines the leading-order contribution to
the far-field flow around a swimmer under a force- and torque-free condition.
The stresslet is defined as

$S_{ij}=\frac{1}{2}\int_{S}[f_{i}x_{j}+f_{j}x_{i}+\frac{2}{3}f_{k}x_{k}\delta_{ij}+2\mu(u_{i}{\color[rgb]{0,0,0}n_{j}}+u_{j}n_{i})]\,dS,$
(19)

where $f_{i}=-\sigma_{ij}n_{j}$ is the surface force on the swimmer [46, 38].
For an axisymmetrical swimmer, the stresslet is represented by a single
parameter $P$ as $S_{ij}=P(d_{i}d_{j}-\frac{1}{3}\delta_{ij})$ with $\bm{d}$
denoting the director vector along the axis of symmetry. The swimmer pattern
is often categorized via the sign of $P$ and a swimmer with $P>0$, $P<0$, or
$P=0$ is called a pusher, puller, or neutral swimmer, respectively. A pusher
swimmer is typically found in a cell with an appendage at its rear, such as
bacteria and sperm cells [47, 48]. A puller often has flagella at the front of
the cell and a Chlamydomonas cell is a popular example [49, 50]. A cell with
uniform distribution of propulsive appendages, such as a ciliate, is well
approximated as a neutral swimmer [51].

##  III Jeffery’s Equation

In this section, we give a derivation of the Jeffery equation, considering a
general rigid body or a squirmer swimmer in a linear background flow.

###  III.1 Linear Background Flow

As explained in the previous section, the background flow in nature and the
laboratory is often locally approximated by a linear flow, which satisfies the
Stokes equations. Let us expand the background flow around the swimmer
position $\bm{X}$. By rewriting the local background linear and angular
velocities as $\bm{U}^{\infty}=\bm{u}^{\infty}_{i}(\bm{X})$ and
$\boldsymbol{{\it\Omega}}^{\infty}=(1/2)\nabla\times\bm{u}^{\infty}_{i}(\bm{X})$,
we obtain up to the first order of the relative position $r_{i}=x_{i}-X_{i}$
as

$u^{\infty}_{i}(\bm{x})=U^{\infty}_{i}+\epsilon_{ijk}{\it\Omega}^{\infty}_{j}r_{k}+E^{\infty}_{ij}r_{j}+O(|\bm{r}|^{2}),$
(20)

where
$E^{\infty}_{ij}=(\partial_{j}u^{\infty}_{i}(\bm{X})+\partial_{i}u^{\infty}_{j}(\bm{X}))/2$
is the local rate-of-strain tensor at the position of the swimmer.

We then substitute the linear expression (20) into Eq. (14) to obtain the
flow-induced force and torque as

$\displaystyle F^{\infty}_{i}$ $\displaystyle=$ $\displaystyle
K^{FU}_{ij}U^{\infty}_{j}+K^{F{\it\Omega}}_{ij}{\it\Omega}^{\infty}_{j}+{\it\Gamma}^{F}_{ijk}E^{\infty}_{jk}$
(21) $\displaystyle M^{\infty}_{i}$ $\displaystyle=$ $\displaystyle
K^{MU}_{ij}U^{\infty}_{j}+K^{M{\it\Omega}}_{ij}{\it\Omega}^{\infty}_{j}+{\it\Gamma}^{M}_{ijk}E^{\infty}_{jk},$
(22)

where the third-rank tensors ${\it\Gamma}^{F}_{ijk}$ and
${\it\Gamma}^{M}_{ijk}$ are respectively called the force-shear tensor and
torque-shear tensor, defined as

${\it\Gamma}^{F}_{ijk}=\int_{S}{\it\Sigma}^{U}_{ji}r_{k}\,dS\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
{\it\Gamma}^{M}_{ijk}=\int_{S}{\it\Sigma}^{\it\Omega}_{ji}r_{k}\,dS.$ (23)

With the symmetric property of the rate-of-strain tensor, these third-rank
tensors are symmetric under the exchanges of the last two subscripts, namely
${\it\Gamma}^{F}_{ijk}={\it\Gamma}^{F}_{ikj}$.

These flow-induced force and torque equations (21) and (22) are further
simplified as

$\mathcal{F}^{\infty}_{a}=\mathcal{K}_{ab}\mathcal{U}^{\infty}_{b}+\mathcal{G}_{ajk}E^{\infty}_{jk}.$
(24)

Here, we have introduced the 6-dimensional background velocity vector
$\mathcal{U}^{\infty}$, $6\times 3\times 3$ tensor $\mathcal{G}_{ajk}$, given
by

$\mathcal{U}^{\infty}_{i}=U^{\infty}_{i},\leavevmode\nobreak\
\mathcal{U}^{\infty}_{i+3}={\it\Omega}^{\infty}_{i},\leavevmode\nobreak\
\mathcal{G}_{ijk}={\it\Gamma}^{U}_{ijk},\leavevmode\nobreak\
\mathcal{G}_{i+3,jk}={\it\Gamma}^{\it\Omega}_{ijk}.$ (25)

The $6\times 3\times 3$ tensor $\mathcal{G}_{ajk}$ is only a function of the
instantaneous shape when represented in the body-fixed coordinates if the
viscosity $\mu$ is a fixed constant.

###  III.2 Jeffery’s Equation

To derive the Jeffery equations, let us first consider the symmetry of the
shape of a swimmer, which in turn determines the swimming dynamics. The number
of constants appearing as entries of the tensors is 51 in total, but the
symmetry of the object can dramatically reduce this number.

Now we consider the angular dynamics of a body of revolution under a linear
shear flow to derive the Jeffery equation. We denote the axis of rotation by a
unit vector $\bm{d}$ and set it as $\bm{d}=\hat{\bm{e}}_{1}$. Because the body
of revolution has an axisymmetric nature, we may represent the resistance
tensors represented in the laboratory frame as

$\displaystyle K^{FU}_{ij}$ $\displaystyle=$ $\displaystyle
K^{U}_{||}d_{i}d_{j}+K^{U}_{\perp}(\delta_{ij}-d_{i}d_{j}),$ (26)
$\displaystyle K^{M{\it\Omega}}_{ij}$ $\displaystyle=$ $\displaystyle
K^{\it\Omega}_{||}d_{i}d_{j}+K^{\it\Omega}_{\perp}(\delta_{ij}-d_{i}d_{j}),$
(27)

where the coefficients $K^{U}_{||},K^{U}_{\perp},K^{\it\Omega}_{||}$, and
$K^{\it\Omega}_{\perp}$ are positive constants depending on the shape of the
object. The coupling terms can be taken as zero, that is,
$K^{F{\it\Omega}}_{ij}=K^{MF}_{ij}=0$, if the origin of the body-fixed frame
is taken to coincide with the hydrodynamic center of resistance [29].

Similar symmetric arguments enable us to reduce the representation of the
shear-torque resistance tensor in the laboratory frame as

${\it\Gamma}^{M}_{ijk}={\it\Gamma}({\color[rgb]{0,0,0}\epsilon_{ik\ell}d_{j}+\epsilon_{jk\ell}d_{i}})d_{\ell}.$
(28)

From the swimming formula (16), we use the diagonal representation of the
grand resistance tensors (26), (27), and (28) to obtain the angular velocity
of the swimmer in the form

${\it\Omega}_{i}={\it\Omega}^{\infty}_{i}-(K^{M{\it\Omega}})^{-1}_{ij}({\it\Gamma}^{M}_{{\color[rgb]{0,0,0}jk\ell}}E^{\infty}_{k\ell}+M^{\textrm{prop}}_{j}).$
(29)

With $\dot{\bm{d}}=\boldsymbol{{\it\Omega}}\times\bm{d}$, we arrive at the
Jeffery equation given by

$\dot{d}_{i}=\epsilon_{ijk}{\it\Omega}^{\infty}_{j}d_{k}+B(\delta_{ij}-d_{i}d_{j})E^{\infty}_{jk}d_{k}+\dot{d}^{\textrm{\,prop}}_{i},$
(30)

where we have introduced $B={\it\Gamma}/K^{\it\Omega}_{\perp}$ and
$\dot{\bm{d}}^{\,\textrm{prop}}=\bm{M}^{\textrm{prop}}\times\bm{d}/K^{\it\Omega}_{\perp}$.
The scalar $B$, known as the Bretherton parameter, is only a factor that
reflects the shape of the object. Indeed, for a spheroidal case, the
resistance tensors are exactly obtained [52] and the Bretherton parameter is
given by $B=(c^{2}-1)/(c^{2}+1)$, with $c$ being the aspect ratio of the
spheroid. The Bretherton parameter becomes $B=0$ for a sphere and approaches
$B\rightarrow 1$ at the slender limit and $B\rightarrow-1$ at the disk limit.
Nonetheless, extreme cases are known in which the Bretherton parameter becomes
$|B|>1$ [8, 53].

\begin{overpic}[width=241.84842pt]{fig_jeffery.eps}
\put(3.0,56.0){{\large(a)}} \put(3.0,24.0){{\large(b)}}
\put(45.0,55.0){{\large(c)}} \end{overpic} FIG. 3:  (Color online) Schematic
of (a) a simple shear, (b) definitions of angles, and (c) sample trajectories
of Jeffery’s orbits. The time evolution of the orientation is illustrated by
blue trajectories on a unit sphere. The plots are made for an elongated object
with $B=0.9$.

To see the precise rotational dynamics, we then focus on the case of simple
shear. Let us fix the background flow as $\bm{u}^{\infty}=\gamma y\bm{e}_{1}$,
so that the background flow possesses uniform vorticity field
$\bm{\omega}^{\infty}=\nabla\times\bm{u}^{\infty}=-\bm{e}_{3}$ (Fig. 3a).

We then rewrite Eq. (30) by assuming that
$\dot{\bm{d}}^{\,\textrm{prop}}=\bm{0}$ for simplicity. With the polar
coordinates (see the schematic in Fig. 3b), we have

$\displaystyle\gamma^{-1}\dot{d}_{1}$ $\displaystyle=$
$\displaystyle(B+1)d_{2}/2-Bd^{2}_{1}d_{2}$ (31)
$\displaystyle\gamma^{-1}\dot{d}_{2}$ $\displaystyle=$
$\displaystyle(B-1)d_{1}/2-Bd_{1}d^{2}_{2}$ (32)
$\displaystyle\gamma^{-1}\dot{d}_{3}$ $\displaystyle=$ $\displaystyle-
Bd_{1}d_{2}d_{3}.$ (33)

These are further simplified by calculating
$\dot{d}_{3}=-\dot{\theta}\sin\theta$ and
$d_{1}\dot{d}_{2}-\dot{d}_{1}d_{2}=\dot{\phi}\sin^{2}\theta$ as

$\frac{d\theta}{dt}=\frac{\gamma}{4}B\sin 2\theta\sin
2\phi\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d\phi}{dt}=-\frac{\gamma}{2}\left(1-B\cos 2\phi\right).$ (34)

For a sphere ($B=0$), the object just rotates by the background angular
velocity $|\boldsymbol{{\it\Omega}}^{\infty}|=\gamma/2$. The second equation
is readily integrated and we obtain the periodic angular dynamics if $|B|<1$.
The polar angle is determined by the azimuthal angle dynamics, and by setting
the initial azimuthal angle as $\phi_{0}=0$, we have

$\displaystyle\tan\theta=\frac{Kc}{\sqrt{\cos^{2}\phi+c^{2}\sin^{2}\phi}},\leavevmode\nobreak\
\tan\phi=-c\tan\left(\frac{\gamma t}{c+c^{-1}}\right),$ (35)

where $K$ is a constant of the integral and acts as a constant of motion.
Indeed, $K$ is determined once the initial angles are set, and $K=0$
corresponds to a rotation along the equator, while $K\rightarrow\infty$ and
$K\rightarrow-\infty$ indicate spinning around the poles. Here,
$c=\sqrt{(1+B)/(1-B)}$ is therefore interpreted as an effective aspect ratio
from the Bretherton parameter. The orientation vector $\bm{d}$ traces a closed
orbit on the unit sphere and determined by the initial value of $K$. These
closed orbits are formed by infinitely many marginally stable periodic orbits
and are called degenerated. The period of one rotation,

$T=\frac{2\pi}{\gamma}\left(c+\frac{1}{c}\right)=\frac{4\pi}{\gamma\sqrt{1-B^{2}}},$
(36)

does not depend on $\theta$ but is determined by the shear strength and the
Bretherton parameter. This non-linear periodic motion is the Jeffery’s orbit,
and sample orbits are shown in Fig. 3c as closed paths on a sphere of the unit
radius with the Bretherton parameter $B=0.9$. Note that when $|B|$ exceeds 1,
the particle will align along a preferred orientation and no longer exhibit
periodic motion.

###  III.3 Hydrodynamic Symmetry

Because the shape only affects the tensors in Eq. (24), we can define the
hydrodynamic shape by using the symmetry of the resistance tensors, rather
than relying on the actual geometrical shape.

The symmetry arguments on the resistance tensors were introduced by Brenner
[54, 55], who followed the idea of hydrokinetic symmetry and argued for its
connection with the kinetic energy of an ideal irrotational fluid in the
nineteenth century [56, 57, 58].

We consider a transformation of the body-fixed frame $\\{\hat{\bm{e}}_{i}\\}$
into another body-fixed frame $\\{\hat{\bm{e}}^{\prime}_{i}\\}$. Let us denote
the translation by ${\bf A}=(a_{ij})\in\textrm{O}(3)$, which includes rotation
and reflection. Now we write the matrix representations of an $r$th-rank
tensor ${\bf X}$ for the body-fixed frames $\\{\hat{\bm{e}}_{i}\\}$ and
$\\{\hat{\bm{e}}^{\prime}_{i}\\}$ as $X_{i_{i}i_{2},\cdots,i_{r}}$ and
$X^{\prime}_{i_{i}i_{2},\cdots,i_{r}}$, respectively, and we obtain the
following relation between the two representations:

$X^{\prime}_{i_{1}i_{2}\cdots i_{r}}=a_{i_{1}j_{1}}a_{i_{2}j_{2}}\cdots
a_{i_{r}j_{r}}X_{j_{i}j_{2}\cdots j_{r}}.$ (37)

Similarly, we have the following relation between the matrix representations
of an $r$th-rank pseudotensor ${\bf Y}$ under the transformation of the body-
fixed frame:

$Y^{\prime}_{i_{1}i_{2}\cdots i_{r}}=\sigma a_{i_{1}j_{1}}a_{i_{2}j_{2}}\cdots
a_{i_{r}j_{r}}Y_{j_{i}j_{2}\cdots j_{r}},$ (38)

where we have introduced the determinant of the matrix
${\color[rgb]{0,0,0}\textrm{det}{\bf A}}=\sigma=\pm 1$.

If the shape of an object is unchanged under a transformation, we usually
argue that the object possesses the geometrical symmetry associated with the
transformation matrix. Alternatively, we now introduce the hydrodynamic
symmetry based on the invariance of the resistance tensors. In other words, if
all the resistance tensors inside $\mathcal{K}$ and $\mathcal{G}$ are
invariant under a change of the body-fixed frame denoted by ${\bf
A}=(a_{ij})\in\textrm{O}(3)$, then the object is considered to possess
hydrodynamic symmetry associated with ${\bf A}$. A transformation ${\bf A}$
induces geometrical symmetry and hydrodynamic symmetry for an object, but
these two symmetries do not necessarily coincide.

For example, if we take ${\bf A}$ as a rotational matrix around an axis by an
arbitrary angle, the geometrical symmetry associated with this transformation
is usually called an axisymmetric object, but more precisely this is a body of
revolution. In contrast, the hydrodynamic symmetry associated with the same
${\bf A}$ is known as helicoidal symmetry [54, 55], which includes a chiral
object, being distinct from an achiral body of revolution. Further details are
provided in Sec. IV.1.

##  IV Theoretical Extensions

In this section, we review some theoretical extensions beyond the periodic
Jeffery orbits of an axisymmetrical body.

###  IV.1 More General Shapes

The axisymmetric nature of the Jeffery equation (30) enables us to project the
angle dynamics onto a unit sphere on which the director vector $\bm{d}$ moves.
This reduction is also available for the helicoidal symmetry introduced in the
previous section. The helicoidal symmetry in the Stokes flow was first
introduced by Brenner [54, 55].

As expected from its name, a simple helix has been often described as a
helicoidal object. This simply allows hydrodynamical coupling between the
translation and rotation in the direction of the symmetrical axis, and
therefore a bacterial flagellum can corkscrew to propel the organism in
viscous environments [33]. This representation has been validated for a
rotating helical object such as a bacterial cell when averaged around the axis
of rotation [59, 60]. A simple helix, however, does not rigorously satisfy the
definition of helicoidal symmetry unless the number of turns is an integer.
Nonetheless, numerical computations have confirmed that a simple helix is well
approximated by a helicoidal object [61].

\begin{overpic}[width=199.16928pt]{fig_helicoid.eps} \put(49.0,64.0){{\large
drift force}} \put(-2.0,33.0){{\large drift torque}} \end{overpic} FIG. 4:
(Color online) Schematic of a helicoidal object under a simple shear. The
chirality-induced forces induce a translational drift (shown with a blue
arrow) and rotational drifts (shown with green arrows).

A chiral object generates a drift force when coupled to an external shear flow
(Fig. 4). This force moves the chiral object away from the flow plane, and
therefore one can use microfluidic devices to separate these particles
depending on the handedness of their chirality [62, 63, 64, 65, 66]. When the
chirality is not uniform along the axis of symmetry, such a chirality-induced
drift then induces torque to orient the object towards the background
vorticity vector.

With helicoidal symmetry, we may reduce the number of non-zero components in
the resistance tensors as we did through Eqs. (26)–(28) for a body of
revolution, which yields 13 non-vanishing constants of shape. Recently,
Ishimoto [31] revisited the notion of Brenner’s helicoidal symmetry and
derived the angular dynamics of objects in this symmetry class, motivated by
experiments of bacterial locomotion under a simple shear [67] (further
discussion of microswimmers in simple shear is provided in Sec. V.1). The
final form of the dynamics [31] results in an additional term on the right-
hand side of Eq. (30), given by

$C\epsilon_{ijm}(\delta_{jk}-d_{j}d_{k})E^{\infty}_{k\ell}d_{\ell}d_{m}.$ (39)

This vector is perpendicular both to $\bm{d}$ and the term with $B$ in Eq.
(30), and thus the trajectories of the director vector on a unit sphere move
across Jeffery’s orbits by this chirality-induced term when the object is
under a simple shear (Fig. 4). Here, the shape-dependent constant $C$ is a new
parameter representing the non-uniform distribution of the chiral coupling
along the axis. Constructive calculations of this parameter were also made
based on a detailed swimmer geometry [68], and recent studies experimentally
estimated the values of Ishimoto’s shape parameter for bacterial cells [69,
70] and swimming ciliates [71], and reported $C\sim O(10^{-2})$. This small
number, however, generates the torque to align the cell towards the background
vorticity vector in a simple shear, which we will see below.

The angle dynamics in a simple shear $\bm{u}^{\infty}=\gamma y\bm{e}_{1}$ are
written for $\theta$ and $\phi$ in Fig. 3b as

$\frac{d\theta}{dt}=h_{1}(\theta,\phi;\gamma,B,C),\leavevmode\nobreak\
\frac{d\phi}{dt}=h_{2}(\theta,\phi;\gamma,B,C),$ (40)

where the functions $h_{1}$ and $h_{2}$ are respectively given by

$\displaystyle\gamma^{-1}h_{1}$ $\displaystyle=$ $\displaystyle\frac{B}{4}\sin
2\theta\sin 2\phi+\frac{C}{2}\sin\theta\cos 2\phi,$ (41)
$\displaystyle\gamma^{-1}h_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(1-B\cos 2\phi)-\frac{C}{2}\cos\theta\sin 2\phi.$
(42)

These equations possess non-linear periodic orbits even for a non-zero $C$ if
$B=0$. The integral curve is given by

$C\sin 2\phi\sin^{2}\theta-2\cos\theta=\textrm{const}.$ (43)

If both $B$ and $C$ are non-zero, the degenerate periodic orbits become
broken. In particular, when $B^{2}+C^{2}<1$, the poles $\theta=0,\pi$ are only
the attractive/repulsive orientation, with the attracting pole being flipped
by the change in the sign of $C$. The orientation vector eventually reaches
the neighborhood of the stable pole and rotates with its attraction speed
being slower than the exponential function. The change of the polar angle
during one rotation near the poles is evaluated as $\Delta\theta\simeq
K\theta^{3}$ with a non-zero constant $K=K(B,C)$.

FIG. 5:  (Color online) Schematics of objects with helicoidal symmetry. (a) A
spheroidal cell body with a slender rod is an example of a body of revolution.
(b) A spheroidal cell body with a left-handed helical flagellum is well
represented by an object with helicoidal symmetry. (c) A spheroidal cell body
with two left-handed helical flagella. This possesses an additional
$\pi$-rotational symmetry around the axis perpendicular to $\hat{\bm{e}}_{1}$
and is an example model of a homochiral object. (d) A spheroidal cell body
with right-handed and left-handed helical flagella. This possesses an
additional reflectional symmetry in a plane perpendicular to the axis of
helicoidal symmetry, and an example model of a hetrochiral object. Figure
modified from Ishimoto [31] under the creative commons license,
http://creativecommons.org/licenses/by/4.0. (copyright 2020 by The Author(s))

By definition, every body of revolution is a helicoidal object, and thus the
helicoidal symmetries belong to a genuinely larger class of symmetry,
including the body of revolution (Fig. 5a,b). Ishimoto [31] also found that
the helicoidal symmetry contains two sub-classes of symmetry, called
homochiral and heterochiral objects. A homochiral object possesses an
additional $\pi$-rotational symmetry around an axis perpendicular to the
symmetry axis (Fig. 5c). This induces a fore-aft symmetric chirality
distribution, yielding zero chiral rotational drift but non-zero chiral
translational drift. An example is a simple helix whose rotational dynamics
are the same as the achiral Jeffery equation [61, 72]. Heterochiral objects
instead possess an additional mirror symmetry to the axis of symmetry, and
therefore the chiral-induced translational drifts cease for these objects, but
they still exhibit non-zero chiral rotational drifts (Fig. 5d). An example may
be a composition of the same two helices but with opposite chirality connected
in series. Kramel et al. [73] considered the dynamics of such particles in
turbulent flow to detect local swirling structures of the flow field.

The concept of helicoidal symmetry was extended to discrete rotational
symmetry. Indeed, Brenner [55] introduced the notion of helicoidal symmetry by
the invariance of the resistance tensors under $\pi/2$-rotational symmetry.
Ishimoto [26] confirmed that the dynamics of any object with $n$-fold
rotational symmetry $(n\geq 4)$ satisfy the same equations under a linear flow
as those for a helicoidal object. As in the continuous case, the rotational
dynamics are reduced to achiral Jeffery equations by an additional mirror
symmetry [74]. A complete classification of the translational and rotational
dynamics is provided using the Schoenflies notions in Ref. [26].

Another interesting class of symmetry in terms of hydrodynamic symmetry is
Lord Kelvin’s isotropic helicoid [57]. This hypothetical object possesses a
triaxial helicoidal symmetry, and its director dynamics are the same as that
of a simple sphere [55, 75] but with coupling between the translational and
rotational motions. Recently, Collins et al. [76] fabricated a 3D-printed
model and found that the particle predicted by Kelvin actually exists, though
the transition-rotation coupling is extremely small.

When an object has three-fold rotational symmetry, however, additional shape
parameters are needed to describe the motion dynamics in a linear flow. The
rotational dynamics are no longer projected onto the unit sphere but require
three angle variables, which may cause chaotic behaviors depending on the
shape parameters [26]. Further less symmetric cases with two-fold rotational
symmetry contain an important class of symmetry, that is, tri-axially
ellipsoidal particles [1, 77]. The angular dynamics of a tri-axially mirror-
symmetric particle are described by three shape parameters reflecting the
length scales of each axis [8, 55], and an ellipsoidal particle can also
exhibit chaotic rotational dynamics in a shear flow [78, 79].

###  IV.2 Asymptotic Jeffery’s Orbits

\begin{overpic}[width=241.84842pt]{fig_yaw.eps} \put(-3.0,37.0){{ (a)}}
\put(43.0,37.0){{ (b)}} \end{overpic} FIG. 6:  (Color online) (a) Swimming
trajectory of a human sperm head as an example of a yawing microswimmer traced
from video microscopic images in Ishimoto et al. [48]. (b) Schematic of a
model microswimmer under simple shear. Figures adapted from Walker et al. [80]
under the creative commons license,
http://creativecommons.org/licenses/by/4.0. (copyright 2022 by The Author(s))

In microswimming dynamics, swimming is usually achieved by non-reciprocal
deformations described by the scallop theorem [33]. In viscous environments,
swimming is typically very inefficient, and large portions of displacements
are canceled even with rapid oscillation, as illustrated by the large yawing
motion of sperm motility shown in Fig. 6a [48].

When two separable timescales exist in the dynamics, one may be more
interested in the slow dynamics after averaging out the fast timescale. Walker
et al. [80] introduced the classical multiple-scale expansions to derive the
asymptotic slow dynamics for rapidly yawing microswimmers in a shear flow.
Here, we provide the essential analyses in Walker et al. [80] for planar
swimming, focusing on a simple sinusoidal yawing motion (Fig. 6b). We denote
the position of the swimmer by $(x,y)$ and its angle from the
$\bm{e}_{1}$-axis by $\phi$. The swimmer is assumed to be self-propelled in
this orientation with a constant velocity $V$. Simple shear is induced in this
plane by $\bm{u}^{\infty}=\gamma y\bm{e}_{1}$, and we write the observed yaw
angle as a sum of the slow dynamics and fast sinusoidal oscillation in the
form of $\phi(t)=\overline{\phi}(t)+A\sin(\omega t)$, where $A$ is the
amplitude of the yaw. Assuming the passive part of the dynamics follows
Jeffery’s orbits (34), we may write the swimmer dynamics as

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle V\cos\phi+\gamma
y,$ (44) $\displaystyle\frac{dy}{dt}$ $\displaystyle=$ $\displaystyle
V\sin\phi,$ (45) $\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\gamma}{2}(1-B\cos 2\phi)+A\omega\cos(\omega t).$ (46)

Following the classical procedures of multiple-scale analysis, we introduce
the fast variable $T:=\omega t$ with $\omega\gg 1$ and the formal
transformation $t\mapsto(t,T)$ and write $x=x(t,T),y=y(t,T)$, and
$\phi=\phi(t,T)$. The angle evolution (46) is then rewritten as

$\frac{\partial\phi}{\partial t}+\omega\frac{\partial\phi}{\partial
T}=-\frac{\gamma}{2}(1-B\cos 2\phi)+A\omega\cos T.$ (47)

We then pose an asymptotic expansion for $\phi$ in terms of inverse series of
$\omega$ in the form of $\phi=\phi_{0}+\omega^{-1}\phi_{1}+\cdots$. The
leading-order problem for $\omega\gg 1$ becomes
$\partial_{T}\phi{{}_{0}=A}\cos T$; hence,
$\phi_{0}=\overline{\phi}_{0}(t)+A\sin T$, where $\overline{\phi}_{0}$ denotes
the slow dynamics at the leading order. The slow dynamics are determined by
the solvability condition that corresponds to the $2\pi$-periodicity in $T$.
We denote the $T$ average by $\langle\leavevmode\nobreak\ \rangle$ to obtain

$\frac{d\overline{\phi}{{}_{0}}}{dt}=-\frac{\gamma}{2}(1-B\langle\cos
2\phi_{0}\rangle),$ (48)

if the next-order dynamics are averaged. Similar analyses for the
translational dynamics enable us to derive the asymptotic behaviors of the
swimmer $(\overline{x}_{0},\overline{y}_{0},\overline{\phi}_{0})$ in the form

$\displaystyle\frac{d\overline{x}_{0}}{dt}$ $\displaystyle=$ $\displaystyle
V_{e}\cos\phi+\gamma\overline{y}_{0},$ (49)
$\displaystyle\frac{d\overline{y}_{0}}{dt}$ $\displaystyle=$ $\displaystyle
V_{e}\sin\phi,$ (50) $\displaystyle\frac{d\overline{\phi}_{0}}{dt}$
$\displaystyle=$ $\displaystyle-\frac{\gamma}{2}(1-B_{e}\cos
2\overline{\phi}_{0}),$ (51)

where $V_{e}=J_{0}(A)V$ and $B_{e}=J_{0}(2A)B$ are the effective swimming
speed and the effective Bretherton parameter for the asymptotic slow dynamics,
respectively, and $J_{0}(A)=\langle\cos(A\sin T)\rangle$ is the zeroth-order
Bessel function of the first kind. Comparing Eqs. (44)–(46) with (49)–(51)
shows that the slow dynamics satisfy the same Jeffery orbits with a
renormalized speed and shape parameter. With this sinusoidal yawing, the
effective shape of the particle approaches a sphere as the function $J_{0}(A)$
gradually damps to zero with oscillation in the increase of $A$. Further
generalizations of the yawing function are made in Ref. [80].

The simple example discussed above assumed that the angular dynamics follow
Jeffery’s orbits in the absence of the yawing motion. Gaffney et al. [81]
relaxed this assumption and generalized the asymptotic Jeffery’s orbits to a
wide class of shape-changing swimmers. They considered a drift-free swimmer
with a periodic shape gait in the plane of a shear flow. Let the shear rate
$\gamma$ and frequency of the shape deformation $\omega$ satisfy
$\omega\gg\gamma$, and then the average rotational motion of the swimmer is
given by a Jeffery’s orbit with a relative asymptotic error of
$O(\gamma/\omega)$. Here, the multiple-scale expansions are again performed in
the resistance tensors $\mathcal{K}_{ab}$ and $\mathcal{G}_{ajk}$, which may
be treated as functions of only the fast-time scale when represented in the
body-fixed frame. The asymptotic angular dynamics are

$\displaystyle\frac{d\overline{\phi}_{0}}{dt}=-\frac{\gamma_{e}}{2}\left[1-B_{e}\cos(2\overline{\phi}_{0}+2\phi_{c})\right],$
(52)

where $\gamma_{e}$ is the effective shear strength, $B_{e}$ is the effective
Bretherton parameter, and $\theta_{c}$ represents the phase shift.

The asymptotic Jeffery’s orbits found in the slow dynamics are essential
features of the microswimmer dynamics. Indeed, numerical simulations of a
sperm cell and flagellate in a shear flow well reproduced Jeffery’s orbits on
average [82, 80]. These results include a rigid body with an asymmetric shape,
and thus the dynamics trace the Jeffery orbits. With a boomerang shape with
two rods, $B_{e}$ may exceed 1 so that the swimmer orientation can be aligned
rather than exhibit rotation[83].

Another type of timescale separation can arise when the swimmer, such as a
bacterium cell, is rapidly spinning along its elongated direction. Dalwadi et
al. [84, 85] considered a helicoidal squirmer with an arbitrary constant
surface slip swimming under a background shear flow, $\bm{u}^{\infty}=\gamma
y\bm{e}_{1}$. We take the axis of the helicoidal symmetry as
$\bm{d}=\hat{\bm{e}}_{1}$, as in Fig. 1, and denote the self-propelled linear
velocity by $\bm{V}$ and the angular velocity by
$\boldsymbol{{\it\Omega}}^{\textrm{prop}}={\it\Omega}_{\|}\hat{\bm{e}}_{1}+{\it\Omega}_{\perp}\hat{\bm{e}}_{2}$.
After introducing the appropriate Euler angles $(\theta,\psi,\phi)$, the
angular dynamics of the swimmer are given by

$\displaystyle\frac{d\theta}{dt}$ $\displaystyle=$
$\displaystyle{\it\Omega}_{\perp}\cos\psi+h_{1}(\theta,\phi;\gamma,B,C),$ (53)
$\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle{\it\Omega}_{\perp}\csc\theta\sin\psi+h_{2}(\theta,\phi;\gamma,B,C),$
(54) $\displaystyle\frac{d\psi}{dt}$ $\displaystyle=$
$\displaystyle{\it\Omega}_{\|}-{\it\Omega}_{\perp}\cot\theta\sin\psi+h_{3}(\theta,\phi;\gamma,B,C,D).$
(55)

Here, the functions $h_{1},h_{2}$, and $h_{3}$ represent the shear-induced
dynamics in each angle variable; $h_{1}$ and $h_{2}$ are provided in Eqs. (41)
and (42); and $h_{3}$ is given by

$\gamma^{-1}h_{3}=-\frac{B}{2}\cos\theta\cos
2\phi+\frac{C}{2}\cos^{2}\theta\sin 2\phi+\frac{D}{2}\sin^{2}\theta\sin
2\phi.$ (56)

${\it\Omega}_{\|}$ affects only the $\psi$ angle, the spinning angle, and when
${\it\Omega}_{\perp}=0$, the equations are closed in the angles $\theta$ and
$\phi$, leading to the dynamics of a helicoidal object , Eqs.(40) . Indeed,
the functions $h_{1}$ and $h_{2}$ contain the shape parameters $B$ and $C$,
whereas the new chirality-induced shape parameter $D$ should appear in the
function $h_{3}$.

Let us introduce the ratio of the two components of the angular velocity $p$
as ${\it\Omega}_{\perp}=p\it\Omega_{\|}$. When $p=0$, the self-propulsion only
induces spinning along the symmetry axis, as often assumed for a model
bacterium [33, 86]. In contrast, when $p$ increases as $p\rightarrow\infty$,
the swimmer trajectory approaches a circle in a plane, as observed in marine
sperm cells [87, 88, 89].

In the absence of a background flow, the trajectory of such a swimmer traces a
single helix, and this behavior is expressed by the helix theorem[20]. These
helical trajectories are reported in many biological microswimmers, such as
microalgae [90, 91, 92] and sperm cells [93], as well as artificial swimmers
[94]. Direct integration of Eqs. (53)–(55) after setting $h_{1}=h_{2}=h_{3}=0$
allows one to derive the radius and pitch of the helical trajectories as

$r_{\textrm{helix}}=\frac{{\it\Omega}_{\perp}}{{\it\Omega}_{\|}^{2}+{\it\Omega}_{\perp}^{2}}V\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
b_{\textrm{pitch}}=\frac{2\pi{\it\Omega}_{\|}}{{\it\Omega}_{\|}^{2}+{\it\Omega}_{\perp}^{2}}V.$
(57)

Here we have taken the swimming direction as $\hat{\bm{e}}_{1}$ so that the
swimming speed $V$ is obtained from $\bm{V}=V\hat{\bm{e}}_{1}$.

Now we consider the timescale separation of self-induced spinning and shear-
induced rotation, assuming ${\it\Omega}_{\|}=O({\it\Omega}_{\perp})\gg\gamma$.
The fast timescale is denoted as
$T={\color[rgb]{0,0,0}(}{\it\Omega}_{\|}^{2}+{\it\Omega}_{\perp}^{2})^{1/2}t$,
where $t$ denotes the slow timescale. As in the rapid yawing case, we expand
the angle variables by the inverse series of
$\lambda:=(1+p^{2}){{}^{1/2}}{\it\Omega}_{\|}/\gamma\gg 1$ and use similar
notation to perform multiple-scale analysis, yielding the slow dynamics with
leading-order correction as

$\displaystyle\frac{d\overline{\theta}_{0}}{dt}$ $\displaystyle=$
$\displaystyle
h_{1}(\overline{\theta}_{0},\overline{\phi}_{0};\gamma,\hat{B},\hat{C}),$ (58)
$\displaystyle\frac{d\overline{\phi}_{0}}{dt}$ $\displaystyle=$ $\displaystyle
h_{\color[rgb]{0,0,0}2}(\overline{\theta}_{0},\overline{\phi}_{0};\gamma,\hat{B},\hat{C}),$
(59) $\displaystyle\frac{d\overline{\psi}_{0}}{dt}$ $\displaystyle=$
$\displaystyle
h_{\color[rgb]{0,0,0}3}(\overline{\theta}_{0},\overline{\phi}_{0};\gamma,\hat{B},\hat{C},\hat{D}).\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (60)

This set of equations (58)–(60) completely reproduce the original angular
dynamics (53)–(55) without the self-propelled rotation terms, after
introducing the effective shape parameters as

$\hat{B}=\frac{(2-p^{2})B}{2(1+p^{2})},\leavevmode\nobreak\
\hat{C}=\frac{C+p^{2}D}{(1+p^{2})^{3/2}},\leavevmode\nobreak\
\hat{D}=\frac{3p^{2}C+(2-p^{2})D}{2(1+p^{2})^{3/2}}.$ (61)

This result validates the descriptions of the helicoidal object (40) by
introducing the effective shape parameters $\hat{B}$ and $\hat{C}$, depending
on the rapid spinning behaviors. Indeed, the asymptotic dynamics recover the
chiral Jeffery equations (40) and (42) as $p\rightarrow 0$. In the opposite
limit of $p\rightarrow\infty$, the trajectory loses its chirality effects as
it traces Jeffery’s orbits but with $\hat{B}\rightarrow-B/2$. Similar
asymptotic behaviors are also analyzed for the translational dynamics [84,
85].

###  IV.3 Breakdown of Degenerate Orbits: Inertia, Rheology, and Other Factors

The breakdown of the degeneracy in Jeffery’s orbits has been studied from
various perspectives, which were reviewed by Leal[95] in the early days of
research in this topic. The effects of small amounts of fluid inertia have
been investigated by perturbation theory with a small Reynolds number, $Re$,
and it has been found that the inertia induces a drift towards rotation in a
flow-gradient plane ($\bm{e}_{1}\bm{e}_{2}$ plane in Fig. 3a) as a stable
limit cycle, whereas the poles become repellers [96]. In addition to the fluid
inertia, analysis of the effects of particle inertia, with its magnitude being
characterized by the Stokes number, $St$, found that the in-plane rotation can
be stabilized or destabilized to flip the stability between the in-plane
rotation and poles, while these depend on the amount of inertia and shape of
the spheroid [97, 98, 99, 100]. It was also found that with an increase of the
inertial effect, the rotational dynamics can exhibit a transition to chaos
[101, 102].

Degeneracy in the Jeffery orbits can also be broken by non-Newtonian rheology
[103, 104]. A classical analysis by Leal [104] showed a similar drift effect
on a fiber in a viscoelastic (second-order) fluid, leading to the orientation
of particles away from in-plane rotation. Several theoretical and
computational analyses on the effect of the viscoelastic model have been
performed with different shapes, rheological models, and parameters [105, 106,
107, 108, 109, 110]. Interested readers are referred to a recent review paper
by Li et al. [111] and references therein.

The breakdown of the degenerate Jeffery’s orbits also occurs by particle
deformability and has been intensively studied for a droplet and a capsule
[112, 113, 114, 115], as well as flexible fibers [116, 117, 118, 119, 120].
The effects of the slip-boundary conditions have been numerically investigated
by Zhang et al.[121], who found that the orientation dynamics follow the
Jeffery orbit under a certain symmetry, whereas recent simulations [122, 123]
report the alignment of a particle in a flow plane.

##  V Microswimmers in Flows

In this section, we review theoretical models and experimental observations in
the studies of the motion of a single microswimmer in simple to complex
background flow environments.

###  V.1 Simple Shear

We start with a self-propelled particle in a simple shear flow, and many parts
of this analysis are shown in the previous sections. Not only a rigid body,
but shape-changing bodies such as sperm cells and Leishmania have been
computationally studied and found to periodically rotate along Jeffery orbits
but with a drift due to the background shear [82, 124, 80]. When a swimmer
possesses chirality, such as bacterial flagella, the additional shear-induced
drift force and torque orient the swimmer towards the background vorticity
vector, which is perpendicular to the flow plane (Sec. IV.1). Bacterial
swimmers, therefore, move with non-zero perpendicular velocity, and this
biased locomotion is known as bacterial bulk rheotaxis [67, 69, 125, 126, 70].

As introduced in Sec. IV.2, a squirmer swimmer with a time-constant surface
profile in general exhibits a helical path. Bearon [90] studied a similar
mathematical model with helical trajectories in the presence of gravitational
torque and a simple shear flow to understand the dynamics of swimming
microalgae such as Chlamydomonas and Heterosigma akashiwo under a turbulent
flow. The study of spherical objects found a novel stable equilibrium in a
vertical shear flow, suggesting a robust upward swimming due to the helical
path.

\begin{overpic}[width=241.84842pt]{fig_rheo_theor.eps}
\put(3.0,31.0){{\large(a)}} \put(55.0,31.0){{\large(b)}} \put(61.0,23.0){{flow
direction}} \end{overpic} FIG. 7:  (Color online) Schematic of a rheotactic
microswimmer under a simple shear flow near a flat wall. (a) Schematic of the
swimmer behavior and flow configuration. (inset) Side view of the swimmer
facing toward the wall with its director being denoted by a unit vector
$\bm{d}$. (b) Projection onto the wall plane ($\bm{e}_{1}\bm{e}_{2}$-plane).
The in-plane angle $\Theta$ is taken from the $\bm{e}_{1}$-axis (flow
direction). By this weather-vane mechanism, a microswimmer with a wall-
attacking orientation turns its swimming direction upstream.

A simple shear flow is also ubiquitous in the vicinity of an external boundary
such as a flat wall (Fig. 7). This geometrical confinement is almost
unavoidable for laboratory experiments and also plays significant roles in
microbial habitats [11] and reproduction processes, in particular, for
internal fertilizers[127, 128]. Indeed, it is well known that these cells
accumulate near boundaries due to contact force and hydrodynamic interactions,
as intensively studied in bacteria [129, 130, 131], sperm [132, 133, 134], and
squirmer models [40, 42]. Gravity also contributes to the sedimentation of
artificial microswimmers and confines them in the vicinity of a bottom
surface.

\begin{overpic}[width=526.37598pt]{fig_rheo_exp.eps}
\put(52.0,88.0){{\large(d)}} \put(52.0,76.0){{\large(e)}}
\put(85.0,88.0){{\large(f)}} \put(-0.5,61.0){{\large(g)}}
\put(18.0,61.0){{\large(h)}} \put(52.0,61.0){{\large(i)}}
\put(-0.5,41.0){{\large(j)}} \put(-1.0,29.0){{\large(k)}}
\put(-0.5,18.0){{\large(l)}} \end{overpic} FIG. 8:  (Color online) (a–c)
Sample trajectories of E. coli bacteria swimming in the vicinity of a flat
wall. Superimposed snapshots are shown to illustrate the trajectories of
individual cells, which are shown by dotted arrows. (a) Bacterial cells swim
in circles due to hydrodynamic interactions between the cell and wall in the
absence of a background flow. (b) Flow induction (from the top of the figure,
shown in blue arrows) triggers bacterial rheotactic behavior, resulting in
upstream migration. (c) At a higher shear rate, cells are swept away by the
flow. Figure adapted from Kaya et al. [135] with permission (copyright 2012 by
Elsevier). (d–f) Sample trajectories of M. pneumoniae cells under a gliding
motility on a glass substrate. Cell trajectories are in the (d) absence and
(e) presence of a flow. The blue arrows indicate the flow direction from right
to left. The positions at the initial time are denoted by colored circles. (f)
Time-lapsed images of a single cell exhibiting rheotactic behavior. Figure
adapted from Nakane et al. [136] with permission under the creative commons
license, http://creativecommons.org/licenses/by/4.0 (copyright 2022 by The
Author(s)) . (g,h) Experimental observations of rheotactic human sperm cells.
(g) A sudden change of swimming orientation by a reversal of the flow
direction is shown by superimposed images of a swimming cell at different
times. The flow is reversed at $t=0$. (h) Similar experimental observations,
but with different viscosity of the swimming medium, show slower responses of
the cells. Figure adapted from Kantsler et al.[137] with permission under the
creative commons license, http://creativecommons.org/licenses/by/4.0
(copyright 2014 by The Author(s)) . (i) Sample trajectory of a rheotactic T.
pyriformis cell with the background flow moving from left to right, whereas
the cell swims upwards along the bottom wall. The scale bar is 200 $\mu$m.
Figure adapted from Ohmura et al. [71] with permission under the creative
commons license, http://creativecommons.org/licenses/by/4.0 (copyright 2021 by
The Author(s)) . (j–l) Various behaviors of a catalytic Janus particle near a
flat wall under a flow with different strengths of shear rate. The flow
direction is indicated by the orange arrow at the bottom (from left to right).
Each row indicates (j) snapshots of optical images, (k) sample trajectories of
Janus particles, and (l) distributions of the directions of motion. At the
intermediate shear strength, particles can migrate upstream, while the high
shear flow sweeps the particles away from the near-wall region. Figure adapted
with permission from Sharan et al.[138] (copyright 2022 by the American
Chemical Society).

A notable phenomenon of a microswimmer near a wall in the presence of a
background shear is the swimming behavior against the flow, which is known as
(surface) rheotaxis. It has been reported that many microswimmers exhibit this
upstream migration ranging from biological cells to artificial active colloids
and that these behaviors are often explained by simple mechanical models (Fig.
8). Theoretical building blocks for the various hydrodynamic and mechanical
interactions are examined in Mathijssen et al. [68]. Here, however, we only
sketch out how these rheotactic behaviors emerge from the shear-induced
hydrodynamic interactions, using a simple model by Kantsler et al.[137].

We consider an object whose rotational dynamics obey the Jeffery equation. The
presence of a rigid flat wall near Jeffery orbits yields only a slight
alternation on the rotational dynamics for a prolate spheroid, although it is
more significant for an oblate spheroid [139]. We let $\bm{d}$ be the unit
vector for the swimmer and consider a simple shear $\bm{u}^{\infty}=\gamma
z\bm{e}_{x}$ with $z=0$ being the flat wall. We assume that the attack angle
towards the flat wall $\cos^{-1}(\bm{d}\cdot\bm{e}_{3})$ is kept constant in
time based on experimental observations, and it reflects steric, hydrodynamic,
and other mechanical interactions. The Jeffery equation is then reduced to

$\displaystyle\gamma^{-1}\dot{d}_{1}$ $\displaystyle=$
$\displaystyle\frac{B}{2}d_{3}(1-2d_{1}^{2})+d_{3}+\lambda d_{1}$ (62)
$\displaystyle\gamma^{-1}\dot{d}_{2}$ $\displaystyle=$ $\displaystyle-
Bd_{1}d_{2}d_{3}+\lambda d_{2},$ (63)

where $\lambda$ is a constant determined by the constraint on the attack
angle. Substituting the value of $\lambda$ further simplifies the angle
dynamics to

$\dot{d}_{1}=\gamma\frac{(1+B)}{2}\frac{d_{2}^{2}d_{3}}{1-d_{3}^{2}},\leavevmode\nobreak\
\dot{d}_{2}=-\gamma\frac{(1+B)}{2}\frac{d_{1}d_{2}d_{3}}{1-d_{3}^{2}}.$ (64)

Introducing an in-plane angle $\Theta$ as
$d_{1}=(1-d_{2}^{3})^{1/2}\cos\Theta$ and
$d_{2}=(1-d_{2}^{3})^{1/2}\sin\Theta$ (see also Fig. 7b), the in-plane
dynamics (64) are thus written as

$\displaystyle\frac{d\Theta}{dt}=-\alpha\sin\Theta,$ (65)

where $\alpha=(\gamma/2)(1+B)d_{3}$. For a typical object, $|B|<1$ is
satisfied, and thus the sign of $\alpha$ is determined by the sign of $d_{3}$.
When the swimmer’s head is facing towards the wall boundary as shown in Fig.
7a, according to Eq. (65), the downstream direction ($\Theta=0$) becomes
unstable and the in-plane orientation eventually evolves to the upstream
direction ($\Theta=\pi$). Note also that the rheotactic response is faster for
an elongated object by an increase of $B$, but even a spherical object ($B=0$)
can exhibit the same rheotactic response. The hydrodynamic interactions that
mechanically cause the rheotactic behavior of microswimmers are sometimes
referred to as weather-vane mechanisms. In the above arguments, note that we
have neglected hydrodynamic wall interactions, steric interactions, and chiral
shape effects.

Surface rheotaxis has been studied in swimming bacteria [140, 135, 68]. Kaya
et al. [135] experimentally demonstrated surface rheotaxis in E. coli (Fig.
8a–c). The bacterial cells swim in circles near a wall boundary via the
hydrodynamic torque generated by their spinning flagella [86] in the absence
of a background flow (Fig. 8a). When external flow is induced, the bacterial
cells turn upstream and exhibit surface rheotaxis at the moderate shear rate
(Fig. 8b). At a higher shear rate, however, bacterial cells are swept away by
the external flow (Fig. 8c). Many bacteria can migrate on the substrate
without using flagella but with other short and thin appendages called type-IV
pili. The upstream migration by pili-mediated motile bacteria has been
reported in various species such as Pseudomonas aeruginosa[141], Acidovorax
citrulli[142], and Xylella fastidiosa[143]. Another type of gliding motility
on the substrate is known in Mycoplasma bacteria, and they also exhibit
upstream migration [144, 136]. Fig. 8d–f show the experimental observation of
Mycoplasma pneumoniae by Nakane et al. [136]. During the gliding motility,
cells adhere to the substrate and exhibit membrane protrusion in the leading
“edge,” forming a wall-attack configuration as in Fig. 7a. Sample trajectories
and time-lapsed images of M. pneumoniae cells in Fig. 8e and f show cell
migration towards the right, facing upstream. This bacterial rheotaxis is also
biologically relevant to understanding the invasion of infectious bacteria in
medical devices and within organs.

Sperm rheotactic behavior was first reported in the 1870s [145] for mammalian
sperm, and its mechanisms were debated for more than a century. Bretherton and
Rothschild [146] examined sperm rheotaxis, but they could not explain the
rheotactic behavior of live cells. More recently, however, hydrodynamic
interaction has successfully explained mammalian sperm rheotactic behaviors
[147, 137, 89, 148, 149]. Fig. 8g and h reproduce images from Kantsler et al.
[137] that show superimposed images and sample trajectories of rheotactic
human sperm cells in different media. Experiments with calcium imaging
analysis also support the mechanical origin of this phenomenon rather than the
hypothesis that cells sense the flow and actively modulate their motility
[150]. The recent development of microfluidics device further advances
quantitative analysis and sperm sorting based on sperm rheotaxis [151, 152,
153, 154, 155]. Bukatin et al. [156] reported two modes of rheotactic turning,
suggesting the roles of flagellar elasticity [156]. These elastic properties
are known to be important for prompting collective cell behavior [157, 158].
Recently, collective sperm rheotaxis has been reported, such as in chicken
sperm [159] and bull sperm [160], suggesting swimming abilities are enhanced
when individual sperm are part of a group.

Other biological cells are known to exhibit surface rheotaxis. Ohmura et al.
[71] recently found that the swimming ciliate Tetrahymena pyriformis exhibits
surface rheotaxis, and their experimental and numerical observations suggest
the kinesthetic sensing system of cilia. Fig. 8i shows a sample trajectory of
T. pyriformis under a simple shear in the vicinity of a bottom wall, as
reported in Ohmura et al. [71]. Attachments of the cilia to the bottom
substrate alter the ciliary beating, resulting in asymmetric propulsion that
maintains the wall-attacking orientation of the cell. Rheotactic behaviors are
also reported in a crawling ciliate, Uronychia setigera [161], that employs
its cilia for locomotion on the substrate. Upstream swimming is also known in
C. elegans [162, 163]. In this review, we have highlighted the physical and
hydrodynamic mechanisms of the rheotaxis; however, note that biological cells
usually possess several mechanosensing channels and can respond to mechanical
cues. Indeed, Dictyostelium rheotaxis is dependent on sensory response [164].
The mechanisms underlying experimental observation should be carefully
analyzed.

Surface rheotaxis is also realized in synthetic active colloids and
microrobots, such as light-activated colloidal particles[165], half-coated
catalytic Janus particles [42, 166, 138], self-propelled Au-Pt nanorods [167,
168, 169], and active droplets [170, 171]. Fig. 8j–l, adapted from Sharan et
al. [138], show the motions of catalytic Janus particles under a background
flow near a plane boundary. The panels represent experimental optical images
(Fig. 8j), sample trajectories (Fig. 8k), and the distribution of the motion
direction (Fig. 8l) under various shear rate strengths. By considering the
surface velocity slip, gravity, and surface repulsion, their numerical
simulations of a squirmer model well explain the various rheotactic behaviors.
Theoretical investigations for the rheotactic response of microswimmers have
been performed for some simple mathematical models, such as a squirmer [42,
172, 173] and a three-sphere model [174]. The rheotactic behavior is affected
by the flow field that the swimmer itself creates because this field alters
the near-surface behavior.

###  V.2 Poiseuille Flow

\begin{overpic}[width=241.84842pt]{fig_pois_theor.eps} \end{overpic} FIG. 9:
(Color online) Schematic of a microswimmer in planar Poiseuille flow.

Next, we proceed to a microswimmer in Poiseuille flow with quadratic flow and
a unidirectional velocity profile (Fig. 9). We assume that the microswimmer is
sufficiently small compared to the width of the chamber so that the background
flow around the swimmer is well approximated by the linear flow. As shown in
Fig. 9, we consider a planar Poiseuille flow with a chamber width $h$ and
focus on the swimmer movements in the $\bm{e}_{1}\bm{e}_{2}$ plane, where the
flow and chamber direction are taken as the $\bm{e}_{1}$-axis. The background
flow field is then written as $\bm{u}^{\infty}=\gamma(h^{2}-y^{2})\bm{e}_{1}$,
with $\gamma$ representing the strength of the flow. Let the model
microswimmer move to the orientation vector $\bm{d}=(\cos\theta,\sin\theta)$
with a constant speed $V$ and assume that the orientation dynamics follow the
Jeffery equation with the local linear flow. The evolutions of the position
and the orientation of the swimmer are then written as

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
V\cos\theta+\gamma(h^{2}-y^{2}),$ (66) $\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle V\sin\theta,$ (67) $\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\gamma y(1-B\cos 2\theta),$ (68)

where $B$ is the Bretherton parameter. We hereafter set $h=1$.

\begin{overpic}[width=113.81102pt]{fig_cont00.eps} \put(-6.0,95.0){{(a)}}
\end{overpic}\begin{overpic}[width=113.81102pt]{fig_cont09.eps}
\put(-6.0,95.0){{(b)}} \end{overpic}

FIG. 10:  (Color online) Contour plots of the constants of motion in the
$(\theta-y)$-phase space. (a) Spherical swimmer ($B=0$) and (b) elongated
swimmer ($B=0.9$), where the flow strength is set as $\gamma=8$ and the length
scale and timescale are non-dimensionalized by $h=V=1$. The separatrices are
shown with thick red lines.

The system above is closed in $y$ and $\theta$, and the constant of motion is
obtained by directly solving Eqs. (67) and (68) [175] as

$H=\frac{\gamma}{2V}y^{2}+g(\theta;B),$ (69)

where the function $g$ is given by

$g(\theta;B)=\tanh^{-1}\left(\sqrt{\frac{2B}{1+B}}\cos\theta\right)/\sqrt{2B(1+B)}$
(70)

for $B>0$. When $B=0$, $H$ becomes $H=\frac{\gamma}{2}y^{2}+\cos\theta$, and
this coincides with the Hamiltonian of the pendulum equation [176]. The form
(70) holds even for an oblate swimmer with $B<0$ if an appropriate branch is
taken. The motion in the $\theta-y$ phase plane is thus confined to the
contour line of $H$, which is shown in Fig. 10 for a spherical swimmer $(B=0)$
and an elongated swimmer $(B=0.9)$ with the flow strength $\gamma=8$. The
behaviors of the swimmers are characterized by inner and outer regions divided
by the separatrices shown with red thick curves in Fig. 10. The swimmer in the
inner region faces upstream with oscillation within the chamber (“swinging”),
whereas the swimmer monotonically rotates in one direction (“tumbling”) in the
outer region.

If the swimmer moves out of the flow plane, the dynamics are then
characterized by two constants of motion $\kappa_{1}(\theta,\phi;B)$ and
$\kappa_{2}(\theta,\phi;B)$, where $\phi$ is the angle made by the swimmer
orientation $\bm{d}$ and the $\bm{e}_{1}\bm{e}_{2}$ plane [175]. The motions
in the 3-dimensional $\theta-\phi-y$ space are confined to the intersections
of the isoplanes of $\kappa_{1}$ and $\kappa_{2}$. Similar analyses were made
for a microswimmer in a pipe with an ellipsoidal cross section, but the
constant of motion was only obtained for a spherical swimmer in a cylindrical
pipe [175].

\begin{overpic}[width=483.69684pt]{fig_pois_exp.eps}
\put(3.0,62.0){{\large(a)}} \put(52.0,57.0){{motile}} \put(53.0,43.0){{non-}}
\put(52.0,40.0){{motile}} \put(54.0,62.0){{\large(b)}}
\put(3.0,32.0){{\large(c)}} \put(3.0,18.0){{\large(d)}} \end{overpic} FIG. 11:
(Color online) Experimental observations of microswimming in Poiseuille flow.
(a) Superimposed snapshots of motile and immotile T. brucei cells in a
microfluidic channel, in which flow is induced from left to right. A motile
cell exhibits a swinging migration (upper panel) with its position moving from
left to right. A non-motile cell, however, does not move in the direction
perpendicular to the flow. Figure adapted from Uppaluri et al. [177] with
permission (copyright 2012 by Elsevier). (b–d) Sample trajectories of motile
and non-motile C. reinhardtii cells in a microchannel. The rightward-plane
Poiseuille flow is generated through a microfluidic device. Figures adapted
from Omori et al. [178] with permission under the creative commons license,
http://creativecommons.org/licenses/by/4.0 (copyright 2022 by The Author(s)) .
(b) Superimposed snapshots of a motile C. reinhardtii cell that turns upstream
(upper panel), whereas the non-motile cell only rotates with a constant height
in the chamber (lower panel). (c) Sample trajectory of the movement of the
cell. (d) Schematic of trajectory and orientation of a motile C. reinhardtii
cell.

The motility-induced swinging behavior was experimentally confirmed by a
Typanosoma cell, a motile unicellular parasite [177]. In Fig. 11a,
superimposed snapshots of both motile and non-motile T. brucei cells are
adapted from Ref. [177]. The upper panel shows the oscillating swinging
behavior in the microchannel under Poiseuille flow from the left to the right
with its orientation heading upstream. In the lower panel, in contrast, a
sample trajectory of a non-motile T. brucei cell illustrates a height relative
to the channel walls, showing a tumbling motion.

Experimental studies of microswimmers in Poiseuille flow were also conducted
for bacteria [179, 180]. Rusconi et al. [179] showed a flow-induced drop in
the population of Bacillus subtilis in the middle region. While flow-induced
trapping in the middle of the chamber was found in the numerical simulation
with negligibly small rotational Brownian noise, in real experiments, the
noise caused jumping across the separatrices, leading to depletion of the
population. Junot et al. used smoothly swimming mutants of E. coli bacteria to
study their 3-dimensional motions in a planar Poiseuille flow. They found that
the theoretical model based on the Jeffery equation can predict the swimmer
dynamics but only for a short time due to rotational Brownian noise.

Similar experimental observations were performed for several motile
phytoplankton. Barry et al. [181] studied the cell population of four
phytoplankton species, Heterosigma akashiwo, Amphidium carterae, Dunaliella
tertiolecta, and Chlamydomonas reinhardtii, in Poiseuille flow. They reported
that the first two species with an elongated flagellum for swimming exhibited
a depletion in the weak-shear region and the last two species with two short
flagella for breaststroke swimming, in contrast, accumulated in the middle of
the channel. They concluded that the effective hydrodynamic aspect ratio
determined these different behaviors and also suggested a possibility of
active cell response to the accumulation behavior.

More recently, Omori et al. [178] studied the motion of C. reinhardtii cells
and observed a similar central accumulation of the cell population in
Poiseuille flow (Fig. 11b–d). The upper panel of Fig. 11b shows the
reorientation of a motile cell towards upstream with accumulation in the
center of the channel. This is in contrast to the behavior of the non-motile
cell shown in the lower panel, which stayed in the same position relative to
the channel wall while exhibiting a tumbling motion. A sample trajectory is
shown with its long-time behavior (Fig. 11c) and a schematic illustration
(Fig. 11d).

To explain the mechanism of the swimmer’s central accumulation with upstream
orientation, Omori et al. [178] proposed a mechanical coupling between the
flow and the unsteadiness of the swimming. Considering time-periodic
deformation $V=V(t)$ and $B=B(t)$, they numerically found that hydrodynamic
interactions caused the microswimmers to turn upstream and accumulate at the
center of the channel. This breakdown of degenerate orbits in the shape space
was then theoretically formulated by using the multiple-scale analysis of
Walker et al. [182].

We let $\omega$ be the beat frequency of the deformation and assume that $B$
and $V$ are periodic functions with this frequency. Introducing the new
variable as $z(t)=y(t)/\omega^{1/2}$ and rescaling the equation in the form

$\displaystyle\dot{z}=\omega^{1/2}V(\omega t)\sin\theta,\leavevmode\nobreak\
\dot{\theta}=\gamma\omega^{1/2}z(1-B(\omega t)\cos 2\theta),$ (71)

Walker et al. performed a multiple-scale analysis for the natural fast
timescale $T:=\omega t$ and intermediate timescale $\tau:=\omega^{1/2}t$ with
formal transformations of $z(t)\mapsto z(T,\tau,t)$ and
$\theta(t)\mapsto\theta(T,\tau,t)$, together with

$\frac{d}{dt}\mapsto\frac{\partial}{\partial
t}+\omega^{1/2}\frac{\partial}{\partial\tau}+\omega\frac{\partial}{\partial
T}.$ (72)

The time evolution of the quantity $H$ in the slow dynamics is then considered
with an expansion in an inverse series of $\omega^{1/2}$ as
$H=H_{0}+\omega^{-1/2}H_{1}+\omega^{-1}H_{2}+\cdots$. The zeroth-order
quantity is then given by

$H_{0}=\frac{\gamma}{2\langle V\rangle}z^{2}_{0}+g(\theta_{0},\langle
B\rangle),$ (73)

where $z_{0}=z_{0}(\tau,t)$ and $\theta_{0}=\theta_{0}(\tau,t)$ are
independent of the fast timescale and the bracketing
$\langle\leavevmode\nobreak\ \rangle$ denotes the average over the fast
timescale $T$. The time evolution with leading-order correction is then
obtained as

$\frac{dH_{0}}{dt}=\gamma f(H_{0})W$ (74)

with a non-positive function $f(H_{0})$ that depends on the swimmer’s speed
and shape only via their fast-time averages $\langle V\rangle$ and $\langle
B\rangle$. The quantity $W$ is defined as $W=\langle I(B-\langle
B\rangle)\rangle$, where $I(T)=\int_{0}^{T}V(\tilde{T})-\langle
V\rangle\,d\tilde{T}$. Thus, if either the swimmer shape or swimmer speed is
constant in time, $W$ vanishes and $H_{0}$ remains a constant of motion,
preserving the periodic orbits in the phase space. When $W\neq 0$, however,
$H_{0}$ monotonically increases or decreases depending on the sign of $W$. For
swimming C. reinhardtii, due to a finite phase shift between the time
oscillation of $V$ and $B$, $H_{0}$ is indeed decreasing in time, and
therefore the swimming orientation and position are attracted by the point of
$(\theta,y)=(\pi,0)$, being oriented against the flow at the channel center.

Unsteady swimming is not the only factor that breaks the degenerate periodic
orbit. One possible factor is the effect of inertia, and Choudhary et al.
[183] numerically studied a spherical active particle in Poiseuille flow and
showed that inertia breaks the conservational structure and leads to
bifurcation in the phase space. Another factor is the hydrodynamic wall
interaction, which depends on the swimming pattern of the swimmer as well as
its shape. Zöttl & Stark [176] found a puller-type spherical swimmer with a
constant velocity and stresslet accumulates at the channel center with its
orientation facing upstream. A similar central focus for a puller swimmer was
reported in numerical simulations of spheroidal squirmers [184, 185], while
these simple time-averaged parameterizations need to be derived from faster-
timescale temporal deformation for a largely shape-changing swimmer [186].

For a spherical swimmer, the constant of motion $H$ is indeed a Hamiltonian
and the swimmer’s unsteadiness may cause Hamiltonian chaos [187]. A similar
Hamiltonian structure is found in the 2-dimensional swimmer model in a simple
shear near a planar wall [173], and the swimmer unsteadiness leads to
Hamiltonian chaos [172]. More generally, a swimmer exhibits a large
deformation, and such swimming dynamics in Poiseuille flow are studied for a
sperm cell with consideration of the elastohydrodynamic coupling [124] and for
a general deforming particle [188].

The conserved quantity may be broken by the fluid rheology as discussed for
Jeffery’s orbits. Mathijssen et al. [189] considered a spherical swimmer in a
channel flow with non-Newtonian fluid rheology and found that the viscoelastic
normal stress difference loses the constancy of the motion and reorients the
swimmers upstream, where they accumulate at the centerline of the channel.

###  V.3 More Complex Flows

\begin{overpic}[width=241.84842pt]{fig_vorti.eps} \put(13.0,90.0){{\large(a)}}
\put(5.0,59.0){{\large(b)}} \put(39.0,59.0){{\large(c)}}
\put(5.0,25.0){{\large(d)}} \put(39.0,25.0){{\large(e)}} \end{overpic} FIG.
12:  (Color online) Swimming in vortical flows. (a) Schematic of a
microswimmer swimming in an axisymmetric vortical flow. (b,c) Theoretically
obtained particle positions a long time after initially uniform distribution
inside the central circle of radius 5. Selected trajectories are also shown
for a trapped cell (blue) and escaping cell (red). Figure adapted from
Arguedas-Leiva & Wilczek [190] under the creative commons license,
http://creativecommons.org/licenses/by/4.0 (copyright 2020 by The Author(s)) .
(b) Plots in the laboratory frame $(x,y)$. (c) Plots in the co-rotating frame
$(X,Y)$. (d,e) Depletion of B. subtilis bacteria around a rotating magnetized
chain. Figure adapted from Sokolov & Aranson [191] under the creative commons
license, http://creativecommons.org/licenses/by/4.0 (copyright 2016 by The
Author(s)) . (d) Initial distribution of bacteria. (e) Distribution after the
depletion zone is created. White corresponds to a lower concentration.

We then move to a more complex flow pattern. In nature, oceanic plankton
typically live in a turbulent flow. The minimum size of eddies is
characterized by the Kolmogorov length scale, which is a few millimeters in
length in the upper mixed layer in the ocean [9]. Thus, most microswimmers
experience unsteady shear flow.

As a theoretical model of the smallest eddy in the environment, a swirling
flow may be the simplest flow model. Arguedas-Leiva & Wilczek [190] studied a
microswimmer in a 2-dimensional axisymmetric vortical flow (Fig. 12a). The
background velocity field is simply given by
$\bm{u}^{\infty}=(u(r)\cos\theta,u(r)\sin\theta)$, where
$r=\sqrt{x^{2}+y^{2}}$. For the motion of a microswimmer with a constant
velocity $V$ in the direction of the swimmer orientation, we employ the
Jeffery equation of an axisymmetric object, assuming the flow is locally
linear.

When the swimmer shape is spherical ($B=0$), Arguedas-Leiva & Wilczek [190]
found a Hamiltonian structure in the swimmer dynamics with co-rotating
coordinates, $X=x\cos\theta+y\sin\theta$ and $Y=-x\sin\theta+y\cos\theta$, as
$\dot{X}=\partial_{Y}H(X,Y)$ and $\dot{Y}=-\partial_{X}H(X,Y)$ with a
Hamiltonian $H(X,Y)=VY+\int^{r}sf(s)\,ds$. For Fig. 12b and c, we adapted the
plots in Arguedas-Leiva & Wilczek [190] to show the swimmer distributions
after a long time in the laboratory frame (Fig. 12b) and in the co-rotating
frame (Fig. 12c), assuming a Gaussian vortex, also known as the steady Lamb-
Oseen vortex, for the background vortical flow. Initially, swimmers are
uniformly distributed within a circle of radius $r=5$ whose center coincides
with the origin. As shown in Fig. 12b, some swimmers are confined near the
origin in the laboratory frame, as shown by the concentration given with a
grey scale. Sample swimmer trajectories are also plotted with blue and red
curves, corresponding to the confined and escaping cells, respectively. This
confinement is captured by the inner region separated by the homoclinic orbit
in the co-rotating frame in Fig. 12c.

While the Hamiltonian structure is broken for a non-spherical swimmer ($B\neq
0)$, the swimmer positions are still separated by the homoclinic orbits (or
separatrix) in the $X-Y$ plane, demonstrating a similar confinement of
microswimmers in a vortical flow. A homoclinic orbit can emerge by a saddle-
node bifurcation for a small effective velocity corrected by the Bretherton
parameter given by $V_{\textrm{eff}}=V/(1-B)$. When rotational noise is added,
the cells escape from the trapping region around the saddle point.

Focusing on a prolate microswimmer $(B>0)$, Tansijević & Lauga [192] found a
constant of motion in the system of microswimming in a vortical flow generated
by 3- and 2-dimentional rotlets, corresponding to the flow around a rotating
sphere and cylinder, respectively. The flow field is explicitly given by
$u(r)=a^{3}\omega/r^{2}$ and $u(r)=a^{2}\omega/r$ for the 3- and 2-dimensional
rotlets, respectively, where $a$ is the radius of the sphere or cylinder and
$\omega$ is the angular velocity of the rotation.

Motions of B. subtilis bacteria around a rotating object were experimentally
studied by Sokolov & Aranson [191], who reported a depletion zone around the
object. In Fig. 12d and e, we show pictures of bacteria concentrations before
and after the depletion zone was formed. This depletion is well explained by
the vortical confinement in numerical computation with the rotational noise of
the cells [192]. This confinement of microswimmers by a vortical flow was also
studied in a dumbbell model [193], a bead-spring system [194], and an abstract
deforming particle model [195].

\begin{overpic}[width=241.84842pt]{fig_hyp.eps} \put(18.0,70.0){{\large(a)}}
\put(-1.0,32.0){{\large(b)}} \put(49.0,32.0){{\large(c)}} \end{overpic} FIG.
13:  (Color online) Swimming in hyperbolic flows. (a) Schematic of a swimming
microswimmer in a hyperbolic flow. (b,c) Sample trajectories of smoothly
swimming B. subtilis bacteria projected on the $x-\phi$ phase space, together
with the swimmer invariant manifold shown with a thick blue line. Figure
adapted with permission from Berman et al. [196] (copyright 2021 by the
American Physical Society). (b) Leftward-escaping and rightward-escaping
trajectories are plotted in green and magenta, respectively. (c) Selected
trajectories of each cell are shown in a different color with their initial
positions being marked by open squares.

The existence of a saddle point in the dynamical system is the key structure
for the escaping behaviors of swimmers. Berman et al. [196] studied the
2-dimensional dynamics of microswimmers in a hyperbolic flow (Fig. 13a), which
is the linear flow around a saddle point with its flow field given by
$\bm{u}^{\infty}=(Ax,-Ay)$, where the flow strength is $A>0$. The swimmer
moves in the direction of the body axis, denoted by $\bm{d}$, with a constant
velocity $V$. We can non-dimensionalize the system by $A=V=1$.

Passive particles are transported along the streamline. The trajectories of
active particles are modified but the swimmers are still swept towards the
$+x$ or $-x$ direction. The paths in the $x-y-\phi$ phase space are therefore
separated by a surface, across which the sign of the final $x$ position is
reversed. This surface, termed a swimmer invariant manifold, acts as a
separatrix. In Fig. 13b and c, its projection to the $x-\phi$ plane is shown
with a thick blue curve. Fig. 13b and c show the sample trajectories
experimentally obtained for smoothly swimming B. subtilis cells in a
hyperbolic flow taken from Berman et al. [196]. The leftward and rightward
swimmers are shown in green and magenta plots in Fig. 13b, respectively, and
are well separated by the swimmer invariant manifold. Selected trajectories in
Fig. 13b are also shown with their initial positions marked by open squares.

Hyperbolic flow induces contractile and extensile forces on the swimmer, and
flexible swimmers such as sperm cells modulate their waveform during their
transport [197]. Similar compression is found in a flexible helix in a
converging flow [198]. Motions of rigid rod swimmers have been studied in a
similar converging chamber [199].

The coexistence of confinement and escape in swimmer dynamics is more clearly
shown in a 2-dimensional vortical flow lattice. In a 2-dimensional
incompressible flow, passive tracers follow a stream function $\Psi(x,y)$
given by $\bm{u}^{\infty}=(\partial_{y}\Psi,-\partial_{x}\Psi)$. This
Hamiltonian structure conserves the volume of the phase space, and thus the
distribution of passive particles is homogeneous. For active particles,
however, this does not hold.

Torney & Neufeld [200] analyzed the microswimmer dynamics in vortical arrays
whose stream function is given by $\Psi=(U_{0}L/[2\pi])\sin(2\pi x/L)\sin(2\pi
y/L)$, where $L$ is the size of the lattice and $U_{0}$ is the maximum
velocity in a vortical lattice. The orientation dynamics again satisfy the
Jeffery equation, under the assumption that the background flow is locally
approximated by a linear flow. An elongated swimmer in a cellular flow can
move across vortical cells if it swims with a speed higher than a critical
value, and the population is then separated into two different groups: one is
accumulating inside a vortical cell and the other moves around cells near the
heteroclinic orbits (or edges of vortical cells). Fig. 14a and b show the
long-time asymptotic distribution of particles initially distributed uniformly
in space, with the two panels corresponding to different shapes and flow
strength ($B=0.9$ in Fig. 14a and $B=0.7$ in Fig. 14b). The trapped swimmers
and non-trapped swimmers are plotted in red and blue, respectively.

Further analysis from the dynamical systems point of view was performed for
vortex lattice and hyperbolic flows in Ref. [201]. The dynamics in the vortex
arrays were also studied with fluid viscoelasticity [202], for gyrotactic
swimmers [203], and for sedimenting cells [204].

\begin{overpic}[width=241.84842pt]{fig_turb.eps} \put(0.0,91.0){{\large(a)}}
\put(40.0,91.0){{\large(b)}} \put(9.0,54.0){{\large(c)}} \end{overpic} FIG.
14:  (Color online) (a,b) Long-term asymptotic distribution of elongated
microswimmers in a vortical lattice flow. The cells are initially uniformly
distributed. With cell motility, the population separates into swimmers
trapped inside the vortical cells (plotted in red) and swimmers moving across
the vortical cells along their edges (plotted in blue). Figure adapted with
permission from Torney & Neufeld [200] (copyright 2007 by the American
Physical Society). (c) Snapshot of the distribution of swimmers in homogeneous
turbulence. The color bar indicates the concentration of cells. Figure adapted
from Cencini et al. [205] with permission (copyright 2019 by Springer-Nature).

Studies of microswimmer dynamics in oscillating cellular flows [200, 206, 207]
suggest a patchy distribution of the cells. Indeed, formations of patches are
found in gyrotactic microalgae, H. akashiwo, in vortical flows [208].
Accumulation was also found for a gyrotactic model swimmer and elongated
swimmer in Navier-Stokes turbulence [208, 209, 210, 205, 211]. For readers
interested in this topic, refer to the recent reviews [205, 212]. Fig. 14c is
a snapshot of the distribution of swimmers in homogeneous turbulence, where
colors indicate the concentration of cells.

Complex flows that microswimmers face are also ubiquitous near external
boundaries. Examples include a sinking sphere [213], obstacles fabricated
within microfluidic devices [214, 215], and surface gravity waves [216].

##  VI Summary and Outlook

The Jeffery equation can be applied to many other interesting problems for
exploring the effect of shape on interactions of microswimmers with external
environments.

###  VI.1 Collective Behavior

For modeling and simulating the collective behavior of microswimmers, the
background flow for individual swimmers is approximated by summations of flow
fields generated by other neighbors. In a dilute population, the far-field
flows are represented by the flow induced by swimmer stresslets [4]. This
method is also used for hydrodynamic interaction for a swimmer near a
boundary, as the flow by its mirror image is taken as a background flow for
the swimmer [217]. In either case, the swimmer’s shape affects whether the
interactions are attractive or repulsive.

These interactions have been scaled up to collective dynamics [218, 219] and
used in modeling and understanding bioconvection patterns [220] and active
turbulence [221]. The orientation dynamics are also significant in the
rheological properties of suspensions of microswimmers[222, 223]. In a dense
suspension, however, the swimmer shape is also significant due to steric and
near-field hydrodynamic interactions [224, 225].

###  VI.2 Smart Swimmers

Control and navigation of a microswimmer under a flow are also active topics.
What is the optimal strategy for a swimmer that can sense the flow around it
[226]? How can we optimize the flow field to control a swimmer’s path inside a
microdevice [227, 228]? Recently, adaptive and tactic swimming behaviors in
smart swimmers have attracted remarkable attention in the context of creating
and designing such artificial smart swimmers [229] and understanding adaptive
behaviors of real biological swimmers in complex environments [230, 231, 232],
where the use of machine learning is rapidly emerging, for example, in active
matter physics [233] and fluid mechanics [234].

In particular, reinforcement learning is used to obtain an optimal swimming
stroke [235, 236], optimal flow mixing [237], and an optimal locomotion
strategy [238]. Many studies have recently been performed on navigation of a
microswimmer through a complex flow environment, such as Taylor-Green vortical
cells [239], a complex potential [240], a chaotic ABC flow [241], in Navier-
Stokes turbulence [242], and in a fluid with spatially varying viscosity
[243], in addition to the gyrotactic stability problem [244] and predation
dynamics [245, 246].

###  VI.3 Concluding Summary

In this review, we have provided a theoretical introduction to microswimmer
hydrodynamics, the dynamics of a self-propelled particle in a low-Reynolds-
number flow, with an extensive focus on the Jeffery equations that govern the
orientation dynamics of a particle in a linear flow. By introducing the
concept of hydrodynamic symmetry, we have discussed the theoretical
backgrounds of the Jeffery equations and their extension to objects with more
general symmetry and objects that may rapidly deform (Sec. IV). The importance
and usefulness of the Jeffery equation are also discussed through its periodic
motion in simple shear, which emerges as a constant of motion. Motions of
microswimmers in simple shear flow, Poiseuille flow, vortical flow, hyperbolic
flow, and other complex flow patterns are summarized in Sec. V, with special
attention to the emergence and breakdown of the constants of motion in
different situations. Since its first derivation by Jeffery a century ago, the
motions of non-spherical particles and swimmers in low-Reynolds-number flows
have opened up extensive research fields in material, life, and engineering
sciences, and at the same time have provided a large variety of non-linear
dynamics solutions from simple equations.

K.I. acknowledges support from the Japan Society for the Promotion of Science
(JSPS), KAKENHI for Young Researchers (Grant No. 18K13456); JSPS, KAKENHI for
Transformative Research Areas (Grant No. 21H05309); and the Japan Science and
Technology Agency (JST), PRESTO (Grant No. JPMJPR1921). K.I. acknowledges M.
P. Dalwadi, E. A. Gaffney, C. Moreau, and B. J. Walker for stimulus
discussions. This study was partially supported by the Research Institute for
Mathematical Sciences, an International Joint Usage/Research Center located at
Kyoto University.

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