We theoretically investigate the motions of an object immersed in a background flow at a low Reynolds number, generalizing the Jeffery equation for the angular dynamics to the case of an object with *n*-fold rotational symmetry (*n* ≥ 3). We demonstrate that when *n* ≥ 4, the dynamics are identical to those of a helicoidal object for which two parameters related to the shape of the object, namely, the Bretherton constant and a chirality parameter, determine the dynamics. When *n* = 3, however, we find that the equations require a new parameter that is related to the shape and represents the strength of triangularity. On the basis of detailed symmetry arguments, we show theoretically that microscopic objects can be categorized into a small number of classes that exhibit different dynamics in a background flow. We perform further analyses of the angular dynamics in a simple shear flow, and we find that the presence of triangularity can lead to chaotic angular dynamics, although the dynamics typically possess stable periodic orbits, as further demonstrated by an example of a triangular object. Our findings provide a comprehensive viewpoint concerning the dynamics of an object in a flow, emphasizing the notable simplification of the dynamics resulting from the symmetry of the object’s shape, and they will be useful in studies of fluid–structure interactions at a low Reynolds number.

## I. INTRODUCTION

Ever since Stokes^{1} first derived an expression for the hydrodynamic drag on a moving sphere in a viscous fluid, the so-called Stokes law of drag, microscopic particles in fluids have usually been modeled as spheres. Microparticles in nature, however, exhibit a tremendous diversity in shape, including not only bodies of revolution such as rods and disks but also objects with discrete rotational symmetry such as snow crystals^{2} and planktonic microorganisms, the latter of which are illustrated in Haeckel’s famous book *Art Forms in Nature*.^{3}

The first extension from a sphere to be considered was a spheroid, the motion of which was analyzed mathematically by Jeffery^{4} who derived the equations for the angular motion of a spheroid and found a nonlinear periodic orbit in a simple shear, which is now known as the “Jeffery orbit.” The concept of periodic Jeffery orbits was later extended to arbitrary bodies of revolution by Bretherton,^{5} with further mathematically rigorous proofs subsequently being provided.^{6,7} The angular dynamics of any body of revolution are then described by an equation with a single parameter related to the aspect ratio of the object, which is now known as the “Bretherton constant.” Such a unified expression for a large class of object geometries suggests that symmetry arguments could allow the classification of particle dynamics in a shear flow into a smaller number of categories.

Recently, with attention focused on the time-averaged nature of the hydrodynamics of microswimmers in a flow, the Jeffery equations of angular dynamics have been extended to helicoidal objects, including chiral objects such as helices.^{8} A helicoidal object is mathematically defined as either a passive particle or an active swimmer whose resistance tensors are unchanged under a *π*/2-rotation around an axis. This is the symmetry referred to as helicoidal symmetry,^{6,9} first introduced in the late nineteenth century in the theory of inviscid flow^{10,11} (see Ref. 12 for more details).

Although, strictly speaking, a simple helix is not a helicoidal object according to this definition, a good description of the associated hydrodynamics can still be obtained by considering it as if it were such an object.^{13,14} Moreover, the rapid rotation of a bacterial flagellar swimmer around its rotation axis allows a description of such a microswimmer as a helicoidal object,^{15–17} and the biased locomotion of a bacterial swimmer in a shear flow, known as bacterial rheotaxis, has been successfully formulated.^{18} Such a representation is also applicable to a large class of microswimmers, considering their time-average motions over a beat cycle.^{19}

Under the assumption of helicoidal symmetry, the Jeffery equations can be derived as follows, with an additional constant arising from the chirality of the object:^{8}

where the director vector ** d** represents the axis of rotation,

**Ω**

^{∞}and

**E**^{∞}are the background rotational velocity vector (equal to half the background vorticity) and the rate-of-strain tensor of the background flow, respectively, and

*is the identity tensor. The shape parameter*

**I***β*is the Bretherton constant reflecting the aspect ratio of the object. The new constant in the last term,

*α*, represents the effects arising from the chirality of the object, and we shall refer to it in this paper as the “chirality parameter.” An equivalent parameter has been introduced in models of chiral objects.

^{17,20}The angular dynamics described in Eq. (1) have been investigated in detail for an object in a simple shear flow, and it has been found that if the Bretherton constant is nonzero, the chirality parameter

*α*can orient the director vector so that it is aligned parallel or antiparallel to the background vorticity vector.

The motions of nonspherical microscopic particles in a turbulent flow have been intensively studied in recent years,^{20–27} and simple equations such as Eq. (1) could be usefully applied not only to passive particles in turbulence but also to active swimmers,^{28–31} for example, in studies of the distribution of planktonic microalgae in the turbulent ocean.^{32,33}

Drift motions in flows are affected by object chirality, and this has been utilized for sorting in microdevices.^{34–36} In the context of active swimmers in a moderately strong background flow, the resulting hydrodynamic interactions could orient the swimming direction.^{17,18,37–39} Another possible application of the associated equations is to the study of Brownian motion and the rheology of nonspherical particles.^{40–43}

In this paper, we extend the Jeffery equations to an object with discrete rotational symmetry, which implies the existence of an *n*-fold rotational axis, where *n* ≥ 3 is an integer.

As presented in Haeckel’s book,^{3} the diatoms, a major group of microalgae producing a significant portion of atmospheric oxygen, are known for their symmetric shell shape.^{44} For example, *Triceratium* and *Trigonium* possess *n*-gonal symmetry, the rod-like species *Thalassiothrix* and *Asterionella* form star-shaped colonies, and species with more complicated triangular shapes, such as *Hydrosera*,^{45} are also known. Some other protozoa, the radiolarians, produce mineral skeletons with discrete rotational symmetry, including a triangular pyramidal shape in the case of *Lithochytris pyramidalis*, threefold rotational symmetry such as in *Thyrsocyrtis* and *Podocyrtis*,^{46,47} and a cell with fourfold rotational symmetry in the case of *Lithoptera muelleri*.^{48} These microorganisms can also be found in fossils and have been used in research on ancient climate and marine environments. Other microorganisms, for example, *discoasters*, distinct star-shaped marine algae (now extinct),^{49} do not necessarily have a mirror symmetry in the plane perpendicular to the rotation axis. Microorganisms with discrete rotational symmetry include other microalgae such as Dictyochales (also known as Silicoflagellates because of their siliceous skeleton).^{50,51}

The Jeffery orbits for a rigid particle with discrete rotational symmetry have been investigated by Fries *et al.*^{52} who studied the angular dynamics of a particle with *n*-fold rotational symmetry and an additional mirror plane to decouple the angular dynamics from the translational motions. The primary aim of our paper is therefore to generalize the Jeffery equations to an arbitrary microscopic object, which can be either a passive particle or an active swimmer, with *n*-fold rotational symmetry but without a mirror plane, where *n* ≥ 3. Following the Schoenflies notation^{53} for specifying the symmetry of an object, such as *C*_{n} for an object with an *n*-fold rotational symmetry, we call the object by its specified symmetry, for example, a *C*_{n}-object. Our secondary aim is to classify the shape of an object based on its dynamics in a flow, showing that the number of parameters in the generalized Jeffery equations is reduced from the original degrees of freedom in the resistance tensors.

Another important type of discrete symmetry can be found in a particle with triaxially mirror-symmetric planes, such as an ellipsoid. The equation for the dynamics of an ellipsoidal particle in a shear flow was also derived by Jeffery,^{4} and numerical studies^{54,55} revealed that such a particle can exhibit quasiperiodic and chaotic rotational motion. More recently, Thorp and Lister^{56} derived the general angular dynamics for non-axisymmetric particles with two planes of symmetry, demonstrating that the angular dynamics are identical to those of ellipsoids with three constants of particle shape.

The contents of the remainder of this paper are as follows: In Sec. II, we introduce hydrodynamics at a low Reynolds number to describe the dynamics of an object in a background flow, together with the symmetry arguments regarding resistance tensors under *n*-fold rotational symmetry, demonstrating that when *n* ≥ 4, the dynamics coincide with those obtained for a helicoidal object, which is a *C*_{4}-object. We also show that when *n* = 3, we need additional terms in the equations that reflect the triangularity of the object shape. Then, in Sec. III, we derive expressions for the linear and angular velocities of an arbitrary *C*_{n}-object, where *n* ≥ 3. We also discuss the reduction of the Jeffery equations by additional symmetries. In Sec. IV, we examine the angular dynamics in more detail for a *C*_{3}-object in a simple shear flow. Since new contributions will appear as the object becomes less symmetric, we will discuss each new term in the Jeffery equations in terms of the symmetry class. We present an example of a triangular object, and we estimate the new constants for the triangularity by detailed calculations in Sec. V. This is followed by our concluding remarks in Sec. VI.

## II. HYDRODYNAMICS AND SYMMETRY

### A. Stokes hydrodynamics

We consider a neutrally buoyant microscopic object immersed in a background flow field *u*^{∞} and assume that the flow field around the object obeys the Stokes equation,

where *p* is the pressure field, the constant *η* is the viscosity, and the velocity field ** u** satisfies the incompressibility condition ∇ ·

**= 0.**

*u*On the surface of the object, *S*, we impose the no-slip boundary condition. To describe the surface of the object, we introduce a laboratory-fixed reference frame {*e*_{i}} (*i* = 1, 2, 3) and a body-fixed reference frame ${e^i}$ whose origin is denoted by *x*_{0}, as illustrated in Fig. 1. We allow the object to be self-propelled by its deformation, and we decompose the velocity into rigid body motion and the deformation velocity as follows:^{57,58}

where ** r** =

**−**

*x*

*x*_{0},

**and**

*U***Ω**are the translational and rotational velocities, respectively, and

**′ represents the surface deformation velocity.**

*u*In the far field away from the object, the velocity fields should approach the background velocity for which we assume a linear flow in this study. The velocity gradient tensor * G* = ∇

*u*^{∞}is therefore uniform in space. Introducing the background linear velocity

*U*^{∞}, the background angular velocity

**Ω**

^{∞}, which is obtained from the antisymmetric part of the velocity gradient tensor, and the rate-of-strain tensor

**E**^{∞}= (

*+*

**G**

**G**^{T})/2, we have the boundary condition in the far field as

where superscript *T* indicates the transpose of a tensor.

From the linearity of the Stokes equations, the hydrodynamic force and torque acting on the object may be written as combinations of drag- and shear-induced effects and propulsion by surface deformation^{6–8} via

The hydrodynamic drag force and torques are

respectively, where the second-rank resistance tensors ** K** and

**, which are symmetric and negative-definite, are known as the translational and rotational tensors, respectively, and the second-rank (pseudo)tensor**

*Q***, which is not necessarily symmetric, is known as the coupling tensor. Using the third-rank shear-force tensor $\Gamma $ and the shear-torque (pseudo)tensor $\Lambda $, we can write the shear-induced force and torque as**

*C*respectively. Here, the double dot products are defined such that $[\Gamma :E\u221e]i=\Gamma ijkEkj\u221e$, where there is a summation over the repeated indices, following the Einstein convention that will be used throughout this paper. From the symmetric property of the rate-of-strain tensor, without loss of generality, we can define $\Gamma $ and $\Lambda $ so that they are symmetric with respect to their second and third indices, i.e., Γ_{ijk} = Γ_{ikj} and Λ_{ijk} = Λ_{ikj}. We use the transposition symbol *T* to indicate the exchange of the last two indices in the third-rank tensors, enabling us to rephrase the relations as $\Gamma $^{T} = $\Gamma $ and $\Lambda $^{T} = $\Lambda $. Note that the tensors ** K**,

**,**

*Q***, $\Gamma $, and $\Lambda $ are determined only by the shape of the object and the viscosity constant.**

*C*In the last terms in Eqs. (5) and (6), the vectors *F*_{prop} and *M*_{prop} represent the propulsive force and torque generated by the surface deformation of the object,^{57–60}

where the second-rank tensors $\Sigma $ and $\Pi $ are the translational and rotational surface force resistance tensors, defined by the decomposition of the local hydrodynamic force on the surface for an arbitrary rigid motion^{61} such that $f^=\Sigma \u22c5U^+\Pi \u22c5\Omega ^$, and these tensors are also dependent only on the shape of the object and the viscosity constant. Henceforth, we focus on their dependence on the object shape.

### B. Hydrokinetic symmetry

We now proceed to symmetry arguments that enable us to reduce the degrees of freedom in the resistance and shear-related tensors, following the approach of Brenner.^{6,9} As repeatedly noted in Sec. II A, these tensors are determined only by the shape of the object, and therefore, their matrix representations with respect to body-fixed coordinates should inherit the symmetry of the object shape. We consider a transformation of the body-fixed frame ${e^i}$ into another body-fixed frame ${e^i\u2032}$ via rotation and/or reflection. Let us write the matrix representations of an *r*th-rank tensor ** X** with respect to the body-fixed frames ${e^i}$ and ${e^i\u2032}$ as $Xiii2,\u2026,in$ and $Xiii2,\u2026,in\u2032$, respectively. The transformation is then given by

using the orthogonal matrix *a*_{ij} with its determinant |*a*_{ij}| = *σ* = ±1. Similarly, we have the following relation between the matrix representations of an *r*th-rank pseudotensor ** Y** under the transformation of the body-fixed frame:

If the shape of an object is unchanged under a transformation, we say that the object possesses the symmetry associated with the transformation matrix. A series of symmetries are discussed by Brenner,^{6,9} and in this paper, we focus on the symmetry with *n*-fold rotation (*n* ≥ 3), which is referred to as “generalized helicoidal symmetry” in Brenner’s papers.

Adopting the Schoenflies notation that is commonly used to categorize point groups in the studies of the molecular structure,^{53} let us call an object with *n*-fold rotational symmetry a *C*_{n}-object. We set the axis of rotation to be $e^1$, and the transformation matrix is then given by

where the angle Θ = 2*π*/*n*. For a *C*_{n}-object, the matrix representations in Eqs. (13) and (14) should be unchanged after the transformation in Eq. (15), and we obtain $Xi1i2\u2026ir\u2032=Xi1i2\u2026ir$ for the three resistance tensors and two shear-related tensors, since the determinant of *a*_{ij} is *σ* = 1.

For the resistance tensor ** K**, its matrix representation with respect to the body-fixed frame satisfies

*K*

_{ij}=

*a*

_{ik}

*a*

_{jl}

*K*

_{kl}, and similar relations should hold for

*Q*

_{ij}and

*C*

_{ij}. When equating the matrices obtained by Θ/2- and −Θ/2-rotations, we may rephrase the relations for each component as

with similar relations holding for *Q*_{ij} and *C*_{ij}. For the symmetric tensors ** K** and

**, the off-diagonal components should vanish from Eq. (17), and thus, the three resistance tensors for a**

*Q**C*

_{n}-object are given by

where I is the second-rank identity tensor, and the constants *K*_{1}, *K*_{2}, *Q*_{1}, and *Q*_{2} are all negative.

For the shear-related tensors, we again consider the matrix representations, which satisfy the relation Γ_{ijk} = *a*_{il}*a*_{jm}*a*_{kn}Γ_{lmn}. When *n* ≥ 3, this may be expressed as the following relations for each component, after equating the two matrix representations transformed by Θ/2- and −Θ/2-rotations:

and

with further constraints on Γ_{222} and Γ_{333} such that

When Θ ≠ 2*π*/3 or, equivalently, *n* ≥ 4, the last constraint [Eq. (28)] gives Γ_{222} = Γ_{333} = 0, and thus, the components in Eqs. (26) and (27) all vanish. When *n* = 3, however, these components generally remain unless the object possesses further symmetries. The components Γ_{111} and Γ_{122} = Γ_{133} are generally left unrestricted from symmetry arguments. The shear-related tensors affect the motion in the form of $\Gamma $ : **E**^{∞}, but the incompressibility condition * I* :

**E**^{∞}= 0 enables us to eliminate one of the components between Γ

_{j11}, Γ

_{j22}, and Γ

_{j33}for each

*j*= 1, 2, 3, if none of three components are left nonzero after the symmetry arguments. These additional degrees of freedom, referred to as the principle of indeterminacy, were discussed in detail by Brenner.

^{6}We then choose Γ

_{122}= Γ

_{133}= 0 and arrive at the expression for the shear-force tensor $\Gamma $, and the same arguments follow for the shear-torque tensor $\Lambda $, yielding the following expressions for the shear-related tensors:

and

Note that the 1/2 prefactors for the terms involving Γ_{2}, Γ_{3}, Λ_{2}, and Λ_{3} are inherited from the helicoidal object.^{8} With regard to the new terms involving Γ_{4}, Γ_{5}, Λ_{4}, and Λ_{5}, which did not appear in the case of a helicoidal object, it is important to note that these constants all become zero when *n* ≥ 4, and we recover the same expressions as for a helicoidal object when *n* ≥ 4. Thus, the dynamics of a *C*_{n}-object with *n* ≥ 4 coincide with those of a helicoidal object derived in the previous study,^{8} which emphasizes the surprisingly wide applicability of the Jeffery equations derived for a helicoidal object, i.e., a *C*_{4}-object.

For simplicity in the following discussions, we assume that the propulsions *F*_{prop} and *M*_{prop} also satisfy the *n*-fold rotational symmetry, considering the propulsion to be generated by the surface deformation of the object under the same rotational symmetry. This assumption is valid for a self-deforming swimmer in a moderately strong background shear, if one approximates the swimming by its time average over the periodic deformation.^{8} Thus, these vectors should be aligned with the rotation axis $e^1$, i.e., $Fprop=F0e^1$ and $Mprop=M0e^1$. From the expressions in Eqs. (29) and (30), this condition is satisfied if the surface velocity possesses the same *n*-fold rotation axis as for a squirmer swimmer with swirling,^{62–64} while the shape of the swimmer is not necessarily a body of revolution but is a *C*_{n}-object.

## III. GENERALIZED JEFFERY EQUATIONS

### A. Dynamics of a *C*_{n}-object

We proceed to derive the expressions for the translational and angular velocities of the object, generalizing the Jeffery equations to a *C*_{n}-body with *n* ≥ 3, noting that the dynamics for the *n* ≥ 4 case have already been presented as the helicoidal Jeffery equations (1).

We start by decomposing the shear-induced force and torque in the form

where **Φ**_{i} (*i* = 1, 2, 3) are given by

and $\Psi $_{i} (*i* = 1, 2, 3) have similar forms with Γ_{1}, …, Γ_{5} replaced by Λ_{1}, …, Λ_{5}.

The force and torque on the object [Eqs. (5) and (6)] must be balanced owing to the negligible inertia, since no other external force and torque are applied. We introduce the 6 × 6 grand resistance tensor

and the balance equations ** F** =

**=**

*M***0**can then be solved as

where it should be noted that the summations must be performed over the repeated label *i* (*i* = 1, 2, 3). The inverse of $K$ can be calculated directly, and some straightforward but rather lengthy calculations lead to an explicit form for the velocities,

where summations are performed over the repeated greek indices *μ* = 1, 2, …, 5. The 27 degrees of freedom in the resistance and shear-related tensors are now reduced to the 10 constants *α*_{μ} and *β*_{μ}, which contain the object’s shape factors Γ_{μ} and Λ_{μ}, and the constants in the resistance tensors [Eqs. (19)–(21)], together with the propulsions *F*_{0} and *M*_{0}, as shown explicitly below.

The vectors *d*_{μ} are functions of the background rate-of-strain tensor and the orientation of the object, given by

From Eqs. (40)–(44), the vectors *d*_{2}, …, *d*_{5} are all perpendicular to *d*_{1}, and it is found that the vectors {*d*_{1}, *d*_{2}, *d*_{3}} and {*d*_{1}, *d*_{4}, *d*_{5}}, respectively, form right-handed orthogonal bases, as

where it should be noted that these vectors are not orthonormal, in general.

With the determinants of the block matrix in the grand resistance matrix $K$,

both of which are positive owing to the negative-definiteness of the tensor, the constants *α*_{μ} and *β*_{μ} may be written as follows:

and

The director vector for the axis of the rotational symmetry, *d*_{1}, thus follows the equation for its time evolution, $d\u03071=\Omega \xd7d1$, given by

which generalizes the Jeffery equations for a *C*_{n}-object. When *n* ≥ 4, we readily recover the helicoidal Jeffery equations (1) from *β*_{4} = *β*_{5} = 0, with the Bretherton constant *β* = −*β*_{2} and the chirality constant *α* = *β*_{3}. However, when *n* = 3, the angular dynamics of the object can be more complicated owing to the vectors *d*_{4} and *d*_{5}, which depend on the instantaneous orientations $e^2$ and $e^3$. The new vectors *d*_{4} and *d*_{5} are different from those found in the angular dynamics of an object with triaxial reflection symmetry, such as an ellipsoid.^{54,65,66}

Note that the vector *d*_{5} is obtained by *π*/2-rotation in the $e^2\u2212e^3$ plane, and the entire dynamics should be identical on rotation of the orientation of the $e^2$ axis. There is only a single parameter that determines the dynamics, which we refer to as the “triangularity strength,” defined as $\gamma =\beta 42+\beta 52$.

When the object is a deformable self-propelled swimmer with discrete symmetry, the shape-dependent coefficients *α*_{μ} and *β*_{μ} are, in general, functions of time. One may, however, approximate them by their time-averaged values, if the timescale of the swimmer deformation is sufficiently faster than the shear-induced motions, as has been shown to be the case for bacterial swimmers^{15–17} and other flagellate swimmers^{19} in a moderate background flow.

The detailed motions of a general *C*_{n}-object in the presence of a simple linear shear will be discussed in Sec. IV. Before proceeding to that discussion, however, we shall present comprehensive arguments for *C*_{n}-objects with further symmetry.

### B. Dynamics of an object with further symmetry

In this subsection, we provide comprehensive reductions of the dynamics from the generalized Jeffery equations (38) and (39) for objects with further symmetry using the Schoenflies notation. The results are summarized in Tables I and II, and more detailed discussions are provided below. It should be noted that the dynamics in a simple shear flow will be discussed in detail in Sec. IV.

. | α_{1}
. | β_{1}
. | α_{2}
. | α_{3}
. | β_{2}
. | β_{3}
. | Dynamical classification . |
---|---|---|---|---|---|---|---|

C_{n} | * | * | * | * | * | * | Helicoidal object |

C_{nv} | * | 0 | 0 | * | * | 0 | Body of revolution |

C_{nh} | 0 | * | 0 | 0 | * | * | Heterochiral object |

D_{n} | 0 | 0 | * | 0 | * | 0 | Homochiral object |

D_{nh} | 0 | 0 | 0 | 0 | * | 0 | Body of revolution |

D_{nd} | 0 | 0 | 0 | 0 | * | 0 | Body of revolution |

S_{2n} | 0 | * | 0 | 0 | * | * | Heterochiral object |

. | α_{1}
. | β_{1}
. | α_{2}
. | α_{3}
. | β_{2}
. | β_{3}
. | Dynamical classification . |
---|---|---|---|---|---|---|---|

C_{n} | * | * | * | * | * | * | Helicoidal object |

C_{nv} | * | 0 | 0 | * | * | 0 | Body of revolution |

C_{nh} | 0 | * | 0 | 0 | * | * | Heterochiral object |

D_{n} | 0 | 0 | * | 0 | * | 0 | Homochiral object |

D_{nh} | 0 | 0 | 0 | 0 | * | 0 | Body of revolution |

D_{nd} | 0 | 0 | 0 | 0 | * | 0 | Body of revolution |

S_{2n} | 0 | * | 0 | 0 | * | * | Heterochiral object |

. | α_{4}
. | α_{5}
. | β_{4}
. | β_{5}
. | Angular dynamics . |
---|---|---|---|---|---|

C_{3} | * | * | * | * | Can be chaotic |

C_{3v} | * | 0 | 0 | * | Can be chaotic |

C_{3h} | * | * | 0 | 0 | Helicoidal Jeffery orbits |

D_{3} | * | 0 | * | 0 | Can be chaotic |

D_{3h} | * | 0 | 0 | 0 | Jeffery orbits |

D_{3d} | * | 0 | 0 | 0 | Jeffery orbits |

S_{6} | 0 | 0 | * | * | Can be chaotic |

. | α_{4}
. | α_{5}
. | β_{4}
. | β_{5}
. | Angular dynamics . |
---|---|---|---|---|---|

C_{3} | * | * | * | * | Can be chaotic |

C_{3v} | * | 0 | 0 | * | Can be chaotic |

C_{3h} | * | * | 0 | 0 | Helicoidal Jeffery orbits |

D_{3} | * | 0 | * | 0 | Can be chaotic |

D_{3h} | * | 0 | 0 | 0 | Jeffery orbits |

D_{3d} | * | 0 | 0 | 0 | Jeffery orbits |

S_{6} | 0 | 0 | * | * | Can be chaotic |

#### 1. *C*_{nv}-object

We start with the dynamics of an object with symmetry *C*_{nv}, which possesses *n* reflection planes containing the axis of *n*-fold rotation. Examples include a regular pyramid with an *n*-sided base. We are able to reduce the number of parameters through symmetry arguments on such an object. Setting $e^2$ to coincide with one of the *n* reflection planes, we have an additional reflection symmetry in the $e^1$–$e^2$ plane, and we obtain *C*_{1} = *C*_{2} = 0 and Γ_{3} = Γ_{5} = 0, Λ_{1} = Λ_{2} = Λ_{4} = 0. We, therefore, have *α*_{2} = 0, *α*_{3} ≠ 0, *β*_{2} ≠ 0, and *β*_{3} = *α* = 0. For a passive particle, we find *α*_{1} ≠ 0 and *β*_{1} = 0. We, thus, recover the angular dynamics, which are the same as those of a body of revolution with zero drift velocity when *n* ≥ 4. From Eq. (38), the terms *α*_{2}, …, *α*_{5} induce a drift velocity, which is perpendicular to the director vector *d*_{1}. For a *C*_{nv}-object, which can be either a passive particle or an active swimmer, however, the drift velocity from the *α*_{3} term will vanish over the periodic motion described by the Jeffery orbit, as obtained for a *C*_{4v}-object.^{8} When *n* = 3, however, we have nonzero contributions from *β*_{5} ≠ 0 and *α*_{4} ≠ 0, while the *α*_{5} and *β*_{4} terms vanish as *α*_{5} = *β*_{4} = 0.

#### 2. *C*_{nh}-object

This class of objects corresponds to the chiral crystals that possess reflection symmetry in the plane perpendicular to the axis of *n*-fold rotation, as studied by Fries *et al.*^{52} We consider the additional reflection symmetry in the $e^2$–$e^3$ plane, and when *n* ≥ 4, we have *C*_{1} = *C*_{2} = *C*_{23} = 0 and Γ_{1} = Γ_{2} = Γ_{3} = 0, but the two parameters in the shear-torque tensor remain nonzero, i.e., Λ_{2} ≠ 0 and Λ_{3} ≠ 0, yielding the relations *α*_{2} = *α*_{3} = 0, *β*_{2} ≠ 0, and *β*_{3} = *α* ≠ 0. These indicate that no drift velocity is generated, but the chiral parameter *α* does not vanish. We refer to the class of objects that exhibit such behavior in a simple shear flow as “heterochiral objects,” following the terminology used for molecules in chemistry,^{67} since this case is a generalization of the dynamics for an object with two helices with opposite chirality connected at one end.^{8,20} This result again recovers the dynamics obtained for a *C*_{4h}-object.^{8} For a passive particle, *α*_{1} = 0 and *β*_{1} ≠ 0 are obtained. When *n* = 3, with this symmetry, we obtain Γ_{4} ≠ 0 and Γ_{5} ≠ 0, while Λ_{4} = Λ_{5} = 0 holds, and we find that *α*_{4} and *α*_{5} do not vanish, but there is no additional effect on the angular dynamics,^{52} since *β*_{4} = *β*_{5} = 0.

#### 3. *D*_{n}-object

The symmetry denoted by *D*_{n} is called “dihedral” in which the object possesses *n* additional twofold axes perpendicular to the *n*-fold rotation axis. We set $e^2$ as one of the *n* twofold axes and consider the twofold rotational axis around it, which enables us to derive the relations *C*_{23} = 0 and Γ_{1} = Γ_{2} = Γ_{5} = Λ_{1} = Λ_{2} = Λ_{5} = 0. We, thus, obtain *α*_{3} = *β*_{3} = 0, and the relation *α*_{1} = *β*_{1} = 0 also holds for a passive particle. When *n* ≥ 4, the nonzero constants are *α*_{2} and *β*_{2}, which generate a nonzero drift velocity, but the angular dynamics follow the Jeffery orbits. We define the class of objects with this dynamics as “homochiral objects,” following again the terminology from chemistry, since such dynamics can be obtained for a uniform helical object.^{14,35} When *n* = 3, however, we need additional nonzero contributions from *α*_{4} and *β*_{4}.

#### 4. *D*_{nh}-object

The class of *D*_{nh}-objects possesses a further reflection symmetry in the $e^2$–$e^3$ plane, which is the plane perpendicular to the *n*-fold rotational axis, in addition to the dihedral symmetry discussed above. Examples of such objects include an *n*-gonal prism and a regular *n*-bipyramid. The additional reflection symmetry yields further relations, Γ_{3} = Λ_{4} = 0, enabling us to set *α*_{2} = *β*_{4} = 0, whereas these remain nonzero for a *D*_{n}-object. Therefore, we have angular dynamics that obey the Jeffery orbits for *n* ≥ 3, and no drift velocities are generated when *n* ≥ 4. However, when *n* = 3, a new drift velocity term remains owing to the nonzero *α*_{4} term.

#### 5. *D*_{nd}-object

An object denoted by *D*_{nd} has further *n* reflectional planes that pass between the twofold axes, in addition to the symmetry of *D*_{n}. An *n*-gonal antiprism belongs to this symmetry class. As in the discussions for a *D*_{n}-object, we set $e^2$ as one of the *n* twofold axes.

When *n* is an odd number, *n* = 3, 5, 7, …, the object possesses reflection symmetry in the $e^1$–$e^2$ plane, and we have an additional constraint on the resistance tensors, *C*_{1} = *C*_{2} = 0, and on the shear-related tensors, Λ_{4} = 0 as for the *C*_{nv}-object. We still have nonzero values of Γ_{4} and Λ_{3}.

When *n* is even but not a multiple of 4, i.e., *n* = 6, 10, 14, …, the additional *n* reflection plane for a *C*_{nd}-object passes through the $e^3$ axis. We, therefore, have further constraints from the reflection symmetry in the $e^1$–$e^3$ plane, which results in the same relations as in the odd-*n* cases, *C*_{1} = *C*_{2} = Λ_{4} = 0, and Γ_{4} = 0 since *n* ≠ 3. Only the constant Λ_{3} is left nonzero. Similar arguments hold when *n* is a multiple of 4, since the object then possesses reflection symmetry in both the $e^1$–$e^2$ and $e^1$–$e^3$ planes. The constraint relations are found to be the same as in the cases above.

From these symmetry arguments, we finally obtain the same constraints on the coefficients in the tensors as in the *D*_{nh} case, yielding angular dynamics coinciding with those for a body of revolution, with an additional drift velocity from the *α*_{4} term that appears only when *n* = 3.

#### 6. *S*_{2n}-object

The symmetry denoted by *S*_{2n} involves only a 2*n*-fold rotation–reflection axis and is not a subgroup of *C*_{2n} but of *C*_{n}, where it should be noted that *S*_{n} = *C*_{nh} for odd *n*. The rotation–reflection symmetry provides further constraints such that *C*_{1} = *C*_{2} = *C*_{23} = Γ_{1} = Γ_{2} = Γ_{3} = Γ_{4} = Γ_{5} = 0 (the detailed calculations are provided in the Appendix). We, therefore, have *α*_{μ} = 0 for *μ* = 1, …, 5, whereas *β*_{μ} remain nonzero, in general, and when *n* ≥ 4, the dynamics are thus identical to those of a *C*_{nv}-object from *β*_{4} = *β*_{5} = 0, and the object is therefore classified as a heterochiral object. When *n* = 3, the additional angular velocity needs to be considered owing to the *β*_{4} and *β*_{5} terms.

## IV. GENERALIZED JEFFERY ORBITS: MOTIONS IN A SIMPLE SHEAR

As shown in Fig. 2(a), we consider the background simple shear given by *u*^{∞} = *Gy**e*_{3}, where *G* > 0 is the shear strength, although we henceforth set *G* = 1 since this factor may be absorbed into the timescale. The background angular velocity vector is then directed toward the *e*_{1} axis, **Ω**^{∞} = (1/2)*e*_{1}, and the rate-of-strain tensor **E**^{∞} is given by **E**^{∞} = (*e*_{2}*e*_{3} + *e*_{3}*e*_{2})/2 We now consider more detailed dynamics, focusing on motions in a simple shear flow that generalize the Jeffery orbits of a body of revolution. If *n* ≥ 4, as the angular dynamics are described by the director vector *d*_{1} = *e*_{1}, which satisfies the helicoidal Jeffery equations,^{8} we focus on the dynamics for *n* = 3 in this section. The dynamics of the director then need three angles to describe its orientation for which we introduce the Euler angles (*ϕ*, *θ*, *ψ*) via the *zxz* convention, where *ϕ* is the azimuthal angle, *θ* is the polar angle, and *ψ* is the roll angle [Fig. 2(b)]. The time evolutions are given by^{68}

using the values of the angular velocity **Ω** in the body-fixed coordinates, $\Omega =\Omega ^1e^1+\Omega ^2e^2+\Omega ^3e^3$. Substituting the representations of **Ω**^{∞} and **E**^{∞} in the body-fixed frame into the expressions for the vectors *d*_{μ} [Eqs. (40)–(44)], we can derive a closed form of Eqs. (60)–(62) in terms of the Euler angles by using Eq. (39). After some straightforward calculations, we can write down the equations as

Here, we have introduced *h*_{p} and *h*_{v} as functions of the Euler angles and the shape-dependent parameters *β*_{2}, …, *β*_{5}, given by

where

From the expressions above, we readily recover the helicoidal Jeffery equations, which are decoupled from the angle *ψ* when *β*_{4} = *β*_{5} = 0. When *n* = 3, however, these two terms are left nonzero, and the angular dynamics can be different from what has been known. We will discuss how each new term contributes to the angular dynamics in detail below, and the results will be summarized at the end of this section.

### A. Motions of *D*_{3h}-, *D*_{3d}-, and *C*_{3h}-objects

We start with the motions of a *D*_{3h}-object, a *D*_{3d}-object, and a *C*_{3h}-object for which the angular dynamics are described by the Jeffery equations of a body of revolution, since *β*_{3} = *β*_{4} = *β*_{5} = 0, although a new drift velocity term appears.

The new drift velocity term for *D*_{3h}- and *D*_{3d}-objects is given by *U*_{4} = *α*_{4}*d*_{4} from Eq. (38) and the results in Sec. III. To clarify the direction of this vector, we consider the dynamics with *θ* = 0 and *π* in which the object is parallel/antiparallel to the background vorticity vector, and the angular dynamics follow

where the upper and lower signs indicate the cases *θ* = 0 and *θ* = *π*, respectively, and the object rotates around the *e*_{1} axis with a constant rotational velocity. The vector *d*_{4} is then found to be

which can be rewritten in terms of the laboratory-fixed frame as

From the constant rotation given by Eq. (72), the drift velocity *U*_{4} induces a threefold horizontal velocity during one period of the rotation around *e*_{1}, reflecting the triangular geometric features of the object.

Since the angles *ϕ* and *θ* are periodic in time following the Jeffery orbit, *ψ* is also periodic, and the time average of the drift velocity ⟨*U*_{4}⟩ is found to vanish, yielding no net drift velocity.

We now proceed to the dynamics of a *C*_{3h}-object for which the *α*_{4} and *α*_{5} terms are nonzero, but the angular dynamics are described by the helicoidal Jeffery equation with nonzero chirality parameter *α* = *β*_{3}. The stable orientation of the object is then either parallel or antiparallel to the background vorticity vector, as follows from analysis of the helicoidal Jeffery equation.^{8} At the stable orientation, the drift velocity *U*_{4}, which is proportional to the vector *d*_{4}, is toward the orientation in Eq. (75). As the drift velocity *U*_{5} = *α*_{5}*d*_{5} is perpendicular to both *e*_{1} and *d*_{4}, we have a similar threefold periodic drift motion from *U*_{5}, which again leads to zero average drift velocity over the period of rotation.

### B. Motions of *C*_{3v}- and *D*_{3}-objects

We now consider *C*_{3v}- and *D*_{3}-objects for which we need an additional term *β*_{4} or *β*_{5} in the angular dynamics. Since the vector *d*_{5} is obtained by *π*/2-rotation of the angle *ψ*, we first consider the dynamics with nonzero *β*_{4} with *β*_{5} being kept zero. From Eqs. (74) and (75), the orientations at *θ* = 0 and *π* are not expected to be equilibria owing to the angular velocity generated by the new terms. Substituting *θ* = *π*/2 into Eqs. (63)–(65) gives the relation

with *dψ*/*dt* = 0. The angle *θ* = *π*/2 is thus found to be stationary when cos 3*ψ* = 0, and we then find that the dynamics of the angle *ϕ* are periodic in time. Nevertheless, the stability of this periodic orbit is nontrivial. We proceed with the numerical solution of Eqs. (63)–(71) for an object with only the parameter *β*_{4} being nonzero.

For the numerical integration, we use different initial values of the angle *θ* but fix the initial values of the angles *ϕ* and *ψ* to be zero, and we perform the numerical integration via the MATLAB ode45 solver from time *t* = 0 to *t* = 10^{5}, which will be fixed throughout the computations shown below.

We plot the sample dynamics of objects with *β*_{4} = −0.5 [Figs. 3(a) and 3(b)] and *β*_{4} = −0.9 [Figs. 3(c) and 3(d)], with the other parameters, including the Bretherton constant, being set to zero (*β*_{1} = *β*_{2} = *β*_{3} = *β*_{5} = 0). In the left panels of Fig. 3, the Poincaré map at $\varphi =2\pi Z$, where $Z={0,\xb11,\xb12,\u2026\u2009}$, is shown for the dynamics with 10 different initial values of the angle *θ*. The right panels show the corresponding time evolutions of *θ* for five different simulations, with the same colors as used in the left panels for the same simulation. The horizontal axes in Figs. 3(b) and 3(d) are normalized by the time period of the Jeffery orbit with the corresponding *β*_{2}, i.e., *T*_{J} = 2*π*(*r* + *r*^{−1}), with $r=(1\u2212\beta 2)/(1+\beta 2)$.

The Jeffery orbit of a body of revolution with zero Bretherton constant *β* = −*β*_{2} = 0 is simple rotation with constant *θ*. As shown in Figs. 3(a) and 3(b), when *β*_{2} = 0 but *β*_{4} ≠ 0, the angle *θ* can vary in time, although the angular dynamics are still quasiperiodic in time as long as the strength of the triangularity, |*β*_{4}|, is small enough. However, as this strength increases, the angular dynamics can become chaotic, in particular, around the angle *θ* ≈ *π*/2, as in Figs. 3(c) and 3(d), in contrast to the helicoidal object (i.e., a *C*_{n}-object with *n* ≥ 4), which always exhibits nonchaotic dynamics. It is therefore found that the symmetries indicated by the Schoenflies notations *C*_{3v} and *D*_{3} are the minimal symmetries that can exhibit chaotic angular dynamics.

We now consider a nonzero value of *β*_{2}, which is the negative of the Bretherton constant *β* = −*β*_{2} in the range −1 < *β* < 1 for a standard object,^{5} although ring-like particles can possess values outside this range.^{69} When *β*_{4} is nonzero, the object exhibits a periodic Jeffery orbit, but when |*β*_{4}| is small, an orbit starting around *θ* ≈ *π*/2 is captured by an attracting periodic orbit at *θ* = *π*/2, which is one of the periodic orbits of cos 3*ψ* = 0. The basin of the attractor becomes wider as the strength of the triangularity increases, accompanied by the localization of the regions of quasiperiodic motions.

The attraction of the dynamics toward the angle *θ* = *π*/2, which is perpendicular to the background vorticity vector, can be found in large parameter ranges of *β* ≠ 0, *β*_{4}, and *β*_{5}. Figure 4 illustrates Poincaré maps and angular evolutions with (*β*, *β*_{4}, *β*_{5}) = (0.1, −0.5, 0) in (a) and (b) and with (0.5, 0, 0.5) in (c) and (d). With a nearly spherical Bretherton constant, *β* = 0.1, the attraction is fairly weak, but the orbit starting near *θ* = *π*/2 is slowly attracted by the stable periodic orbit at *θ* = *π*/2, as indicated by the arrow in Fig. 4(a). Noting that the vector *d*_{5} is obtained by *π*/2-rotation in the plane perpendicular to *d*_{1}, we can interpret the exchange between *β*_{4} and *β*_{5} as a *π*/2-shift of the angle *ψ* [Fig. 4(c)]. The attractive behavior of the orbits toward the angle *θ* = *π*/2 is found also for the parameters in Figs. 4(c) and 4(d), but more complex structures of the quasiperiodic orbits are observed.

### C. Motions of *S*_{6}- and *C*_{3}-objects

We now proceed to more general situations for less-symmetric objects such as *S*_{6}- and *C*_{3}-objects for which *β*_{1} and *β*_{3} can take nonzero values.

We first consider the effects of *β*_{3}, which is equivalent to the chirality parameter *α* = *β*_{3}. When *β* = 0, there is still a nonlinear periodic orbit for a helicoidal object,^{8} despite the nonzero chirality term *α* ≠ 0, and strong attractive behavior around *θ* = *π* is not found even when both the chirality and triangularity terms are considered. Figure 5 shows the Poincaré maps for the angular dynamics with parameters (*β*, *β*_{4}, *β*_{5}) = (0, −0.5, 0) as in Figs. 3(a) and 3(b), but with nonzero-*α* effect. It can be seen from the figure that as the chirality strength increases, the quasiperiodic orbits collapse into chaotic dynamics.

When the Bretherton constant is nonzero, *β* ≠ 0, however, depending on the sign of the parameter *α*, the axis of rotational symmetry, $e^1$, is attracted to align parallel or antiparallel to the background vorticity vector, if the triangularity is not considered, as for a helicoidal object. In Figs. 6(a)–6(c), the angular dynamics are shown, with the Poincaré maps and time evolutions of the angle *θ* for the same parameter set (*β*, *β*_{4}, *β*_{5}) = (0.5, 0, 0.5) as in Figs. 4(c) and 4(d), but with the nonzero chirality term *α* = 0.1. The time evolution shows two different attractive periodic orbits [Figs. 6(b) and 6(c)]: one is the stable periodic orbit at *θ* = *π*/2 generated by the triangularity term, and the other is located near *θ* ≈ 0, where the stable orientation is created by the chirality term with *α* > 0. From the three spiral structures in the Poincaré map in Fig. 6(a), the stable periodic dynamics near *θ* = 0 are found to have two periods with threefold difference, which reflects the triangular geometric features of the object. As the chirality term is strengthened to *α* = 0.3, the periodic orbit at *θ* = *π*/2 becomes less attractive and the dynamics falls into a doubly periodic stable orbit [Fig. 6(d)].

We finally consider the effects of the *β*_{1} term, which generates spinning motion around the axis of rotation and is decoupled from the angular dynamics in a helicoidal object. With triangularity, however, this term is coupled with the rotation dynamics. The parameter *β*_{1} can also be generated by self-propulsion, and this effect needs to be considered for a general active particle with self-spinning motion.

When only the *β*_{1} term and the triangularity terms are nonzero, it is found, for *β*_{1} > 0, that the *β*_{1} term can generate chaotic orbits in the region with *θ* > *π*/2, although the chaotic motions are suppressed as *β*_{1} increases. These tendencies are also found in larger parameter regions.

When the Bretherton constant is nonzero, *β* ≠ 0, with the existence of the stable orbits at *θ* = *π*/2, it is less likely that chaotic dynamics will be realized. With the *β*_{1} term, however, when the triangularity effect is comparable in size to the *β*_{1} term, chaotic orbits are possible in either hemisphere *θ* > *π*/2 or *θ* > *π*/2, depending on the sign of *β*_{1}. Figure 7 shows the Poincaré maps and time evolutions of the angle *θ* for dynamics with fixed values of the parameters *β* = 0.9 and *β*_{1} = 0.2 and different values of *β*_{5}: 0.1 in (a) and (b) and 0.3 in (c) and (d). With an intermediate size of the *β*_{1} effect, chaotic behavior can be realized in the region with *θ* > *π*/2, as is found in the Poincaré map in Fig. 7(a). The chaotic angular dynamics, however, disappear as *β*_{1} increases, with attraction by a periodic orbit occurring at a certain angle *θ* > *π*/2 [Figs. 7(c) and 7(d)].

### D. Summary of results for the generalized Jeffery orbits

In this section, we have considered the generalized Jeffery orbits, which represent the angular dynamics in a simple shear, where a body of revolution follows nonlinear periodic orbits, parameterized by the Bretherton constant *β*. When *β* = 0, as is the case for a sphere, an object will rotate around a background vorticity vector with a constant rotational angular velocity. Even if a chirality constant *α* is added, there still exist nonlinear periodic orbits. However, if triangularity is considered for a *C*_{3}-object, the nonlinear orbits are found to collapse into chaotic angular dynamics.

Although the equations are not integrable, the chaotic behavior looks very similar to that in Hamiltonian chaos and can be compared to the angular dynamics of objects with triaxial reflection symmetry, such as ellipsoids and cuboids.^{55} These objects are denoted in the Schoenflies notation by *D*_{2h} and are known to exhibit chaotic angular dynamics. Since no chaotic angular motions are possible for an object with *n*-fold rotational symmetry (*n* ≥ 4), *n* = 3 is the maximum number for the emergence of chaotic Jeffery orbits.

A Jeffery orbit with a nonzero Bretherton constant exhibits nonuniform rotation in a simple shear flow, accompanied by longer stays along the background velocity vector. The coupling between this nonuniform rotation and the triangularity of the object shape can allow a stable periodic orbit in which the axis vector of the threefold rotation periodically rotates in a plane perpendicular to the background vorticity vector without spinning motion in the plane. When a chirality constant *α* is added, the object orientation is attracted by a stable orbit that is perpendicular or parallel to the background vorticity vector, depending on the initial orientation and the strengths of the chirality and triangularity.

Finally, we have considered the effects of nonzero spinning rotation along the axis of *n*-fold rotation. This spinning term can also be generated by self-spinning propulsion, and its magnitude can be changed by controlling the propulsion for biological and artificial microswimmers. From numerical computations, the spinning term is found to yield chaotic angular dynamics when *β* ≠ 0 if the size of this term is comparable to that of the triangularity terms.

In the absence of the *β*_{1} term, the angle *ψ* typically increases when *θ* < *π*/2 and decreases when *θ* > *π*/2. A positive *β*_{1} term is related to the angle *ψ* according to Eq. (65), and so the angular motion is accelerated when *θ* < *π*/2 but decelerated by the *β*_{1} term when *θ* > *π*/2. This asymmetric contribution could be key to the emergence of chaotic motion as a result of the *β*_{1} term. Nonetheless, the mechanisms underlying the chaotic motion and its suppression require further theoretical analysis.

We do not discuss in detail other symmetric objects, such as those possessing tetrahedral and icosahedral symmetries: *T*, *T*_{d}, *T*_{h}, *O*, *O*_{h}, *I*, and *I*_{h}. These higher symmetry constraints may impose further relations on the Bretherton constant, as in the case of an isotropic helicoid,^{70} which possesses a triaxial helicoidal symmetry.

## V. EXAMPLE OF A TRIANGULAR OBJECT

In this section, we discuss an example of an object that possesses threefold rotational symmetry by considering a set of slender rods spatially arranged to form a regular triangular pyramid (Fig. 8). Such a tetrahedral object could be used as a simplified model of radiolarians with threefold rotational symmetry such as *Thyrsocyrtis* and *Podcrytis*.^{46,47}

One of the rods of the model object is directed toward the axis of threefold rotational symmetry $e^1$, and the remaining rods have a polar angle of *θ* = 2*π*/3 from the axis of rotation. We label these three rods by *k* = 1, 2, 3, with rod *k* being obtained by a 2*πk*/3-rotation around the $e^1$ axis after a 2*π*/3-rotation around the $e^3$ axis. The combined rotation matrix, which maps from the rod parallel to the $e^1$ axis to one of the others, is then given by

where we have introduced the notations *s*_{k} = sin(2*πk*/3) and *c*_{k} = cos(2*πk*/3). From the method of construction, we could straightforwardly generalize this example to an object consisting of *N* + 1 rods with *C*_{N}-symmetry.^{15} As a rod is a body of revolution, this example of an object is a *C*_{3v}-object. It is straightforward to extend the model to general *C*_{n}-objects by the chirality effect, replacing a simple rod by a chiral object such as a helix. Nonetheless, our current focus is on providing an example of an object with triangular symmetry and estimating the values of the triangularity strength, and therefore, we do not pursue any further generalization in this paper.

From symmetry, we can write the resistance tensors for the slender rod in the $e^1$ axis, labeled by superscript (0), as $K11(0)=K1(0)$ and $K22(0)=K33(0)=K2(0)$ and similarly for the rotational resistance tensor ** Q** and the coupling tensor $C23(0)=\u2212C32(0)$, represented in body-fixed coordinates. We can also write the nonzero components of the shear-force and shear-torque matrices as

and

We approximate the resistance tensors of a rod complex by summing the contribution from each rod *k*, neglecting the hydrodynamic interactions between the rods. Following the transformation of the matrix representation in Eqs. (13) and (14), we obtain

With the identity $\u2211k=1N\u2061sin2(2\pi k/N)=\u2211k=1N\u2061cos2(2\pi k/N)=N/2$ for *N* ≥ 3, these are calculated as

and similar relations also follow for *Q*_{1} and *Q*_{2}. The representation of the coupling tensor is obtained as

We now consider the coefficients in the shear-related tensor. A similar calculation leads to the following expressions for the shear-force tensor components:

and

The new term from the triangular geometry remains nonzero and is given by

which we can further reduce to

We note here that the identities $\u2211k=1N\u2061sin3(2\pi k/N)=\u2211k=1N\u2061cos3(2\pi k/N)=0$ hold only for *N* ≥ 4 and that for *N* = 3, the sums $\u2211k=13ck3=3/4$ and $\u2211k=13sk2ck=\u22123/4$ do not vanish, yielding a particular contribution for a *C*_{3}-object while this term does not appear for a helicoidal object.

Similar calculations lead to the following expressions for the shear-torque tensor components for Λ_{3}:

For Λ_{5}, however, we have

which should, in general, be nonzero from symmetry arguments but perhaps vanishes as a result of the neglect of the hydrodynamic interactions between the rods.

If we take *L* = 1 as the unit of length and *C*_{T} = −1 as the unit of force, then the parameters of the object under consideration are *c*/*L* and *ξ* = *C*_{N}/*C*_{T}, the latter of which typically ranges from 1.5 to 2 for cilia and flagella but can be less than unity for some flagellated protists with mastigonemes.^{71} The scalar components of the resistance tensors for a single rod are given^{8,72} by $K1(0)=\u22121$, $K2(0)=\u2212\xi $, $Q1(0)=0$, $Q2(0)=\u2212\xi (c2+c+1/3)$, and $C23(0)=\u2212\xi (c+1/2)$, and those for the shear-related tensors are similarly obtained^{8} as $\Gamma 1(0)=\u2212(c+1/2)$, $\Gamma 2(0)=\u2212\xi (c+1/2)$, and $\Lambda 2(0)=\u2212\xi (c2+c+1/3)$.

As the object under consideration possesses *C*_{3v}-symmetry, the nonzero constants in the angular dynamics are the *β*_{2} and *β*_{5} terms. Using the expressions above and substituting these values into Eqs. (55) and (58), we can calculate the values of *β*_{2} and *β*_{5}, depending on the geometrical parameters *c*/*L* and *ξ*. The results are shown in Fig. 9 using the Bretherton constant *β* (= −*β*_{2}). The tetrahedral rod complex has a positive Bretherton constant, which is around *β* ≈ 0.25, although it can be varied slightly by changing the morphology. The effective aspect ratio is then estimated as $c\u0303=(1+\beta )/(1\u2212\beta )\u22481.3$, which is the aspect ratio of a spheroid with the corresponding Bretherton constant. The value of the triangularity strength is obtained as *β*_{5} ≈ − 0.04 for a typical slender rod with *ξ* = 1.5–2.

Taking these values as representing the typical dynamics of the example of a tetrahedral rod complex, we have calculated the angular dynamics as in Sec. IV, and the Poincaré map and time evolution of the angle *θ* thus obtained are shown in Fig. 10. Even though the triangularity strength is relatively small owing to the nonzero Bretherton constant, there are attracting stable periodic orbits at *θ* = *π*/2. For a periodic orbit, the angle *ψ* satisfies sin 3*ψ* = 0, and two attracting orbits with *ψ* = *π*/3 and 5*π*/3 can be found in Fig. 10(a). The stable periodic motion is then a rotation around the background vorticity vector, with one of the rods being perpendicular to it, while the projection onto the *e*_{2}–*e*_{3} plane is a regular triangle with two of the rods facing away from the background vorticity vector. As the attraction is weak owing to the small value of |*β*_{5}|, the object exhibits quasiperiodic motions if the initial configuration is away from the stable point at *θ* = *π*/2.

## VI. CONCLUDING REMARKS

We have theoretically investigated the dynamics of a microscopic object, which can be either a passive particle or an active swimmer, with a *n*-fold rotational symmetry (*n* ≥ 3), which we have referred to in this paper as a *C*_{n}-object, using the Schoenflies notation. When *n* ≥ 4, the angular motions of such an object in a background linear flow are found to be described by the helicoidal Jeffery orbit, which is determined by the two shape parameters: the Bretherton constant and a constant associated with the object’s chirality.

Further symmetries can simplify the helicoidal Jeffery equations, and we have found that *C*_{nh}- and *S*_{2n}-objects can be categorized into a class of heterochiral objects, while *D*_{n}-objects belong to the class of homochiral objects, each of which classes contains a subclass of bodies of revolution. *C*_{nv}-, *D*_{nh}-, and *D*_{nd}-objects follow the Jeffery orbits of a body of revolution. These results illustrate that a wide variety of shapes of microscopic objects can be classified into a small number of hydrodynamic shapes that share the same dynamics in a flow. Even a discrete symmetry can be identical to a continuous symmetry as far as the dynamics in a flow are considered.

The unified description of the dynamics of a *C*_{n}-object as those of a helicoidal is only possible for *n* ≥ 4, and it is found that the motions of a *C*_{3}-object require further parameters in the dynamics, which reflect the triangularity of the object’s shape. In turn, this new effect of a *C*_{3}-object leads to a mechanical coupling between the direction of the axis of rotation and the other two perpendicular axes, and thus, the orientation dynamics need to be specified by three angles.

Owing to triangularity, the angular dynamics of a *C*_{3}-object can be chaotic, whereas the orbits typically possess stable periodic orbits when the Bretherton constant is nonzero. The existence of a stable periodic orbit has been confirmed for the example of an object consisting of four slender rods in a tetrahedral configuration. The basin of attraction is fairly small, and only slow attractions are possible owing to the small values of the triangularity constant for this particular object. The triangularity also generates a mechanical coupling between the self-propelling spin and the entire rotation induced by background shear, and it is found that the spin of the swimmer can trigger chaotic angular dynamics, which emphasizes the quantitative difference emerging from the triangularity of self-propelling swimmers.

As discussed in Sec. I, many microorganisms, such as diatoms and radiolarians, have shape with discrete rotational symmetry. The dynamics in a flow can be simply described by helicoidal Jeffery equations if the object has an *n*-fold rotational axis with *n* ≥ 4. Triangularity, however, will affect the dynamics of cells with a threefold rotational axis but without extra mirror symmetry, such as *Lithochytris pyramidalis*, *Thyrsocyrtis*, and *Podocyrtis*,^{46,47} for which the example of a tetrahedral rod complex presented here could be useful.

The inertia of particles could lead to chaotic motion for a spheroidal particle,^{73,74} although motion is stabilized in the ellipsoidal case.^{75} The notable difference between the angular dynamics of spheroids and ellipsoids suggests that small shape perturbations could dramatically affect the entire dynamics.^{55,56} The integrable Jeffery orbits are modulated by such small effects of geometry, inertia, or non-Newtonian rheology^{76,77} and could lead to chaotic motion. This is, however, in contrast to the case where the dynamical system possesses an orbit that is robustly stable under perturbations.^{38,78} This suggests that the stable orbits induced by triangularity and chirality with a nonzero Bretherton constant would not be collapsed by geometrical perturbations, and the equations of motion derived in this study should be useful for a biological particle or swimmer, which are not often exactly symmetric and can change their shape in time, although further studies will be needed to confirm this.

The classification of objects in terms of symmetries of hydrodynamic resistance tensors has been examined in the presence of gravity,^{79,80} which can be understood in terms of the hydrodynamic response of an object to a vectorial external force, while the classification of dynamics in shear is connected with the response to the external second-rank tensors. More recently, such symmetry-based arguments have been made in the context of a magnetic swimmer propelled by an external rotating magnetic field.^{81,82} Since symmetry arguments can be systematically applied, it is expected that classifications of objects in terms of their hydrodynamic response to higher-rank tensorial actions can be developed.

In conclusion, we have provided a general Jeffery equation for an object with *n*-fold rotational symmetry (*n* ≥ 3) with a new triangularity parameter in addition to the Bretherton constant and chirality parameter, and we have demonstrated that microscopic objects can be categorized into a small number of classes based on their dynamics in a background flow. Triangularity can lead to chaotic Jeffery orbits in a simple shear flow, although the dynamics typically possess stable periodic orbits. These findings emphasize the significance of the symmetry classification of an object for fluid–structure interactions at a low Reynolds number.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## ACKNOWLEDGMENTS

K.I. acknowledges JSPS-KAKENHI for Young Researchers (Grant No. 18K13456) and JST, PRESTO, Japan (Grant No. JPMJPR1921). This work was partially supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located at Kyoto University.

### APPENDIX: SYMMETRY ARGUMENTS FOR AN *S*_{2n}-OBJECT

In this appendix, we provide a derivation of the symmetric reductions of the shape-related hydrodynamic tensors for an *S*_{2n}-object. As an *S*_{2n}-object contains the symmetry denoted by *C*_{n}, we can parameterize the tensors as in Eqs. (19)–(30) in the main text. Let the angle Θ′ = *π*/*n* represent the 2*n*-fold rotation–reflection symmetry. We proceed by equating the representation by a Θ′/2-rotation together with reflection in the $e^2$–$e^3$ plane and the representation by a −Θ′/2-rotation without reflection. Noting that the coupling tensor ** C** needs an additional minus sign when the reflection operation is applied as in Eq. (14), we obtain the following relations for the matrix representation:

from which we have *C*_{1} = *C*_{2} = *C*_{23} = 0, whereas the rotation–reflection transformation does not affect the translational and rational resistance tensors ** K** and

**.**

*Q*We now proceed to the constraint imposed by the rotation–reflection symmetry on the shear-force tensor $\Gamma $. Equating the two representations leads to the following relations:

and

as in Eq. (28). However, the factor 4 cos^{2}(Θ′/2) − 1 does not vanish, since the angle Θ′/2 ≤ *π*/6 for *n* ≥ 3, and we obtain Γ_{4} = Γ_{5} = 0 and Γ_{1} = Γ_{2} = Γ_{3} = 0 from Eqs. (A4)–(A6).

For the shear-torque tensor, on the other hand, noting the additional sign in Eq. (14) from the reflection transformation, we find that no further constraints are obtained from the comparison between the two representations, except that

Nonetheless, 4 sin^{2}(Θ′/2) − 1 = 0 for *n* = 3, and thus, Λ_{4} and Λ_{5} are still nonzero for an *S*_{6}-object, while these vanish when *n* ≥ 4.