Physics of Fluids

ARTICLE

scitation.org/journal/phf

Reciprocal microswimmers in a viscoelastic fluid
Cite as: Phys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540
Submitted: 15 June 2020 • Accepted: 21 August 2020 •Published Online: 8 September 2020
Kento Yasuda (安田健人), Mizuki Kuroda (黒田瑞季), and Shigeyuki Komura (好村滋行)a)
AFFILIATIONS
Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Note: This paper is part of the Special Topic, Fluid-Structure Interactions: From Engineering to Biomimetic Systems.
a)Author to whom correspondence should be addressed: komura@tmu.ac.jp

ABSTRACT

We suggest several reciprocal swimming mechanisms that lead to locomotion only in viscoelastic fluids. In the first situation, we consider athree-sphere microswimmer with a difference in oscillation amplitudes for the two arms. In the second situation, we consider a three-spheremicroswimmer in which one of the frequencies of the arm motion is twice as large as the other one. In the third situation, we consider atwo-sphere microswimmer with a difference in size for the two spheres. In all these three cases, the average velocity is proportional to theimaginary part of the complex shear viscosity of a surrounding viscoelastic medium. We show that it is essential for a micromachine to break
its structural symmetry in order to swim in a viscoelastic fluid by performing reciprocal body motions.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0018540., s

I. INTRODUCTION
Microswimmers are small machines that swim in a fluid andhave potential applications in microfluidics and microsystems.1
Over the length scale of microswimmers, the fluid forces acting onthem are dominated by the frictional viscous forces. By transforming
chemical energy into mechanical energy, however, microswimmers
change their shape and move efficiently in viscous environments.
According to the scallop theorem suggested by Purcell, reciprocal body motion cannot be used for locomotion in a Newtonian
fluid.2–4 As one of the simplest models exhibiting non-reciprocal
body motion, Najafi and Golestanian proposed a three-sphere swimmer5,6 in which three in-line spheres are linked by two arms ofvarying length. Recently, such a swimmer has been experimentallyrealized by using colloidal beads manipulated by optical tweezers,7 ferromagnetic particles at an air–water interface,8,9 or neutrally
buoyant spheres in a viscous fluid.10
For many microswimmers in nature, however, the surrounding
fluid is not necessarily purely viscous, but, in general, viscoelastic.
Several studies have discussed the swimming behaviors of micromachines in different types of viscoelastic fluids.11–18 In particular, Lauga showed that the scallop theorem in a viscoelastic fluid
breaks down if the squirmer has a fore-aft asymmetry in its surface velocity distribution.13 In our recent study, we have discussed
the locomotion of a three-sphere microswimmer in a viscoelastic medium.19 Here, a relationship linking the average swimming

Phys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540
Published under license by AIP Publishingvelocity to the frequency-dependent viscosity of the surrounding
medium was derived. We demonstrated that the absence of the timereversal symmetry of the body motion (i.e., non-reciprocal motion)is reflected in the real part of the frequency-dependent complex
viscosity, whereas the absence of the structural symmetry of theswimmer shape is reflected in its imaginary part.19
Later, we investigated the locomotion of a three-spheremicroswimmer in a viscoelastic structured fluid, characterized bytypical length and time scales.20 The competition between the swimmer size and the characteristic length scale associated with the fluid
internal structure gives rise to the rich dynamics.21,22 The present
authors have also proposed a generalized three-sphere microswimmer model in which the spheres are connected by two harmonic
springs, i.e., an elastic microswimmer.23–26 It has been shown that anelastic microswimmer in a purely viscous fluid exhibits “viscoelastic”
effects as a whole.23,24
In this paper, employing either a three-sphere or a two-spheremicroswimmer, we suggest several swimming mechanisms that
include only reciprocal (rather than non-reciprocal) body motions
and can lead to locomotion only in viscoelastic fluids. According
to the scallop theorem,2–4 the considered reciprocal body motions
cannot be used for locomotion in a purely viscous fluid. For a threesphere swimmer in a viscoelastic fluid, the simplest reciprocal body
motion has been proposed in our previous work.19 This is possible when the two amplitudes of the oscillatory arm motion are
different, namely, when the structural symmetry of a three-sphere32, 093102-1

Physics of Fluids

ARTICLEmicroswimmer is broken. For the illustration of the calculation
scheme, we first explain this reciprocal motion even though theresult is a part of the calculation in Ref. 19.
We then suggest two other reciprocal swimming mechanisms
in a general viscoelastic fluid: a three-sphere microswimmer inwhich one of the frequencies of the arm motion is twice as large
as the other one and a two-sphere microswimmer with a difference in size for the two spheres. In all these three cases, we showthat the average velocity is proportional to the imaginary part ofthe complex shear viscosity that characterizes the elasticity of thesurrounding fluid. The suggested body motions highlight the essential swimming mechanism of a micromachine in viscoelastic fluids.
For the sake of clarity, we do not include any non-reciprocal body
motions of a microswimmer, as discussed in Ref. 19. Moreover, we
assume that the surrounding viscoelastic fluid is homogeneous anddo not consider any fluid internal structures as in Ref. 20.
In Sec. II, we briefly review Ref. 19 to show the basic equations
for the motion of a three-sphere swimmer in a general viscoelastic fluid. In Sec. III, we discuss the locomotion of a three-sphereswimmer when the two arm amplitudes are asymmetric, as alreadydiscussed in Ref. 19. In Sec. IV, we explain the case of asymmetricarm frequencies for a three-sphere swimmer. The generalization forhigher frequencies of the arm motion is also discussed. In Sec. V, we
present the result for an asymmetric two-sphere microswimmer in a
viscoelastic fluid. Finally, a summary of our work and a discussion isprovided in Sec. VI.
II. THREE-SPHERE MICROSWIMMERIN A VISCOELASTIC FLUID
The general equation that describes the hydrodynamics of a
low-Reynolds-number flow in a viscoelastic medium is given by thefollowing generalized Stokes equation:27
�

t

−∞

dt ′ η(t − t ′ )∇2 v(r, t ′ ) − ∇p(r, t) = 0.

(1)

Here, η(t) is the time-dependent shear viscosity, v is the velocity
field, p is the pressure field, and r stands for a three-dimensional
positional vector. The above equation is further subjected to theincompressibility condition,∇ ⋅ v = 0.

(2)

From these equations, one can obtain a linear relation between thetime-dependent force F(t) acting on a hard sphere of radius a and its
time-dependent velocity V(t). In the Fourier domain, this relation
can be represented asV(ω) =1F(ω),
6πη[ω]aPublished under license by AIP Publishing

Fig. 1, this model consists of three spheres of the same radius a. They
are connected by two arms of lengths L1 (t) and L2 (t), which undergo
time-dependent motion, as we will discuss separately in Secs. III and
IV. Moreover, the radius of the two arms is assumed to be negligibly
small. If we define the velocity of each sphere along the swimmer
axis as V i (t) (i = 1, 2, 3), we haveL̇1 (t) = V2 (t) − V1 (t),L̇2 (t) = V3 (t) − V2 (t),

(4)
(5)

where L̇1 and L̇2 indicate the time derivatives of L1 and L2 ,respectively.
Owing to the hydrodynamic effect, each sphere exerts a force
F i on the viscoelastic medium and experiences a force −F i fromit. To relate the forces and the velocities in the frequency domain,
we use the GSR in Eq. (3) and the Oseen tensor in which thefrequency-dependent viscosity η[ω] is used instead of a constantone.33,34 Assuming that a � L1 , L2 , we can write the three velocities
V i (ω) as5,6
V1 (ω) =V2 (ω) =V3 (ω) =F1 (ω)
F2 (ω) ∗ L−1
1
1 (ω)
+
6πη[ω]a 4πη[ω]2πF3 (ω) ∗ (L1 + L2 )−1 (ω)
1,4πη[ω]2π(6)F3 (ω) ∗ L−1
1
2 (ω)
,
4πη[ω]2π(7)

F2 (ω) ∗ L−1
F3 (ω)12 (ω)+,
4πη[ω]
2π
6πη[ω]a(8)+

F1 (ω) ∗ L−1
F2 (ω)11 (ω)+
4πη[ω]
2π
6πη[ω]a+F1 (ω) ∗ (L1 + L2 )−1 (ω)
14πη[ω]
2π+

where we have used bilateral Fourier transforms such as L−11 (ω)−1 −iωt= �∞. Furthermore, the convolution of two func−∞ dt [L1 (t)] e′
tions is generally defined by g1 (ω) ∗ g2 (ω) = �∞−∞ dω g1 (ω′′− ω )g2 (ω ) in the above equations.
Since we are interested in the autonomous net locomotion of
the swimmer, there are no external forces acting on the spheres.

(3)

where we use a bilateral Fourier transform for V(ω)
−iωt
−iωt
= �∞
and F(ω) = �∞, while we employ−∞ dt V(t)e−∞ dt F(t)e∞−iωta unilateral one for η[ω] = �0 dt η(t)e . Equation (3) is the generalized Stokes-Einstein relation (GSR), which has been successfully
used in active microrheology experiments,28–30 and its mathematical
validity has also been discussed.31,32
Next, we briefly explain the three-sphere micromachine model
proposed by Najafi and Golestanian.5,6 As schematically shown inPhys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540

scitation.org/journal/phf

FIG. 1. Najafi–Golestanian three-sphere swimmer model. Three identical spheresof radius a are connected by arms of lengths L1 (t) and L2 (t), and they undergo
time-dependent cyclic motions [see Eqs. (10) and (11) or Eqs. (18) and (19)].
Such a microswimmer is embedded in a viscoelastic medium characterized bya frequency-dependent complex shear viscosity η[ω]. In this work, we consider
only reciprocal body motions.

32, 093102-2

Physics of Fluids

ARTICLE

Neglecting the inertia of the surrounding fluid, we require thefollowing force balance condition:
F1 (ω) + F2 (ω) + F3 (ω) = 0.

(9)Since Eqs. (6)–(8) involve convolutions in the frequency
domain, we cannot solve these equations for arbitrary L1 (t) andL2 (t). In Secs. III–V, we assume three different reciprocal arm
motions for L1 (t) and L2 (t) and obtain the average velocity of a
microswimmer in a viscoelastic fluid.
III. ASYMMETRIC ARM AMPLITUDES
We first consider the case when the amplitudes of the two arms
are different. We assume that the two arms undergo the following
reciprocal periodic motion:
L1 (t) = ` + d1 cos(Ωt),
L2 (t) = ` + d2 cos(Ωt).

(10)
(11)

In the above equations, ` is the constant length, d1 and d2 are theamplitudes of the oscillatory motion, and Ω is the common arm
frequency. It should be emphasized that, in contrast to Ref. 19, we
do not include any difference in the phases between the two arms,
and hence, the whole body motion is reciprocal. On the other hand,
we characterize the structural symmetry of the swimmer by d1 andd2 . The whole micromachine is symmetric when d1 = d2 , while it isasymmetric when d1 ≠ d2 .
Since the arm frequency is Ω, we assume that the velocities andthe forces of the three spheres can generally be written as∞

Vi (ω) = Vi,0 δ(ω) + �[Vi,n δ(ω + nΩ) + Vi,−n δ(ω − nΩ)], (12)n=1∞

Fi (ω) = Fi,0 δ(ω) + �[Fi,n δ(ω + nΩ) + Fi,−n δ(ω − nΩ)].
n=1

(13)

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Since Eq. (15) involves η′′ [Ω], it can be regarded as an elastic
contribution that exists when the structural symmetry of the swimmer is broken, i.e., d1 ≠ d2 . In other words, a reciprocal three-spheremicromachine uses the elastic degree of freedom of the surrounding
viscoelastic medium for its locomotion. The structural asymmetry,d1 ≠ d2 , is necessary for a microswimmer to determine its moving
direction. For a purely Newtonian fluid, namely, for a medium characterized by a constant viscosity, Eq. (15) vanishes even when d1 ≠ d2
because η′′ [Ω] = 0. The above result also implies that a three-sphereswimmer cannot move in a purely elastic medium for which we have
η0 → ∞.
When the arm motion is non-reciprocal, such as by introducing
a phase difference between the two arms, a different term arises.19,20
This term includes η′ [Ω] and hence can be regarded as the viscous
contribution. Because Eq. (15) contributes to the average velocity
even for a reciprocal body motion, the scallop theorem should be
generalized for a three-sphere swimmer in a viscoelastic medium.13
To illustrate the above result, we assume that the surrounding
viscoelastic medium is described by a simple Maxwell model.19 In
this case, the frequency-dependent complex viscosity can be writtenasη[ω] = η0

1 − iωτ,
1 + ω2 τ 2(16)

where τ is the characteristic time scale. Within this model, themedium behaves as a viscous fluid for ωτ � 1, while it becomes
elastic for ωτ � 1. Using Eq. (16), we can easily obtain the averageswimming velocity in Eq. (15) as19
V=

5(d12 − d22 )aΩ Ωτ
.
48`2
1 + Ω2 τ 2(17)Here, V increases as V ∼ Ω2 for Ωτ � 1, and it approaches aconstant for Ωτ � 1. In Fig. 2, we plot the dimensionless average

Substituting Eqs. (12) and (13) into the six coupled equations (4)–(9), we obtain, in general, a matrix equation with infinite dimensions.
Under the conditions d1 , d2 � ` and a � `, we are allowed
to consider only n = 0, ±1, and we further use the approximation
F i,±2 ≈ 0. Then, we can solve for the six unknown functions V i (ω)
and F i (ω) and also calculate the total swimming velocity,
1
V = (V1 + V2 + V3 ).
3(14)

Up to the lowest order terms in a, the average swimming velocity
over one cycle of motion becomes19
V =−5a(d12 − d22 )Ω ′′η [Ω],
48`2 η0(15)

where η′′ [Ω] is the imaginary part of the complex shear viscosity,
η[Ω] = η′ [Ω] + iη′′ [Ω], and η0 = η[Ω → 0] is the constant zerofrequency viscosity. A detailed derivation of Eq. (15) is given in the
Appendix. Note that η′′ [Ω] is taken to be negative in our notation.
Hence, V > 0 when d1 > d2 .
Phys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540
Published under license by AIP Publishing

FIG. 2. Average swimming velocity V as a function of Ωτ, where Ω is the arm
frequency and τ is the characteristic time scale in the Maxwell model. Here, V is
scaled by 5(d12 − d22 )a�(48`2 τ) assuming that d 1 ≠ d 2 . V increases as V ∼ Ω2
for Ωτ � 1.

32, 093102-3

Physics of Fluids

ARTICLEswimming velocity V as a function of the dimensionless arm frequency Ωτ when d1 ≠ d2 .
IV. ASYMMETRIC ARM FREQUENCIESAs the second case, we consider the situation where the frequencies of the two arms are different. For the sake of simplicity,
we consider here the following time dependencies:
L1 (t) = ` + d cos(Ωt),

L2 (t) = ` + d cos(2Ωt).

(18)
(19)

In the above equations, the frequency of L2 is twice as large as thatof L1 , whereas the amplitude of oscillation d is taken to be the same.
Since the arm frequencies are different, a phase shift does not play
any role, and the overall arm motion can be regarded as reciprocal
for Eqs. (18) and (19).
The procedure to obtain the average velocity is essentially thesame as in Sec. III. We assume that the velocities and the forces of thethree spheres are also expressed by Eqs. (12) and (13), respectively.
Under the conditions d � ` and a � `, we consider only n = 0, ±1,±2 and use the approximation F i,±3 ≈ 0 because of Eq. (19). Aftersome calculation, the average swimming velocity can be obtained as
5ad2 Ω ′′�η [Ω] − 2η′′ [2Ω]�.
V =−48`2 η0(20)

Similar to Eq. (15), only the imaginary part of the complex
shear viscosity appears in the above expression, and the two terms
in Eq. (20) are the elastic contributions. The above result meansthat a micromachine can swim as long as η′′ [Ω] ≠ 2η′′ [2Ω], whichusually holds for viscoelastic fluids. It is interesting to note thatthe direction of locomotion is determined by the relative magnitude between η′′ [Ω] and 2η′′ [2Ω]. When the arm amplitudes aredifferent and characterized by d1 and d2 , as in Eqs. (10) and (11),
we have confirmed that the average velocity is then proportional tod12 η′′ [Ω] − 2d22 η′′ [2Ω], as one can expect from Eqs. (15) and (20).
In general, the motions of the two arms can be given by
L1 (t) = ` + d cos(Ωt),

L2 (t) = ` + d cos(mΩt),

(21)
(22)

where m is an integer. Note that the average velocity vanishes form = 1 even in a viscoelastic fluid because the arm amplitudes are thesame in Eqs. (21) and (22). Although we have explicitly calculated
only up to m = 3, we speculate that the average velocity can be givenbyV =−5ad2 Ω ′′�η [Ω] − mη′′ [mΩ]�,48`2 η0

Published under license by AIP Publishing

this case, we predict, in general, that the first term in Eq. (23) will
be replaced by Mη′′ [MΩ], which results from the symmetry of oursystem.
V. ASYMMETRIC TWO-SPHERE MICROSWIMMERAs the third reciprocal body motion, we consider a two-sphereswimmer consisting of two hard spheres having different sizes. As
shown in Fig. 3, these two spheres are connected by a single arm thatcan vary its length. The radii of the two spheres are denoted by a1
and a2 , respectively, and the distance between them is L(t). As theequations of motion for the two spheres are even simpler than those
for a three-sphere swimmer, we shall explicitly write them below.
Similar to Eqs. (4) and (5), the time derivative of L is given byL̇(t) = V2 (t) − V1 (t).

(24)

Corresponding to Eqs. (6)–(8), the relations between the velocities
and the forces in the frequency domain can be written as
V1 (ω) =V2 (ω) =F1 (ω)
F2 (ω) ∗ L−1 (ω)
1+,6πη[ω]a1 4πη[ω]2πF1 (ω) ∗ L−1 (ω)
F2 (ω)
1+.
4πη[ω]
2π
6πη[ω]a2

(25)
(26)

Finally, the force balance equation now becomes
F1 (ω) + F2 (ω) = 0.

(27)

The periodic arm motion is assumed to have the following
simple form:
L(t) = ` + d cos (Ωt).

(28)

Since there is only one arm, it is obvious that any periodic arm
motion is inevitably reciprocal. Under the conditions d � ` anda1 , a2 � `, we consider only n = 0, ±1 and use the approximation
F i,±2 ≈ 0 in Eqs. (12) and (13). Calculating the total swimming velocity V = (V 1 + V 2 )/2, we finally obtain the average swimming velocity
over one cycle of motion asV=

3a1 a2 (a1 − a2 )d2 Ω ′′
η [Ω].
4`2 (a1 + a2 )2 η0

(29)

(23)

which is a natural generalization of Eq. (20). When m is very large,
the first term becomes negligible, and the whole locomotion isdominated by η′′ [mΩ].
One can further generalize Eq. (21) to L1 (t) = ` + d cos(MΩt),where M is another integer, while L2 is still given by Eq. (22), but
M ≠ m. Then, the least common multiple of M and m determines
the period of the overall reciprocal motion of a micromachine. In

Phys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540

scitation.org/journal/phf

FIG. 3. Asymmetric two-sphere swimmer model. Two spheres of different radii
a1 and a2 (a1 < a2 ) are connected by an arm of length L(t), and it undergoes
a time-dependent periodic motion [see Eq. (28)]. The swimmer is embedded ina viscoelastic medium characterized by a frequency-dependent complex shear
viscosity η[ω].

32, 093102-4

Physics of Fluids

ARTICLEThis result shows that a reciprocal two-sphere micromachine
can swim in a viscoelastic fluid when the sphere sizes are different, i.e., a1 ≠ a2 . Similar to the previous cases, the average velocity
depends only on η′′ [Ω], and it is due to the elastic contribution.
Hence, the elasticity of a viscoelastic medium is responsible for the
locomotion of a reciprocal microswimmer as long as its structure
is asymmetric. This statement does not contradict with the original
scallop theorem that holds only for purely viscous fluids.2–4 Whenthe surrounding fluid is purely elastic, however, the average velocity
V vanishes because η0 → ∞.
In the limit of a1 � a2 , for example, Eq. (29) further reduces toV ≈−3a1 d2 Ω ′′
η [Ω].
4`2 η0(30)This result shows that the average velocity of a two-sphere swimmer
is proportional to the radius of the smaller sphere, a1 . Since η′′ [Ω]< 0 by definition, V > 0 in the limit of Eq. (30).
Here, we discuss the connection between a three-spheremicroswimmer and a two-sphere microswimmer considered inSecs. IV and V, respectively. According to the average velocity in
Eq. (23) for a three-sphere microswimmer, its locomotion is dominated by η′′ [mΩ] when m � 1. In such a situation, the motion of
the first arm L1 appears to be stagnant when compared with that of
the second arm L2 . Note that the limiting expression of Eq. (23) form � 1 is similar to the average velocity in Eq. (30) for a highly
asymmetric two-sphere microswimmer, i.e., a1 � a2 . Although thenumerical factors are different between these two limiting expressions, their dependence on the structural and dynamical parameters
is identical. Such a similarity between a three-sphere microswimmer and a two-sphere microswimmer is an interesting feature ofreciprocal micromachines in a viscoelastic fluid.
VI. SUMMARY AND DISCUSSION
In this paper, employing either a three-sphere or a twosphere microswimmer, we have suggested three reciprocal swimming mechanisms that can lead to locomotion only in viscoelastic
fluids. In the first situation, we consider a three-sphere microswimmer with a difference in oscillation amplitudes for the two arms.19
In the second situation, we consider a three-sphere microswimmer in which one of the frequencies of the arm motion is twice aslarge as the other one. In the third situation, we consider a twosphere microswimmer with a difference in size for the two spheres.
In all these three cases, the average velocity is proportional to theimaginary part of the complex shear viscosity that characterizes theelastic property of the surrounding viscoelastic fluid. Hence, it isessential for a micromachine to break its structural symmetry inorder to swim in viscoelastic fluids by performing reciprocal body
motions. Our result also indicates that the scallop theorem shouldbe generalized for microswimmers in a viscoelastic fluid.
Lauga considered an axisymmetric squirming motion of aspherical squirmer embedded in an Oldroyd-B fluid, which represents a typical polymeric fluid.13 It was reported that the scallop
theorem in a viscoelastic fluid breaks down if the squirmer has afore-aft asymmetry in its surface velocity distribution, which is inaccordance with our result. On the other hand, Curtis and Gaffney

Phys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540
Published under license by AIP Publishingscitation.org/journal/phf

showed that the swimming velocity in a viscoelastic medium is thesame as that in a Newtonian fluid.15 Recently, the motion of a twosphere swimmer in viscoelastic fluids has been discussed by Dattet al.18 However, their calculations are limited to an Oldroyd-B fluid.
Our treatment using the GSR in Eq. (3) is more general because we
do not specify any frequency dependence of the complex shear viscosity. We emphasize that our theory applies for all types of linear
viscoelastic fluids.
The scallop theorem states that a microswimmer cannot gain
any net displacement after one cycle of reciprocal body motion when
the surrounding fluid is purely viscous.2,3 It should be noted that
this theorem is correct only when the Reynolds number strictly vanishes.4 Lauga showed that oscillatory reciprocal forcing of a solid
body leads to net translational motion when the Reynolds number
is nonzero even when the fluid is purely viscous.35 It was further
predicted that the scallop theorem breaks down with inertia in a continuous manner as long as there are some spatial broken symmetries
that govern the direction of the net motion. In the future, it would
be interesting to see the effects of inertia for a reciprocal microswimmer in a viscoelastic fluid and to elucidate how the scallop theorem
needs to be extended in more general situations.
Even though the argument in this work is restricted to an artificial microswimmer, we expect that the basic concept can be applied
to more complex biological processes such as the motion of bacteria,
flagellated cellular swimming, and the beating of cilia. Since most ofthese phenomena take place in a viscoelastic environment, we hopethat the suggested mechanisms in this paper will be applicable formore complex biological swimming objects.
ACKNOWLEDGMENTS
We thank T. Kato and Y. Hosaka for useful discussions. We
also thank S. Al-Izzi for his critical reading of this manuscript. K.Y.
acknowledges support from Grants-in-Aid for JSPS Fellows (Grant
No. 18J21231) from the Japan Society for the Promotion of Science(JSPS). S.K. acknowledges support from the Grants-in-Aid for Scientific Research (C) (Grant Nos. 18K03567 and 19K03765) from the
JSPS and also from Grants-in-Aid for Scientific Research on Innovative Areas “Information Physics of Living Matters” (Grant No.
20H05538) from the Ministry of Education, Culture, Sports, Scienceand Technology of Japan.
APPENDIX: DERIVATION OF EQ. (15)In this appendix, we show the detailed derivation of Eq. (15).
Substituting Eqs. (10) and (12) into Eq. (4), we obtain
V2,0 − V1,0 = 0,

V2,1 − V1,1 = −iπd1 Ω,V2,−1 − V1,−1 = iπd1 Ω,

V2,n − V1,n = 0 for �n� ≥ 2.

(A1)
(A2)
(A3)
(A4)

Similarly, substituting Eqs. (11) and (12) into Eq. (5), we obtain
V3,0 − V2,0 = 0,

(A5)32, 093102-5

Physics of Fluids

ARTICLEV3,1 − V2,1 = −iπd2 Ω,V3,−1 − V2,−1 = iπd2 Ω,V3,n − V2,n = 0 for �n� ≥ 2.

(A6)

DATA AVAILABILITY(A7)

The data that support the findings of this study are available
from the corresponding author upon reasonable request.

(A8)

REFERENCES

Next, we expand Eqs. (6)–(8) in terms of the small quantities
d1 /` and d2 /` while keeping only the lowest order terms. Substituting
Eqs. (12) and (13) into these three equations, we obtain
V1,n ≈

V2,n ≈

V3,n ≈

F1,n1
d1 F2,n+1+
�F2,n −
6πη[−nΩ]a 4πη[−nΩ]`2`
d1 F2,n−1
1
F3,n d1 F3,n+1
−
�+
�
−
2`
4πη[−nΩ]` 2
8`
d1 F3,n−1 d2 F3,n+1 d2 F3,n−1
−
−
−
�,
8`
8`
8`

1
d1 F1,n+1 d1 F1,n−1
�F1,n −−
�
4πη[−nΩ]`
2`
2`
F2,n
1
d2 F3,n+1++
�F3,n −
6πη[−nΩ]a 4πη[−nΩ]`2`d2 F3,n−1
−
�,
2`

1
F1,n d1 F1,n+1 d1 F1,n−1�−−
4πη[−nΩ]` 2
8`8`
d2 F1,n+1 d2 F1,n−1
1
−
−
�+
8`
8`
4πη[−nΩ]`d2 F2,n+1 d2 F2,n−1
F3,n× �F2,n −
−
�+
.
2`
2`
6πη[−nΩ]a

(A9)

(A10)

(A11)

Note that the couplings between different n-modes are involvedin these equations. Finally, substituting Eq. (13) into Eq. (9), we
obtainF1,n + F2,n + F3,n = 0.

(A12)

The above set of equations constitutes a matrix equation with
infinite dimensions and cannot be solved, in general. Under theassumption of a � `, however, we are allowed to consider onlyn = −1, 0, 1 and further approximate as F i,±2 ≈ 0. The justification of
the latter approximation is also seen by solving Eqs. (A4) and (A8)–(A12) for n = ±2 and taking the limit of a � `. Hence, the above setof equations can be solved for 18 unknowns, i.e., V i,n and F i,n for
i = 1, 2, 3 and n = −1, 0, 1.
The velocity of each sphere is simply obtained by the inverseiωtFourier transform, Vi (t) = (2π)−1 �∞−∞ dω Vi (ω)e . The averageswimming velocity over one cycle of motion is then calculated
by
V=

2π�Ω
Ω
dt [V1 (t) + V2 (t) + V3 (t)]�3.
�
2π 0

(A13)

Up to the lowest order terms in a, we finally obtain Eq. (15). In orderto obtain more accurate higher order terms in a, one needs to take
into account the higher order n-modes (|n| ≥ 2). Equations (20) and(29) can be obtained similarly.

Phys. Fluids 32, 093102 (2020); doi: 10.1063/5.0018540
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