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Journal of the Physical Society of Japan 88, 054804 (2019)https://doi.org/10.7566/JPSJ.88.054804

Hydrodynamic Interaction between Two Elastic Microswimmers
Mizuki Kuroda, Kento Yasuda, and Shigeyuki Komura+
Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan(Received January 24, 2019; accepted March 11, 2019; published online April 25, 2019)

We investigate the hydrodynamic interaction between two elastic swimmers composed of three spheres and two
harmonic springs. In this model, the natural length of each spring is assumed to undergo a prescribed cyclic change,
representing the internal states of the swimmer [K. Yasuda et al., J. Phys. Soc. Jpn. 86, 093801 (2017)]. We obtain the
average velocities of two identical elastic swimmers as a function of the distance between them for both structurally
asymmetric and symmetric swimmers. We show that the mean velocity of the two swimmers is always smaller than thatof a single elastic swimmer. The swimming state of two elastic swimmers can be either bound or unbound depending on
the relative phase diﬀerence between them.

1.

Introduction

Microswimmers are small machines that swim in a ﬂuid
and they are expected to be used in microﬂuidics andmicrosystems.1) Over the length scale of microswimmers, theﬂuid forces acting on them are dominated by the frictional
viscous forces. By transforming chemical energy intomechanical energy, however, microswimmers change theirshape and move eﬃciently in viscous environments.
According to Purcell’s scallop theorem, time-reversible body
motion cannot be used for locomotion in a Newtonian
ﬂuid.2,3) As one of the simplest models exhibiting broken
time-reversal symmetry, Najaﬁ and Golestanian proposeda three-sphere swimmer (NG swimmer),4,5) in which three
in-line spheres are linked by two arms of varying length.
Recently, such a swimmer has been experimentally realized
by using colloidal beads manipulated by optical tweezers,6)
ferromagnetic particles at an air-water interface,7,8) or
neutrally buoyant spheres in a viscous ﬂuid.9)
Using the NG swimmer model, Pooley et al. showed thatthe interaction between two swimmers depends on theirrelative displacement, orientation, and phase, leading tomotion that can be either attractive, repulsive, or oscillatory.10) Scattering of two NG swimmers was also investigatedon the basis of the time-reversal invariance of the Stokes
equation.11) Later Farzin et al. reexamined the hydrodynamic
interaction between two NG swimmers and concluded thatthe long-time swimming states are diﬀerent between moving
in the same and opposite directions.12) To understandhydrodynamic coupling for stochastic swimmers, on theother hand, Najaﬁ and Golestanian studied the correlated
motion of a three-sphere swimmer and a two-spheresystem.13) They calculated the swimming velocities asfunctions of the statistical transition rates for the conformational changes.
Recently, the present authors have proposed a generalized
three-sphere microswimmer model in which the spheres areconnected by two harmonic springs, i.e., an elastic microswimmer.14) Compared with the NG swimmer, the main
diﬀerence of the elastic swimmer is that the natural length ofeach spring (rather than the arm length) oscillates in time andis assumed to undergo a prescribed cyclic change. As a result,the sphere motion in our model is determined by the natural
spring lengths, representing the internal states of a swimmer,
and also by the force exerted by the ﬂuid. We haveanalytically obtained the average swimming velocity as afunction of the frequency of the cyclic change in the naturallength.14) In the low-frequency region, the swimming
velocity increases with frequency and it reduces to that of
the NG swimmer.4,5) Conversely, in the high-frequency
region, the velocity decreases with increasing frequency. We
note that similar models were proposed by other people,15–17)
while our elastic swimmer model was further extended tothermally driven elastic micromachines.18)
In this work, we investigate the hydrodynamic interaction
between two elastic three-sphere swimmers that are conﬁned
in one-dimensional space and moving in the same direction.
We ﬁrst derive a general expression for the average velocities(over a period of one cycle) of two hydrodynamicallyinteracting three-sphere swimmers as a function of the
distance between them. Using this general expression, wethen calculate the explicit forms of the average velocities of
two identical elastic microswimmers. We show that the meanof the two average velocities is always smaller than that ofa single elastic swimmer, whereas the velocity diﬀerence
depends on the relative phase diﬀerence in the natural lengths
between the two swimmers. As a result, the swimming stateof two elastic swimmers can be either bound or unbounddepending on the relative phase diﬀerence.
In Sect. 2, we ﬁrst discuss the motion of two interacting
three-sphere microswimmers. In Sect. 3, we calculate theaverage velocities of two interacting elastic swimmers, andfurther discuss the mean and the diﬀerence between the two
average velocities. The average velocities of two symmetricelastic swimmers is discussed in Sect. 4. In Sect. 5, we
discuss the interaction of two NG swimmers by considering
the low-frequency limit of our results. Finally, a summary of
our work and some discussion are given in Sect. 6.
2.

Two Interacting Three-sphere Swimmers

As shown in Fig. 1, we consider two general three-sphereswimmers in a viscous ﬂuid characterized by shear viscosity
η. Each swimmer consists of three hard spheres of the sameradius a connected either by two arms (NG swimmer) or by
two harmonic springs (elastic swimmer explained in the next
section) A and B. The positions of the three spheres in theleft (L) swimmer are denoted by x1 , x2 , and x3 in a onedimensional coordinate system, while those in the right (R)swimmer are denoted by x4 , x5 , and x6 . We also assume
x1 < x2 < x3  x4 < x5 < x6 without loss of generality. The

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J. Phys. Soc. Jpn. 88, 054804 (2019)

ηx1 KA x2 KB x3
a

L
A

L
B

M. Kuroda et al.

suﬃcient equations to solve the twelve unknowns, namely, x_i
and fi (i ¼ 1; . . . ; 6). Finally the average velocities of the left
and right swimmers can be obtained byx4 KA x5 KB x6
a

R
A

R
B

VL ¼
Fig. 1. (Color online) Two elastic three-sphere microswimmers in a
viscous ﬂuid characterized by shear viscosity η. The positions of the threespheres in the left (L) swimmer are denoted by x1 , x2 , and x3 in a onedimensional coordinate system, while those in the right (R) swimmer are
denoted by x4 , x5 , and x6 . The distance between these two swimmers is
deﬁned by D ¼ x5  x2 . In each elastic swimmer, three identical spheres ofradius a are connected by two harmonic springs A and B characterized byelastic constants KA and KB , respectively. The four natural lengths of thesprings, ‘LA , ‘LB , ‘RA , and ‘RB , depend on time and are assumed to undergo
cyclic changes as given by Eqs. (13)–(16).

distance between the two swimmers is deﬁned by the
positions of the middle spheres, i.e., D ¼ x5  x2 .
Owing to the hydrodynamic interaction, each sphere exerts
a force on the viscous ﬂuid and is subjected to an opposite
force from it. Denoting the velocity of each sphere by
x_i ¼ dxi =dt and the force acting on each sphere by fi(i ¼ 1; . . . ; 6), we can write the equations of motion of eachsphere as
6
X
x_i ¼
Mij fj ;ð1Þ
j¼1

where the details of the hydrodynamic interactions are taken
into account through the mobility coeﬃcients Mij . Within
Oseen’s approximation, which is justiﬁed when the spheresare considerably far from each other (a  jxi  xj j), theexpressions for the mobility coeﬃcients Mij can be written
as
8 1>
i ¼ j,>< 6a ;
ð2Þ
Mij ¼
>
1
>
:
; i ≠ j.
4jxi  xj j
We remark here that although the two swimmers are aligned
along the one-dimensional axis, the spheres are interacting
through the three-dimensional hydrodynamic interaction.
Furthermore, we require two force-free conditions of thetwo swimmers, i.e.,
f1 þ f2 þ f3 ¼ 0;

f4 þ f5 þ f6 ¼ 0:

ð3ÞLet us denote the average arm length (NG swimmer) or the
average natural length (elastic swimmer) by ‘, and also
introduce the four displacements of the springs with respectto ‘ for the left and right swimmers as
uLA ¼ x2  x1  ‘; uLB ¼ x3  x2  ‘;ð4ÞuRA ¼ x5  x4  ‘;

uRB ¼ x6  x5  ‘:

ð5Þ

Then the four kinematic constraints are given by taking thetime derivative of the above relations:u_LA ¼ x_2  x_1 ; u_LB ¼ x_3  x_2 ;ð6Þ
u_RA ¼ x_5  x_4 ;u_RB ¼ x_6  x_5 :ð7ÞWe note here that Eq. (1) implies six coupled equations.
Together with the two force-free conditions in Eq. (3) and thefour kinematic constraints in Eqs. (6) and (7), we have1
hx_1 þ x_2 þ x_3 i;
3

VR ¼

1
hx_4 þ x_5 þ x_6 i;3ð8Þwhere averaging h  i is performed by time integration in afull cycle.
Under the condition that the two swimmers are far fromeach other and the deformations are small compared with theaverage arm length ‘, i.e., a  uL,RA,B  ‘  D, one canperform a perturbative calculation to obtain the averagevelocities as
7a
VL ¼
huL u_L  uLB u_LA i24‘2 A Ba‘ 3 huRA u_RB  uRB u_RA  uLA u_RA  uLA u_RB þ uLB u_RA þ uLB u_RB i;
D
ð9Þ
7a
VR ¼
huR u_R  uRB u_RA i24‘2 A Ba‘ 3 huLA u_LB  uLB u_LA  uRA u_LA  uRA u_LB þ uRB u_LA þ uRB u_LB i:
Dð10Þ
Note that we have kept only up to second-order terms in uL,R
A,B
as in Ref. 13, meaning that we are also assuming thecondition uL,R
A,B =‘  ‘=D. The ﬁrst terms on the right-handside of the above equations represent the average swimming
velocity of a single three-sphere swimmer, as previously
obtained by Golestanian and Ajdari.5) These terms indicatethat the average velocity of an isolated three-sphere swimmer
is determined by the area enclosed by the orbit of the periodic
motion in the conﬁguration space.
The second terms on the right-hand side of Eqs. (9) and(10) are due to the hydrodynamic interaction between the two
swimmers. These correction terms decay as ð‘=DÞ3 withincreasing distance because they result from force quadrupoles rather than force dipoles.5) In fact, such a cubic
dependence originates from the symmetry such that the
motion of three-sphere swimmers is invariant under acombined time-reversal and parity transformation.10) The
correction terms huRA u_RB  uRB u_RA i in V L and huLA u_LB  uLB u_LA i
in V R are both passive terms because they correspond to theswimming of only the second swimmer. The other correction
terms are due to the simultaneous motion of the two
swimmers and hence are called active terms.10,12) We show
later that only the active terms depend on the phase diﬀerence
between the two swimmers.
3.

Two Interacting Elastic Swimmers

In this section, we consider two interacting elastic threesphere swimmers, as schematically shown in Fig. 1, andcalculate their average velocities. We ﬁrst assume that these
two elastic swimmers have identical structures, whereas the
structure of each swimmer can be either asymmetric orsymmetric (as separately discussed in Sect. 4). For eachswimmer, the two spring constants of harmonic springs A
and B are denoted by KA and KB , respectively. Thenthe total energy of these two elastic swimmers is givenby054804-2©2019 The Physical Society of Japan

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J. Phys. Soc. Jpn. 88, 054804 (2019)

E¼

M. Kuroda et al.

KAKBðx2  x1  ‘LA Þ2 þ
ðx3  x2  ‘LB Þ2
2
2
KA
KB
ðx5  x4  ‘RA Þ2 þ
ðx6  x5  ‘RB Þ2 :
þ
2
2

ð11Þ

In the above, ‘LA , ‘LB , ‘RA , and ‘RB are the natural lengths of
the respective harmonic springs and generally depend ontime. Hence, the six forces in Eq. (1) are given byfi ¼ 

@E
:
@xi

ð12Þ

For these two elastic swimmers, we assume that the four
natural lengths of the springs undergo the following periodic
changes in time:12)‘LA ðtÞ ¼ ‘ þ dA cosðtÞ;
ð13Þ
‘LB ðtÞ ¼ ‘ þ dB cosðt  Þ;

ð14Þ‘RA ðtÞ ¼ ‘ þ dA cosðt  Þ;

ð15Þ

ð16Þ
‘RB ðtÞ ¼ ‘ þ dB cosðt    Þ:Here, ‘ is the common average length as introduced in
Eqs. (4) and (5), dA and dB are the amplitudes of theoscillatory change, Ω is the common frequency, ϕ is therelative phase diﬀerence between the two springs within theswimmers, and Ψ is the relative phase diﬀerence between theleft and right swimmers. For each swimmer to move by itself,
the time-reversal symmetry of the spring dynamics should be
broken, i.e.,  ≠ 0. In the absence of the hydrodynamic
interaction between the two swimmers, they move in thesame direction with the same velocity. Although thisassumption can be relaxed, the current situation alreadyprovides us with very rich dynamical behaviors when theyinteract hydrodynamically. We also note that the frequency Ω
can be diﬀerent between the two swimmers, but such a study
is left as a future work.
It is convenient to introduce a characteristic time scaledeﬁned by14)
¼

6a
:
KA

ð17Þ

Then we use ‘ to scale all the relevant lengths and employ τ^ ¼ . By further deﬁning theto scale the frequency, i.e., ratio between the two spring constants as  ¼ KB =KA , thecoupled equations can be made dimensionless. Theseequations can be solved in the frequency domain, and wefurther obtain uL,R
A,B in Eqs. (6) and (7) after an inverse Fouriertransform.14) Since their explicit expressions are somewhat
lengthy, we give them in Appendix A. Finally, using Eqs. (9)and (10), we calculate the average velocities V L and V R of
the two elastic swimmers. Their full expressions are given in
Appendix B.
Instead, we show here the mean and the diﬀerence between
the two average velocities V L and V R . The former is given byVR þ VL
48‘3 2¼ V0 1 
sin ð=2Þ ;
ð18Þ
2
7D3
where the average velocity of a single elastic swimmer was
obtained before as14)
V0 ¼

7dA dB a^ Þ sin 
F1 ð;
24‘2 

7ð1  ÞdA dB a
^ Þ cos 
F2 ð;
12‘2 
7ðd2A  d2B Þa^ Þ:þF2 ð;
24‘2 

þ

ð19Þ

On the other hand, the velocity diﬀerence between the two
swimmers is given bya‘ ^ Þ sin 
V R  V L ¼ 3 2dA dB ð1 þ ÞF2 ð;D ^ Þ 3ðd2A  d2B ÞF3 ð;^ Þ sin :ð20Þ ðd2A  d2B 2 ÞF4 ð;In the above equations, we have introduced four scaling
functions deﬁned by^ Þ ¼
F1 ð;^^ 2Þ
3ð3
þ

^2þ^4
92 þ 2ð2 þ  þ 22 Þ

;

^2
3
;
^2þ
^4
92 þ 2ð2 þ  þ 22 Þ^
32 ^ Þ ¼;F3 ð;
2^2þ^4
9 þ 2ð2 þ  þ 22 Þ^ Þ ¼
F2 ð;^ Þ ¼
F4 ð;

^3
3
:
^2þ
^4
92 þ 2ð2 þ  þ 22 Þ

ð21Þ
ð22Þ
ð23Þ
ð24Þ

These are the main results of this paper.
Equation (18) indicates that, owing to the hydrodynamic
interaction between the two elastic swimmers, the mean
velocity is always smaller than V0 irrespective of the relative
phase diﬀerence Ψ. Using the formula sin2 ð=2Þ ¼
ð1  cos Þ=2 in Eq. (18), we point out that the Ψ-independent contribution to the correction is due to the passive
terms in Eqs. (9) and (10), whereas the Ψ-dependent
contribution comes from the active terms. The correction to
V0 vanishes only when  ¼ 0, and the mean velocity is
minimized when  ¼ . In contrast, the velocity diﬀerence in
Eq. (20) can be either positive or negative depending on theconditions. Obviously, we have V R ¼ V L when  ¼ 0. Thisis reasonable because the two swimmers should move withthe same velocity when the relative phase diﬀerence
vanishes. A more detailed discussion concerning the velocity
diﬀerence will be given in the next section for symmetricelastic swimmers.
In Fig. 2, we plot the scaling functions F1 and F2 as^ for  ¼ 0:1, 1, and 10.14) Note, however, thatfunctions of 
the cases of  ¼ 0:1 and 10 are essentially equivalent because
we can always exchange springs A and B, whereas we havedeﬁned the relaxation time τ through KA as in Eq. (17). As
shown in Eq. (19) and previously discussed in Ref. 14, thefrequency dependence of the average velocity V0 for an
isolated elastic swimmer is essentially determined by^ Þ and F2 ð;^ Þ. Notice that F1   and F2  2
F1 ð;^^  1.
for   1, whereas F1  1 and F2  2 for ^Hence the average velocity increases for   1, whereas it^  1 when the frequency is increased.14) To
decreases for 
ensure the validity of our elastic swimmer model, we assume
here that a low-Reynolds-number ﬂow ﬁeld is justiﬁed even^  1.
in the high-frequency regime 
In Fig. 3, we plot the scaling functions F3 and F4 as^ for  ¼ 0:1, 1, and 10. As shown in Eq. (20),functions of ^ Þ and F4 ð;^ Þ as well as
the scaling functions F3 ð;054804-3©2019 The Physical Society of Japan

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J. Phys. Soc. Jpn. 88, 054804 (2019)

100

100

(a)

1

λ = 0.1
λ=1
λ = 10

10-2

-3
1010
-2

100

F3

−1

10-1

100
Ω̂

101

100

F4

2

−2

10-1

100
Ω̂

10-1

100
Ω̂

101

101

102

−1

(b)

1

10-4
3
10

λ = 0.1
λ=1
λ = 10−3

λ = 0.1
λ=1
λ = 1010-2

F21010-2
-6

10-4

-8
1010
-2

102

10-2

10

1

10-6

(b)

-4

−1

(a)

10-2

-1

F1

10

M. Kuroda et al.

λ = 0.1
λ=1
λ = 10

-6

10-8
102

10-2

10-1

100
Ω̂

101

102

^ Þ andFig. 2. (Color online) Plots of the scaling functions (a) F1 ð;^ Þ deﬁned in Eqs. (21) and (22), respectively, as functions of(b) F2 ð;^ ¼  for  ¼ KB =KA ¼ 0:1, 1, and 10. The numbers indicate the slope,
representing the exponent of the power-law behaviors.

^ Þ andFig. 3. (Color online) Plots of the scaling functions (a) F3 ð;^ Þ deﬁned in Eqs. (23) and (24), respectively, as functions of(b) F4 ð;^ ¼  for  ¼ KB =KA ¼ 0:1, 1, and 10. The numbers indicate the slope,
representing the exponent of the power-law behaviors.

^ Þ characterize the frequency dependence of theF2 ð;hydrodynamic interaction between two elastic swimmers.
^  1, whereas
Here, we have F3   and F4  3 for 
31^F3   and F4   for   1. When the swimmers areasymmetric such as when  ¼ 0:1 and 10, on the other hand,
there are intermediate regions where the scaling functions
behave as F3  1 and F4  . Note that the velocity
diﬀerence also decreases in the high-frequency regime.

In Eqs. (25) and (26), the Ψ-independent terms ofsin2 ð=2Þ ¼ ð1  cos Þ=2 correspond to the passive terms
as before. In Fig. 4, we plot the Ψ-dependences of V L  V0
^ ¼ 1. We see that both V L and V R canand V R  V0 when ^ ¼ 1, as
be larger than V0 for certain ranges of Ψ. For 
Lshown in Fig. 4, we have V > V0 for 0:927 <  < 0 and
V R > V0 for 0 <  < 0:927. However, as we have alreadyexplained with Eq. (18) for the general asymmetric case, themean of V L and V R is always smaller than V0 .
Furthermore, the velocity diﬀerence is now given by^2
4d2 a‘
3
VR  VL ¼
sin  sin :
ð28Þ
^2þ
^4
D3  9 þ 10

4.

Two Symmetric Elastic Swimmers

Having discussed the case of two general (asymmetric)
elastic swimmers, we now discuss the case when both elastic
swimmers have symmetric structures, i.e., dA ¼ dB ¼ d andKA ¼ KB (or  ¼ 1). In this case, the two average velocities
can be simply written as



^

48‘3
2
V L ¼ V0 1 
sin
ð=2Þ
þ
sin

; ð25Þ
^2
7D3
3þ



^

48‘3
2
V R ¼ V0 1 
sin
ð=2Þ

sin

; ð26Þ
^2
7D3
3þwhere the average velocity of a single elastic swimmer now
becomes14)
^ þ
^ 2Þ
7d2 a 3ð3
V0 ¼
sin :
ð27Þ
2
2
^4
^ þ
24‘  9 þ 10

This is an interesting result because, for 0 <  <  and hence
V0 > 0, we have V L < V R for 0 <  <  or V L > V R for <  < 0. In the former case, the interaction between thetwo swimmers is repulsive and the distance between themincreases as they move, i.e., an unbound state. In the latter
case, on the other hand, the interaction is attractive and they
form a moving hydrodynamic bound state.
^  1, theIt is worthwhile noting that, in the case of ^
average velocity in Eq. (27) behaves as V0  , whereas the^ 2 forvelocity diﬀerence in Eq. (28) scales as V R  V L  
two symmetric swimmers. Such a diﬀerence arises from thepresence of the active terms in Eqs. (9) and (10) [or the last

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M. Kuroda et al.

48‘3 2sinð=2Þ
:v L ¼ vR ¼ v 0 1 
7D3(V L,R − V0 )

7D3
24 3

0

This result means that the average velocities vL and vR of thetwo symmetric NG swimmers are always smaller than thatof an isolated NG swimmer, i.e., vL,R < v0 . Hence, thepossibility of V L,R > V0 under certain conditions, as shown in
Eqs. (25) and (26), is a unique feature of two elastic
swimmers. Since vL ¼ vR for two symmetric NG swimmers,
the distance between them remains constant, which is incontrast to the case of two symmetric elastic swimmers [seeEq. (28)]. Such a diﬀerence arises from the internal relaxation
dynamics of the spheres in elastic swimmers, leading toasymmetric motion of the two springs in each swimmer.

VL
VR

−1
−π

0
Ψ

π

Fig. 4. (Color online) Average velocities V L (black) and V R (red) of twosymmetric elastic swimmers with respect to V0 as a function of the relative^ ¼ 1. See Eqs. (25) and (26).
phase diﬀerence Ψ between them when ^-dependent
terms in Eqs. (25) and (26)] owing to thesimultaneous motion of the two swimmers. According to
the above frequency dependences, the velocity diﬀerence
V R  V L (for ﬁnite D) becomes much smaller than V0 in^ ! 0, and the two velocities turn out to be
the limit of 
identical, as shown later in Eq. (33). A similar argument^  1 because we have V0  ^ 1 andholds also for RL2^V  V   , the latter being the higher-order active
contribution.
5.

Limit of Two NG Swimmers

The interaction between two asymmetric NG swimmers^ ¼  ! 0.
can be recovered simply by taking the limit of This is because the spring constants KA and KB are inﬁnitely
large and the characteristic time scale  ¼ 6a=KA is
inﬁnitely small for NG swimmers. In this limit, the two^ ! 0Þ become
average velocities deﬁned by vL,R ¼ V L,R ða‘ vL ¼ v 0 
4dA dB sin2 ð=2Þ sin 2D3
 ðd2A  d2B Þ sin  ;
ð29Þ

a‘
vR ¼ v0 
4dA dB sin2 ð=2Þ sin 2D3
þ ðd2A  d2B Þ sin  ;ð30Þ
where the average velocity of a single NG swimmer is5)
7dA dB asin :
ð31Þ
24‘2
Hence, the mean of vL and vR is again given by Eq. (18) inwhich V0 is replaced by v0 . As mentioned in the previous
section, both vL and vR are proportional to Ω. The velocity
diﬀerence, on the other hand, becomesv0 ¼

vR  vL ¼ a‘ 2
ðdA  d2B Þ sin :
D3

ð33Þ

ð32Þ

This result indicates that the velocity diﬀerence depends not
only on Ψ but also on the relative magnitude between dA anddB for asymmetric NG swimmers.
For symmetric NG swimmers, i.e., dA ¼ dB , vL and vR are
identical and are given by

6.

Summary and Discussion

We have investigated the hydrodynamic interaction
between two elastic swimmers consisting of three spheres
and two harmonic springs. In this model, the natural length ofeach spring is assumed to undergo a prescribed cyclic change
in time, reﬂecting the internal states of an elastic swimmer.
For two interacting three-sphere microswimmers, we ﬁrst
obtained their average velocities in terms of the distance D
between them [see Eqs. (9) and (10)]. Using these expressions, we further obtained the explicit forms of the averagevelocities of two identical elastic swimmers. The mean of thetwo average velocities was shown to be always smaller thanthat of a single elastic swimmer [see Eq. (18)]. On the other
hand, the velocity diﬀerence depends on the relative phase
diﬀerence Ψ between the two elastic swimmers [seeEqs. (20) and (28)]. As a result, the swimming state of twoelastic swimmers can be either bound or unbound dependingon the relative phase diﬀerence.
In this paper, the hydrodynamic interaction was considered
only between two three-sphere microswimmers, although
there are several other model swimmers. For example, two
rigid helices neither attract nor repel each other when they
are rotating with zero phase diﬀerence,19) two puller-typesquirmers undergo a signiﬁcant change in their orientations
after an encounter,20) and two spherical swimmers withspatially conﬁned circular trajectories cause either attractive
or repulsive interaction.21) Using the Quadroar model,
Mirzakhanloo et al. showed that two swimmers, which
generate ﬂow ﬁelds mimicking that of Chlamydomonas
reinhardtii, exhibit very rich behaviors.22) The three-sphereswimmer model in one-dimensional space is especiallysuitable for analytical analysis because it is suﬃcient toconsider only the translational motion, and the tensorial
structure of the ﬂuid motion can be neglected.
In our work, we have assumed that the two elastic threesphere swimmers are conﬁned in one-dimensional space andmoving in the same direction. For two NG swimmers, on theother hand, it was shown before that the interaction between
them depends on their relative orientation.10,12) The main
reason that we have investigated only the one-dimensionalcase is that our primary interest is to analytically obtainthe frequency dependence of the hydrodynamic interaction
between two elastic swimmers, which was not studied before.
Another motivation to restrict our study to one-dimensional
space is to clarify how the correlation between a three-sphereswimmer and a two-sphere system, as reported in Ref. 13, can054804-5©2019 The Physical Society of Japan

J. Phys. Soc. Jpn.
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J. Phys. Soc. Jpn. 88, 054804 (2019)

M. Kuroda et al.

be generalized for two three-sphere swimmers [see Eqs. (9)and (10)]. The future study of the hydrodynamic interaction
between two elastic swimmers having diﬀerent orientations
would require a numerical treatment. For instance, theoscillatory motion reported in Refs. 10 and 12, would beobservable only when the space dimension is higher than one.
We have shown analytically that even the interaction
between two elastic microswimmers can be complicated,depending on the relative displacement, structure, and phase
diﬀerence. Nevertheless, it is possible and straightforward toincrease the number of interacting swimmers as long as theassumption of low-Reynolds-number hydrodynamics is valid
and the swimmers are conﬁned in one-dimensional space. We
believe that the present analysis of the hydrodynamic
interaction between two swimmers will be useful in studying
the collective behavior of a large number of self-propelled
microswimmers immersed in a viscous ﬂuid.23,24)

Appendix B:

The average velocities V L and V R of two interacting
elastic swimmers are given as follows:a‘ ^ Þ sin 
V L ¼ V0  3 2dA dB sin2 ð=2ÞF1 ð;D ^ Þ sin  sin þ dA dB ð1 þ ÞF2 ð;^ Þ sin2 ð=2Þ cos þ 4dA dB ð1  ÞF2 ð;^ Þ sin2 ð=2Þ
þ 2ðd2A  d2B ÞF2 ð;1^ Þ
½3ðd2A  d2B ÞF3 ð;
2^ Þ sin  ;þ ðd2A  d2B 2 ÞF4 ð;a‘ ^ Þ sin 
V R ¼ V0  3 2dA dB sin2 ð=2ÞF1 ð;D ^ Þ sin  sin  dA dB ð1 þ ÞF2 ð;Acknowledgements We thank S. Al-Izzi, H.-Y. Chen, Y. Hosaka, T. Kato,
H. Kitahata, Y. Koyano, and R. Okamoto for fruitful discussions and helpful
suggestions. K.Y. acknowledges support by a Grant-in-Aid for JSPS Fellows(Grant No. 18J21231) from the Japan Society for the Promotion of Science(JSPS). S.K. acknowledges support by a Grant-in-Aid for Scientiﬁc Research (C)(Grant No. 18K03567) from the JSPS.

Appendix A:^ 2 Þd^ A sinðtÞþ 2ð32 þ ^ 2 dB cosðt  Þ
 2ð1 þ Þ^ 2 Þd^ B sinðt  Þ ; ð3 þ 
uLB ¼

ðA:1Þ

1

^2þ
^4
92 þ 2ð2 þ  þ 22 Þ
^ 2 dA cosðtÞ 2ð1 þ Þ^ 2 Þd^ A sinðtÞþ ð3  

^ 2 dB cosðt  Þ
þ ½9 þ ð1 þ 4Þ
^ 2 Þd^ B sinðt  Þ ;
þ 2ð3 þ 
uRA ¼

ðA:2Þ

1

^2þ
^4
92 þ 2ð2 þ  þ 22 Þ
^ 2 dA cosðt  Þ
 ½92 þ ð4 þ Þ^ 2 Þd^ A sinðt  Þ
þ 2ð32 þ ^ 2 dB cosðt    Þ
 2ð1 þ Þ^ 2 Þd^ B sinðt    Þ ; ð3 þ 
uRB ¼

ðA:3Þ

1

^2þ
^4
92 þ 2ð2 þ  þ 22 Þ
^ 2 dA cosðt  Þ
 2ð1 þ Þ^ 2 Þd^ A sinðt  Þ
þ ð3  
^ 2 dB cosðt    Þ
þ ½9 þ ð1 þ 4Þ
^ 2 Þd^ B sinðt    Þ :
þ 2ð3 þ 

ðA:4Þ

ðB:1Þ

^ Þ sin2 ð=2Þ cos þ 4dA dB ð1  ÞF2 ð;^ Þ sin2 ð=2Þ
þ 2ðd2A  d2B ÞF2 ð;1^ Þ
½3ðd2A  d2B ÞF3 ð;
2^ Þ sin  ;þ ðd2A  d2B 2 ÞF4 ð;þR
Displacements uLA , uLB , uR
A , and uB

The four displacements uLA , uLB , uRA , and uRB of two
interacting elastic swimmers are given as follows:
1
uLA ¼
2
^2þ
^4
9 þ 2ð2 þ  þ 22 Þ
 2^ 2 dA cosðtÞ
 ½9 þ ð4 þ Þ

Average Velocities V L and V R

ðB:2Þwhere V0 is given by Eq. (19) and the four scaling functions
are given by Eqs. (21)–(24).

+komura@tmu.ac.jp
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