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Letters

Journal of the Physical Society of Japan 86, 093801 (2017)https://doi.org/10.7566/JPSJ.86.093801

Elastic Three-Sphere Microswimmer in a Viscous Fluid
Kento Yasuda, Yuto Hosaka, Mizuki Kuroda, Ryuichi Okamoto, and Shigeyuki Komura+
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan(Received June 28, 2017; accepted July 12, 2017; published online August 9, 2017)

We discuss the dynamics of a generalized three-sphere microswimmer in which the spheres are connected by twoelastic springs. The natural length of each spring is assumed to undergo a prescribed cyclic change. We analyticallyobtain the average swimming velocity as a function of the frequency of cyclic change in the natural length. In the lowfrequency region, the swimming velocity increases with frequency, and its expression reduces to that of the original
three-sphere model by Najaﬁ and Golestanian. Conversely, in the high-frequency region, the average velocity decreases
with increasing frequency. Such behavior originates from the intrinsic spring relaxation dynamics of an elastic swimmer
moving in a viscous ﬂuid.

Microswimmers are tiny machines that swim in a ﬂuid,
such as sperm cells or motile bacteria, and they are expected
to be applied to microﬂuidics and microsystems.1) By
transforming chemical energy into mechanical work, microswimmers change their shape and move in viscous environments. Over the length scale of microswimmers, the ﬂuid
forces acting on them are governed by the eﬀect of viscous
dissipation. According to Purcell’s scallop theorem,2) timereversal body motion cannot be used for locomotion in a
Newtonian ﬂuid.3) As one of the simplest models exhibiting
broken time-reversal symmetry, Najaﬁ and Golestanian
proposed a three-sphere swimmer,4,5) in which three in-line
spheres are linked by two arms of varying length. This model
is suitable for analytical analysis because it is suﬃcient toconsider only the translational motion, and the tensorial
structure of the ﬂuid motion can be neglected. Recently, sucha swimmer has been experimentally realized by using
ferromagnetic particles at an air–water interface and by
applying an oscillating magnetic ﬁeld.6)
The original Najaﬁ–Golestanian model has been further
extended to various situations, such as the case when oneof the spheres has a larger radius7) or when three spheres
are arranged in a triangular conﬁguration.8) Montino and
DeSimone considered the case in which one arm is periodically actuated while the other is replaced by a passive elastic
spring.9) It was shown that such a swimmer exhibits adelayed mechanical response of the passive spring withrespect to the active arm. More recently, they analyzed the
motion of a three-sphere swimmer with arms having active
viscoelastic properties mimicking muscular contraction.10)
Recently, Nasouri et al. discussed the motion of an elastic
two-sphere swimmer, in which one of the sphere is a neoHookean solid.11)
Another approach for extending the Najaﬁ–Golestanianmodel is to consider the arm motions as occurring
stochastically,12,13) rather than assuming a prescribed sequence of deformations.4,5) In these models, the conﬁguration
space of a swimmer generally consists of a ﬁnite number ofdistinct states. A similar idea was employed by Sakaue et al.,
who discussed the propulsion of molecular machines or
active proteins in the presence of hydrodynamic interactions.14) Later, Huang et al. considered a modiﬁed threesphere swimmer in a two-dimensional viscous ﬂuid.15) In
their model, the spheres are connected by two springs, thelengths of which are assumed to depend on the discrete statesthat are cyclically switched. As a result, the dynamics of aswimmer consists of the spring relaxation processes, which
follow after each switching event.
In this letter, we discuss a generalized three-sphereswimmer in which the spheres are simply connected by
two harmonic springs. The main diﬀerence between thismodel and the previous models is that the natural length ofeach spring depends on time and is assumed to undergo aprescribed cyclic change. Whereas the arms in the Najaﬁ–
Golestanian model undergo a prescribed motion regardless ofthe force exerted by the ﬂuid, the sphere motion in our model
is determined by the natural spring lengths representing
internal states of a swimmer and also by the force exerted by
the ﬂuid. In this sense, our model is more realistic to study
the locomotion of active microswimmers. We analyticallyobtain the average swimming velocity as a function of thefrequency of cyclic change in the natural length. In order tobetter illustrate our result, we ﬁrst explain the case in which
the two spring constants are identical and the two oscillation
amplitudes of the natural lengths are the same. Then, we shalldiscuss a general case in which these quantities are diﬀerent
and the phase mismatch between the natural lengths isarbitrary.
The introduction of harmonic springs between the spheres
leads to an intrinsic time scale of an elastic swimmer thatcharacterizes its internal relaxation dynamics. When thefrequency of cyclic change in the natural lengths is smaller
than this characteristic time, the swimming velocity increases
with frequency, as in the previous works.5) In the highfrequency region, on the other hand, the motion of spheres
cannot follow the change in the natural length, and theaverage swimming velocity decreases with increasing
frequency. Such a situation resembles the dynamics of the
Najaﬁ–Golestanian three-sphere swimmer in a viscoelastic
medium.16) We also show that, owing to the elasticity that has
been introduced, the proposed micromachine can swim even
if the change in the natural lengths is reciprocal as long as its
structural symmetry is violated. Although the consideredswimmer appears to be somewhat trivial, it can be regarded
as a generic model for microswimmers or protein machinessince the behaviors of the previous models can be deduced
from our model by taking diﬀerent limits.
We generalize the Najaﬁ–Golestanian three-sphereswimmer model to take into account the elasticity in thesphere motion. As schematically shown in Fig. 1, the present

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Letters

J. Phys. Soc. Jpn. 86, 093801 (2017)

η

x1

a

KA
A

x2

KB

x3

B

Fig. 1. (Color online) Elastic three-sphere microswimmer in a viscous
ﬂuid characterized by the shear viscosity η. Three identical spheres of radiusa are connected by two harmonic springs with elastic constants KA and KB .
The natural lengths of the springs, ‘A ðtÞ and ‘B ðtÞ, depend on time and areassumed to undergo cyclic change [see Eqs. (6) and (7)]. The time-dependent
positions of the spheres are denoted by x1 ðtÞ, x2 ðtÞ, and x3 ðtÞ in a onedimensional coordinate system.

model consists of three hard spheres of the same radius aconnected by two harmonic springs A and B with spring
constants KA and KB , respectively. We assume that thenatural lengths of these springs, denoted by ‘A ðtÞ and ‘B ðtÞ,undergo cyclic time-dependent change. Their explicit timedependences will be speciﬁed later. The total energy of anelastic swimmer is then given by
E¼

KA
KB
ðx2  x1  ‘A Þ2 þ
ðx3  x2  ‘B Þ2 ;
2
2

ð1Þ

where xi ðtÞ (i ¼ 1; 2; 3) are the positions of the three spheresin a one-dimensional coordinate system. We also assume
x1 < x2 < x3 without loss of generality. Owing to thehydrodynamic interaction, each sphere exerts a force on theviscous ﬂuid of shear viscosity η and experiences an opposite
force from it. In general, the surrounding medium can beviscoelastic,16) but such an eﬀect is not considered in thisletter.
Denoting the velocity of each sphere by x_i , we can write
the equations of motion of the three spheres asKAKA ðx2  x1  ‘A Þ
ðx2  x1  ‘A Þ 
x_1 ¼
x2  x1
6a4
KB ðx3  x2  ‘B Þ KB ðx3  x2  ‘B Þ
þ

;
ð2Þ
x2  x1
x3  x1
4
4
KA ðx2  x1  ‘A Þ
KA
ðx2  x1  ‘A Þ
x_2 ¼
x2  x1
4
6a
KB
KB ðx3  x2  ‘B Þ
ðx3  x2  ‘B Þ þ;ð3Þx3  x2
6a4
KA ðx2  x1  ‘A Þ KA ðx2  x1  ‘A Þ
x_3 ¼x3  x1
x3  x2
4
4
KB ðx3  x2  ‘B Þ
KB
ðx3  x2  ‘B Þ;þ
ð4Þx3  x24
6awhere we have used the Stokes’ law for a sphere and theOseen tensor in a three-dimensional viscous ﬂuid. The
swimming velocity of the whole object can be obtained byaveraging the velocities of the three spheres:V¼

1
ðx_1 þ x_2 þ x_3 Þ:3ð5ÞOne of the advantages of the present formulation is that the
motion of the spheres is simply described by coupled
ordinary diﬀerential equations. Moreover, the force-free
condition for the whole system4,5) is automatically satisﬁedin the above equations.
Next, we assume that the two natural lengths of the springs
undergo the following periodic changes:

K. Yasuda et al.

‘A ðtÞ ¼ ‘ þ dA cosðtÞ;

ð6Þ

‘B ðtÞ ¼ ‘ þ dB cosðt  Þ:ð7ÞIn the above, ‘ is the common constant length, dA and dB
are the amplitudes of the oscillatory change, Ω is thecommon frequency, and ϕ is the mismatch in phase between
the two cyclic changes. The time-reversal symmetry of thespring dynamics exists only when  ¼ 0 or π; otherwise,
the time-reversal symmetry is broken. In the following
analysis, we generally assume that dA ; dB ; a  ‘ and focus
on the leading-order contribution. It is convenient tointroduce a characteristic time scale  ¼ 6a=KA . Then weuse ‘ to scale all the relevant lengths (xi , a, dA , and dB ) and^ ¼ . By further
employ τ to scale the frequency, i.e., deﬁning the ratio between the two spring constants as ¼ KB =KA , the coupled Eqs. (2)–(4) can be made dimensionless.
In order to discuss the essential outcome of the present
model, we shall ﬁrst consider the simplest symmetric case,i.e.,  ¼ 1, dA ¼ dB ¼ d, and  ¼ =2. Hence, Eq. (7) now
reads ‘B ðtÞ ¼ ‘ þ d sinðtÞ. For our later calculation, it isuseful to introduce the following spring lengths with respectto ‘:
uA ¼ x2  x1  ‘;

uB ¼ x3  x2  ‘:

ð8Þ

Notice that these quantities are related to the sphere velocities
in Eqs. (2)–(4) as
u_A ¼ x_2  x_1 ;u_B ¼ x_3  x_2 :ð9ÞUsing Eqs. (2)–(4) and solving Eq. (9) in the frequency
domain, we obtain the following expressions after inverse
Fourier transformuA ðtÞ ^ þ 5
^2þ
^3
9  3
d cosðtÞ^2þ
^4
9 þ 10þ^  4
^ 2 þ 2^3
6d sinðtÞ;2^^
9 þ 10 þ 4

uB ðtÞ  ^ þ 4
^ 2 þ 2^3
6d cosðtÞ2^ þ
^4
9 þ 10þ^ þ 5
^2
^3
9 þ 3
d sinðtÞ;^2þ
^4
9 þ 10

ð10Þ

ð11Þ

where we have used a=‘  1.
According to the calculation by Golestanian and Ajdari,5)the average swimming velocity of a three-sphere swimmer
can generally be expressed up to the leading order in uA =‘and uB =‘ asV¼

7ahuA u_B  u_A uB i;
24‘2

ð12Þ

where the averaging h  i is performed by time integration ina full cycle. The above expression indicates that the averagevelocity is determined by the area enclosed by the orbit ofperiodic motion in the conﬁguration space.5) Using Eqs. (10)and (11) for an elastic microswimmer with d=‘; a=‘  1, we
obtain the lowest-order contribution as
V¼

^ þ
^ 2Þ
7d2 a 3ð3
;
^2þ
^4
24‘2  9 þ 10

ð13Þ

which is an important result of this letter.

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Letters

J. Phys. Soc. Jpn. 86, 093801 (2017)

K. Yasuda et al.

^  1.
We ﬁrst consider the small-frequency limit of 
Physically, this limit corresponds to the case when the spring
constant KA is very large. We easily obtainuB ðtÞ  d sinðtÞ;

(a)

ð14Þ

-1

10

and
V

7d2 a
;
24‘2

ð15Þ

V

21d a
:
24‘2 21-2

10

λ = 0.1
λ=1
λ = 10which exactly coincides with the average velocity of the
Najaﬁ–Golestanian swimmer with identical spheres.4,5) Thisis reasonable because the two spring lengths uA and uB are inphase with their respective natural lengths ‘A and ‘B , as we
can see from Eqs. (6), (7), and (14). Notice that the averagevelocity increases as V   in this limit, while it does notdepend on the ﬂuid viscosity η.4,5)^  1, on the
In the opposite large-frequency limit of other hand, we havepﬃﬃﬃ
5d
cos½t  arctan 2;ð16Þ
uA ðtÞ pﬃﬃﬃ
5duB ðtÞ 
sin½t  ð  arctan 2Þ;
ð17Þ
where arctan 2  1:107 and
2−1F1 (Ω̂)uA ðtÞ  d cosðtÞ;

0

10

-3

10 -2
10

-1

10

1

2

10

10

Ω̂
0

10

(b)

-2

F2 (Ω̂)

10

−2

2

-4

10

λ = 0.1
λ=1
λ = 10

ð18Þ

We see here that uA and uB are out of phase with respect tothe natural lengths ‘A and ‘B , while the average velocity
decreases as V  1 when Ω is increased. When the spring
constant KA is small, it takes time for a spring to relax to its
natural length, which leads to a delay in the mechanicalresponse. The crossover frequency between the above two^  1. The general frequency
regimes is determined by 
dependence of Eq. (13) is shown in Fig. 2(a) for  ¼ 1 (black^  1, as expected.
line). It shows a maximum around 
Recently, we investigated the motion of the Najaﬁ–
Golestanian three-sphere swimmer in a viscoelastic medium.16) We derived a relation that connects the averageswimming velocity and the frequency-dependent viscosityof the surrounding medium. In this relation, the viscous
contribution can exist only when the time-reversal symmetryis broken, whereas the elastic contribution is present only
when the structural symmetry of the swimmer is broken. In
particular, we calculated the average swimming velocity
when the surrounding viscoelastic medium is described by asimple Maxwell ﬂuid with a characteristic time scale M .
It was shown that the viscous term increases as V  for M  1, while it decreases as V  1 for M  1.
This is a unique feature of a swimmer in a viscoelastic
medium,16–18) and such a reduction occurs simply because
the medium responds elastically in the high-frequency
regime. We note that the frequency dependence of V for anelastic three-sphere swimmer, as obtained in Eq. (13), isanalogous to that for the Najaﬁ–Golestanian swimmer ina viscoelastic Maxwell ﬂuid. In other words, an elastic
microswimmer in a viscous ﬂuid exhibits “viscoelastic”
eﬀects as a whole.
Having discussed the simplest situation of the proposedelastic swimmer, we now show the result for a general casewhen KA ≠ KB (or  ≠ 1), dA ≠ dB , and the phase mismatch

0

10

-6

10

-2

10

-1

0

10

10

1

10

2

10

Ω̂
^ Þ andFig. 2. (Color online) Plots of the scaling functions (a) F1 ð;^ Þ deﬁned in Eqs. (22) and (23), 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.

ϕ in Eq. (7) is arbitrary. By repeating the same calculation as
before, the spring lengths in Eq. (8) now become1uA ðtÞ ^2þ^4
92 þ 2ð2 þ  þ 22 Þ
n^ 2 dA cosðtÞ½92 þ ð4 þ Þ^ 2 Þd^ A sinðtÞþ 2ð32 þ ^ 2 dB cosðt  Þ
 2ð1 þ Þo^ 2 Þd^ B sinðt  Þ ; ð3 þ 
uB ðtÞ 

ð19Þ

1

^2þ
^4
92 þ 2ð2 þ  þ 22 Þ
n^ 2 dA cosðtÞ
2ð1 þ Þ^ 2 Þd^ A sinðtÞþ ð3  
^ 2 dB cosðt  Þ
þ ½9 þ ð1 þ 4Þo^ 2 Þd^ B sinðt  Þ ;
þ 2ð3 þ 

ð20Þ

respectively, where we have used a=‘  1. Using Eq. (12)again, we ﬁnally obtain the lowest-order general expressionof the average velocity as

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J. Phys. Soc. Jpn. 86, 093801 (2017)

7dA dB a^ Þ sin 
F1 ð;
24‘2 
7ð  1ÞdA dB a
^ Þ cos F2 ð;
12‘2 
7ðd2A  d2B Þa^ Þ;þF2 ð;
24‘2 
where the two scaling functions are deﬁned by^^ 2Þ
3ð3þ^ Þ ¼;
F1 ð;^2þ^4
92 þ 2ð2 þ  þ 22 Þ^23
^ Þ ¼:
F2 ð;^2þ^4
92 þ 2ð2 þ  þ 22 Þ

Letters

V¼

ð21Þ

ð22Þ
ð23Þ

In Fig. 2, we plot the above scaling functions as functions of^ for  ¼ 0:1 and 10. Notice, however, that these two casesare essentially equivalent because we can always exchange
the springs A and B, whereas we have deﬁned the relaxation
time τ by using KA .
When  ¼ 1, dA ¼ dB , and  ¼ =2, only the ﬁrst term
remains, and Eq. (21) reduces to Eq. (13), as expected. When ≠ 1, on the other hand, the second term is present even if
 ¼ 0. The third term is also present when d2A ≠ d2B ,regardless of the phase mismatch ϕ. Notice that the ﬁrst term
represents the broken time-reversal symmetry for  ≠ 0,
whereas both the second and third terms reﬂect the structural
asymmetry of an elastic three-sphere swimmer.16) To be more
precise, the second term is due to the diﬀerence between therelaxation times of the two springs, and the third term reﬂects
the asymmetric changes of their natural lengths.
It is interesting to note that the frequency dependence ofthe second and third terms in Eq. (21), represented by^ Þ, is diﬀerent from that of the ﬁrst term, represented
F2 ð;^ Þ. According to Eq. (23), V due to the second andby F1 ð;^  1, whereas it
third terms increases as V  2 for 
2^decreases as V   for   1. In general, the overallswimming velocity depends on various structural parametersand exhibits a complex frequency dependence. For example,^ Þ in Fig. 2(a) exhibits a non-monotonic frequency
F1 ð;
dependence (two maxima) for  ¼ 0:1 or 10 (namely, when ≠ 1). On the other hand, an important common feature in^ 1, which
all the terms in Eq. (21) is that V decreases for 
is characteristic of elastic swimmers.
We conﬁrm again that Eq. (21) reduces to the result byGolestanian and Ajdari,5) i.e., V ¼ 7dA dB a sin =ð24‘2 Þ,
when the two spring constants are inﬁnitely large, i.e.,
KA ; KB ! 1 and  ¼ 1. The third term in Eq. (21) vanishes^ ! 0 holds in this limit. In
even if dA ≠ dB because 
the modiﬁed three-sphere swimmer model considered byMontino and DeSimone, one of the two arms was replaced bya passive elastic spring.9) Their model can be obtained fromthe present model simply by setting one of the spring
constants to be inﬁnitely large, say KA ! 1, and by
regarding the natural length of the other spring as a constant,say ‘B ¼ ‘ (or dB ¼ 0). The continuous changes of thenatural lengths introduced in Eqs. (6) and (7) are astraightforward generalization of cyclically switched discrete

K. Yasuda et al.

states considered in the previous studies.12–15) We ﬁnally notethat a model similar to the present one was considered inRef. 19, although that study focused only on the lowfrequency region and did not discuss the entire frequency
dependence. Using coupled Langevin equations, the authorsof Ref. 19 mainly investigated the interplay between selfdriven motion and diﬀusive behavior,19) which is also an
important aspect of microswimmers.
To summarize, we have discussed the locomotion of ageneralized three-sphere microswimmer in which the spheresare connected by two elastic springs and the natural lengthof each spring is assumed to undergo a prescribed cyclicchange. As shown in Eqs. (13) and (21), we have analyticallyobtained the average swimming velocity V as a function of
the frequency Ω of cyclic change in the natural length. In the
low-frequency region, the swimming velocity increases withfrequency and reduces to the original three-sphere model by
Najaﬁ and Golestanian.4,5) Conversely, in the high-frequency
region, the velocity decreases with increasing frequency. This
property reﬂects the intrinsic spring relaxation dynamics ofan elastic swimmer in a viscous ﬂuid.
Acknowledgments S.K. and R.O. acknowledge support from a Grant-inAid for Scientiﬁc Research on Innovative Areas “Fluctuation and Structure”(Grant No. 25103010) from the Ministry of Education, Culture, Sports, Scienceand Technology of Japan and from a Grant-in-Aid for Scientiﬁc Research (C)(Grant No. 15K05250) from the Japan Society for the Promotion of Science(JSPS).

+

komura@tmu.ac.jp
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Note added in proof—After this letter was accepted, we became aware that asimilar model was considered in [J. Pande and A.-S. Smith, Soft Matter 11,
2364 (2015)] and [J. Pande et al., New J. Phys. 19, 053024 (2017)].
Although the three spheres are driven by known forces in their model, thefrequency dependence of the average velocity is the same as our result. We
thank the authors of these articles for pointing out their works.

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