August 2018
EPL, 123 (2018) 34002
doi: 10.1209/0295-5075/123/34002

www.epljournal.org

A three-sphere microswimmer in a structured fluid
Kento Yasuda1 , Ryuichi Okamoto2 and Shigeyuki Komura1
1
2

Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University - Tokyo 192-0397, Japan
Research Institute for Interdisciplinary Science, Okayama University - Okayama 700-8530, Japan
received 21 May 2018; accepted in final form 27 July 2018
published online 27 August 201847.63.Gd – Swimming microorganisms
47.63.mf – Low-Reynolds-number motions
PACS 82.70.Gg – Gels and sols
PACS

PACS

Abstract – We discuss the locomotion of a three-sphere microswimmer in a viscoelastic structured
fluid characterized by typical length and time scales. We derive a general expression to link theaverage swimming velocity to the sphere mobilities. In this relationship, a viscous contribution
exists when the time-reversal symmetry is broken, whereas an elastic contribution is present when
the structural symmetry of the microswimmer is broken. As an example of a structured fluid,
we consider a polymer gel, which is described by a “two-fluid” model. We demonstrate in detailthat the competition between the swimmer size and the polymer mesh size gives rise to the rich
dynamics of a three-sphere microswimmer.
c EPLA, 2018
Copyright Introduction. – Microswimmers are tiny machines,
such as sperm cells or motile bacteria, that swim in a
ﬂuid and are expected to be relevant to microﬂuidics andmicrosystems [1]. By transforming chemical energy intomechanical work, microswimmers can change their shapes
and move in viscous environments. The ﬂuid forces acting on the length scale of microswimmers are governed by
the eﬀect of viscous dissipation. According to Purcell’sscallop theorem [2], time-reversal body motion cannot be
used for locomotion in a Newtonian ﬂuid. As one of thesimplest models exhibiting broken time-reversal symmetry, Najafi and Golestanian proposed a three-sphere microswimmer [3,4] in which three in-line spheres are linked
by two arms of varying lengths. This model is suitable foranalytical studies because the tensorial structure of theﬂuid motion can be neglected in its translational motion.
Recently, such a microswimmer has been experimentallyrealized [5,6].
For microswimmers in general situations, however, thesurrounding ﬂuid is not necessarily purely viscous but viscoelastic. Several studies have discussed the swimming
behaviors of micromachines in diﬀerent types of viscoelastic ﬂuids [7–11]. In particular, Lauga showed that the
Scallop theorem in a viscoelastic ﬂuid breaks down if thesquirmer has a fore-aft asymmetry in its surface velocity
distribution [7]. In a recent study, we discussed the locomotion of a three-sphere microswimmer in a viscoelastic medium and derived a relationship linking the averageswimming velocity to the frequency-dependent viscosityof the surrounding medium [12]. We demonstrated thatthe absence of the time-reversal symmetry of the swimmer
motion is reﬂected in the real part of the viscosity, whereas
the absence of the structural symmetry of the swimmer isreﬂected in its imaginary part.
So far, investigations into the swimming behaviors of
micromachines have been limited to homogeneous viscoelastic ﬂuids without any internal structures. However,
one of the fundamental and characteristic features of viscoelastic soft matter is that it contains various intermediate mesoscopic structures and behaves as a structured
ﬂuid [13]. The existence of such internal length scales
significantly aﬀects the rheological properties of soft matter [14]. In this letter, we address the eﬀects of the intermediate structures of the surrounding viscoelastic ﬂuid on
the locomotion of a three-sphere microswimmer. Becausea three-sphere microswimmer is also characterized by its
own size, our main interest is to find out how the averageswimming velocity depends on the relative magnitudes ofthe swimmer’s size and the characteristic length of thesurrounding ﬂuid.
Generalizing our previous work [12], we first obtain the
average velocity of a three-sphere microswimmer moving
in a structured ﬂuid, which is characterized by typicallength and time scales. As an example of a structured
ﬂuid, we employ a “two-ﬂuid” model that has been broadly
used to describe the dynamics of polymer gels [15–17]. Recently, the response of a polymer network to the motion ofa rigid sphere has been investigated within this two-ﬂuid

34002-p1

Kento Yasuda et al.
L2 (t) = ℓ + u2 (t), where ℓ is a constant length that satisfies ℓ ≫ u1 (t), u2 (t). We consider the case when the
two arms undergo the simplest periodic motions [3,4]:
3
1
2
u1 (t) = d1 cos(Ωt) and u2 (t) = d2 cos(Ωt − φ), where d1aand d2 are the amplitudes of the oscillatory motions, Ω isthe common arm frequency, and φ is the mismatch in theL1 (t)
L2 (t)
phases between the two arms. When the arm motions are
given, the above set of equations is suﬃcient to solve forξthe six unknown quantities Vi and Fi . The swimming velocity is obtained by averaging the velocities of the threespheres, i.e., V = (V1 + V2 + V3 )/3.
Fig. 1: (Color online) A three-sphere micromachine swimming
Consider a viscoelastic structured ﬂuid that is characin a structured fluid such as a polymer gel. Three identical terized by a characteristic length scale ξ and a characterspheres of radius a are connected by arms with lengths L1 (t)istic time scale τ . We assume that the above-mentioned
and L2 (t) and undergo time-dependent cyclic motions. Acmobilities are expressed by the following scaling forms:

Gη

cording to the two-fluid model, the polymer gel consists of anelastic network characterized by a constant shear modulus G
and a viscous fluid characterized by a constant shear viscosity η. The elastic and fluid components are coupled via mutual
friction. The length scale ξ characterizes the typical internal
structure of the elastic network, e.g., its mesh size.

model [18–20]. We calculate the frequency dependence of
the average velocity of a three-sphere microswimmer in a
two-ﬂuid gel and obtain its various asymptotic expressions
by changing the swimmer size. The competition between
the swimmer size and the polymer mesh size gives rise to
the rich dynamics of microswimmers. Even though we primarily discuss the two-ﬂuid model here, our result can beapplied to various types of structured ﬂuids.
Microswimmer in a structured fluid. – As shownin fig. 1, we consider a microswimmer consisting of threerigid spheres of the same radius a that are connected by
two arms of variable lengths L1 and L2 [3,4]. We assume
that the motion of the arms is prescribed by two timedependent functions L1 (t), L2 (t) ≫ a. Then the velocityof each sphere Vi (i = 1, 2, 3) should satisfy the conditions
L̇1 (t) = V2 (t) − V1 (t) and L̇2 (t) = V3 (t) − V2 (t), where thedot indicates the time derivative. The surrounding ﬂuid
exerts a force Fi (i = 1, 2, 3) on each sphere, which we assume to be along the swimmer axis. Because we are interested in the autonomous net locomotion of the swimmer,
there are no external forces acting on the spheres. This
leads to the force-free condition: F1 (t) + F2 (t) + F3 (t) = 0.
Within the linear response theory, the velocity andforce acting on a sphere of radius a are related in thedomain by Vi (ω) = μ[a, ω]Fi (ω), where V (ω) =Fourier∞dtV(t)e−iωt (with the same form for F (ω)) denotes−∞∞the Fourier transform and μ[a, ω] = 0 dt μ(a, t)e−iωtgives the frequency-dependent self-mobility. Similarly,
the force Fj acting on the j-th sphere at xj and the induced velocity Vi of the i-th sphere at xi are related by
Vi (ω) = M [r, ω]Fj (ω), where r = xi − xj ≫ a and M [r, ω]
is the frequency-dependent longitudinal coupling mobility.
We further assume that the arm deformations are
relatively small, and given by L1 (t) = ℓ + u1 (t) andμ[a, ω] =μ̂[a/ξ, ωτ ],
6πη0 aM [r, ω] =M̂ [r/ξ, ωτ ],
4πη0 ℓ(1)where μ̂ and M̂ are the dimensionless scaling functionsand η0 is the zero-frequency shear viscosity. Even if
there are more than two length or time scales, the above
assumption is still valid because only the dimensionless ratios between the diﬀerent scales enter into thescaling functions. In other words, if there are several
length scales ξ1 , ξ2 , ξ3 , · · · and several time scales τ1 ,
τ2 , τ3 , · · · , the dimensionless mobility can be expressedas μ̂[a/ξ1 , ωτ1 ; ξ2 /ξ1 , ξ3 /ξ1 , · · · ; τ2 /τ1 , τ3 /τ1 , · · · ] and similarly for M̂ . Under this assumption, we perform an expansion of the swimming velocity to the leading order in
a/ℓ, d1 /ℓ, and d2 /ℓ. After performing the time integration
over a full cycle, we obtain the average swimming velocity:see eq. (2) on the next page
where ∂ M̂ [ℓ/ξ, 0] = limωτ →0 (∂ M̂ [r/ξ, ωτ ]/∂r)r=ℓ (see theSupplementary Material Supplementarymaterial.pdf(SM) for the full derivation). This is a generalization of
our previous result [12] and is the main result of this letter.
The first term in eq. (2) can be regarded as a viscous
contribution, V v , and is present only if the time-reversal
symmetry of the arm motion is broken, i.e., φ = 0, π. The
second term, conversely, corresponds to an elastic contribution, V e , and exists only when the structural symmetryof the swimmer is broken, i.e., d1 = d2 . In other words,
even if the time-reversal symmetry of the swimmer motion is not broken, i.e., φ = 0, π, the swimmer can stillmove in a viscoelastic medium because of the second elastic term as long as its structural symmetry is broken, i.e.,d1 = d2 . We used the condition a ≪ ℓ when deriving
eq. (2), but nothing has been assumed concerning the relative magnitudes between the swimmer size, a and ℓ, andthe characteristic length of the ﬂuid, ξ. Therefore, eq. (2)oﬀers a very general velocity expression for a three-spheremicroswimmer moving in a structured ﬂuid.
Although eq. (2) is applicable to any structured ﬂuid,
some special cases are worth discussing. For a purely viscous ﬂuid, the scaling functions in eq. (1) are given by

34002-p2

A three-sphere microswimmer in a structured ﬂuid
d1 d2 aΩ 
−1
−1
8
M̂
[ℓ/ξ,
0]
−
M̂
[2ℓ/ξ,
0]+2ℓ(−4∂
M̂[ℓ/ξ,0]+∂M̂
[2ℓ/ξ,
0])sin φμ̂[a/ξ,Ωτ]+μ̂[a/ξ,
−Ωτ
]
48ℓ2


i(d21 − d22 )aΩ +
μ̂[a/ξ, Ωτ ]−1 − μ̂[a/ξ, −Ωτ ]−1 4M̂ [ℓ/ξ, 0] + M̂ [2ℓ/ξ, 0] − 2ℓ(2∂ M̂[ℓ/ξ, 0] + ∂ M̂ [2ℓ/ξ, 0]) , (2)
2
96ℓ

V ≈

μ̂ = 1 and M̂ = 1; therefore, eq. (2) reduces to the averagevelocity obtained by Golestanian and Ajdari [4]. Further,
for a viscoelastic ﬂuid without any internal structure, thescaling functions are simply given by μ̂ = η0 /η[ω] andM̂ = η0 /η[ω], where η[ω] is the frequency-dependent complex viscosity. In such a homogeneous but viscoelastic
ﬂuid, eq. (2) reduces to eq. (13) in ref. [12].
We note that the above derivation has been limited
within the linear response theory because linear relationships between forces and velocities have been assumed. Such an assumption of the linear viscoelasticity
is generally justified when the strain amplitude is smallenough [21]. For a three-sphere microswimmer, this condition is given by d1 /ℓ ≪ 1 and d2 /ℓ ≪ 1 which have been
indeed used in the derivation of eq. (2). Otherwise, one
needs to take into account nonlinear viscoelastic eﬀects
such as a shear-thinning behavior [14].
Two-fluid model for a gel. – As a simple example
of various structured ﬂuids, we consider here a polymer
gel described by the two-ﬂuid model. As schematicallydescribed in fig. 1, there are two dynamical fields in thismodel: the displacement field u(r, t) of the elastic network and the velocity field v(r, t) of the permeating ﬂuid.
When inertial eﬀects are neglected, the linearized coupledequations for these two field variables are given by∂u− v , (3)0 = G∇2 u + (K + G/3)∇(∇ · u) − Γ
∂t


∂u
0 = η∇2 v − ∇p − Γ v −
+ f.
(4)
∂t

a stick boundary condition is used for the ﬂuid while the
network does not exchange stress with the sphere. In otherwords, the network moves only because of its coupling to
the ﬂuid. The full expression of the self-mobility μ is givenin the SM [18], and here we show only its limiting behaviors. In the low-frequency limit, ωτ → 0, μ becomes96πηa(9 + 9â + â2 )iωτ (4 + 3ǫ2 )â3 + 36ǫ2 â2 + 162ǫ2â + 81ǫ2
+
, (5)
6πηξ
2(9 + 9â + â2 )2 ǫ2μ[a, ω] ≈where â = a/ξ and one can take further limits depending on the magnitude of â. In the high-frequency limit,ωτ → ∞, μ becomes
μ[a, ω] ≈1
1−.
6πηa 6πηa(iωτ )(6)This expression is equivalent to writing the mobility as
μ[a, ω] ≈ 1/(6πηb a), with an eﬀective viscosity of ηb =
η[1 + 1/(iωτ )]. Therefore, the above two-ﬂuid model reduces to the Kelvin-Voigt model at high frequencies [14].
We previously obtained a general expression for the coupling mobility M connecting the velocity v and the force
f in the two-ﬂuid model [22,23], which is also given in the
SM. For small distances, r/ξ → 0, the coupling mobility is
M [r, ω] ≈1
1−,
4πηr 6πηξ(1 + iωτ )1/2

(7)

and consequently, the gel is nearly purely viscous. Conversely, for suﬃciently large distances, r/ξ → ∞, the couHere, G and K are the shear and compression moduli ofpling mobility is
the elastic network, respectively, η is the shear viscosityof the ﬂuid, p(r, t) is the pressure field, and f (r, t) is the
1
ξ2
1
.
(8)
+
M
[r,
ω]
≈
external force density acting on the ﬂuid component. The
34πηb r 2πηr (1 + iωτ )2elastic and ﬂuid components are coupled via the mutual
Average velocities in a two-fluid gel. – Next wefriction, which is characterized by the friction coeﬃcient Γ.
When the volume fraction of the elastic component is discuss the average velocity of a three-sphere microswimsmall, we further require the incompressibility condition: mer in a two-ﬂuid gel and investigate its frequency as well
∇ · v = 0. The above two-ﬂuid model contains the char- as size dependences. Recall that there are two lengths
acteristic length ξ = (η/Γ)1/2 and the characteristic time that measure the size of a three-sphere microswimmer:τ = η/G. The former length scale roughly corresponds to the sphere radius a and the average arm length ℓ, withthe mesh size of a polymer network, and the latter time the condition a ≪ ℓ. Because the surrounding gel is charscale sets the viscoelastic time. Hereafter, we introduce a acterized by the network mesh size, ξ, the following three
dimensionless ratio defined as ǫ = [(K + 4G/3)/G]1/2 .
diﬀerent situations can be distinguished: i) a large swimDiamant calculated the self-mobility of a sphere in a mer when a ≫ ξ and ℓ ≫ ξ, ii) a medium swimmer when
two-ﬂuid gel, which depends on the choice of the boundary a ≪ ξ and ℓ ≫ ξ, and iii) a small swimmer when a ≪ ξcondition at the surface of the sphere [18]. Here we con- and ℓ ≪ ξ. In each case, following the procedure in thesider the case of “a sticking ﬂuid and a free network”, i.e., SM, we numerically solve for Vi and Fi to calculate the34002-p3

Kento Yasuda et al.

-2

10

(a)

1

1

2

10

(e)

1

10

0

1

a/ξ = 10a/ξ = 102a/ξ = 103-1a/ξ = 10−3a/ξ = 10−2a/ξ = 10−1

-2

10

-6
-4

-2

10

10

0

10

10

-2

-1

10

-3

10

10

1

10

(b)

2

-6

10

-3

a/ξ = 10a/ξ = 102a/ξ = 103

-6

-6

10

-4

-2

10

0

10

2

-9

10

0

10

-3

10

0

10

Ωτ

10

3

10

1

2

10

(f)
1/2

-3

-6

6

10

3/2
-9

10

a/ξ = 10−3a/ξ = 10−2a/ξ = 10−1

10

-12

0

10

10

10

10

-1

10

10

1/2

10

10

-2

10

Ωτ

(d)

0

10

-9

10

2

10

Ωτ
V e × [32 4 τ /17(d21 − d22 )ξ 2 a]

0

0

10

Ωτ
10

a/ξ = 10−5a/ξ = 10−4a/ξ = 10−3

-1

-2

-3

10

V e × [48 2 τ /5(d21 − d22 )a]

-6

10

1

0

10

-3

10

V e × [32 4 τ /17(d21 − d22 )ξ 2 a]

10

10

10

1

10

10

-4

2

(c)

V v × [24 2 τ /7d12 a]0

10

V v × [16 4 τ /31d21 ξ 2 a]

V v × [16 4 τ /31d21 ξ 2 a]

10

-12

10

9

10

a/ξ = 10−5a/ξ = 10−4a/ξ = 10−3

2
0

10

3

10

6

10

9

10

Ωτ

Ωτ

Fig. 2: (Color online) Plots of the average swimming velocity V as a function
 of the scaled frequency Ωτ for a three-spheremicromachine swimming in a polymer gel with ǫ = [(K + 4G/3)/G]1/2 = 14/3 ≈ 2.16. The solid lines are the numericalresults explained in the text, whereas the dotted lines (with the same colors) are the analytical results obtained from eq. (2).
The analytical dotted curves are invisible when they coincide with the numerical solid curves. The numbers indicate the slopes
representing the exponents of the power-law behaviors. (a) The scaled viscous contribution V v and (b) the scaled elastic
contribution V e for a large swimmer (a ≫ ξ, ℓ ≫ ξ). The diﬀerent colors correspond to the diﬀerent a/ξ values shown in thegraphs; here, we have chosen ℓ/ξ = 104 . The other parameters are φ = π/2 and d1 /ℓ = d2 /ℓ = 10−2 in (a) and d1 = 2d2 andd2 /ℓ = 10−2 in (b). (c) The scaled V v and (d) scaled V e for a medium swimmer (a ≪ ξ, ℓ ≫ ξ). The diﬀerent colors correspondto the diﬀerent a/ξ values shown in the graphs; here, we have chosen ℓ/ξ = 104 . The other parameters are the same as thosein (a) and (b). (e) The scaled V v and (d) scaled V e for a small swimmer (a ≪ ξ, ℓ ≪ ξ). The diﬀerent colors correspond to thediﬀerent a/ξ values shown in the graphs; here, we have chosen ℓ/ξ = 10−2 . The other parameters are the same as those in (a)and (b). Notice that in all these plots, a/ξ and ℓ/ξ are chosen in such a way that a/ℓ ≪ 1 is always satisﬁed.

average velocity V without making an expansion in terms
of a/ℓ. These numerical results are plotted by the solidlines in fig. 2, whereas the analytical results obtained fromeq. (2) are plotted by the dotted lines with the same colors.
In fig. 2, the parameters a/ξ and ℓ/ξ are chosen in sucha way that the condition a/ℓ ≪ 1 is always satisfied for athree-sphere microswimmer. Moreover, d1 /ℓ and d2 /ℓ arealso small enough to ensure that the assumption of linear
viscoelasticity is appropriate.
In figs. 2(a) and (b), we separately plot the viscous V v
and elastic V e contributions, respectively, for a large swimmer as a function of the scaled frequency Ωτ . The behavior
of V v in fig. 2(a) is remarkable because it exhibits a nonmonotonic dependence on Ω. A careful analysis reveals
that it behaves as V v ∼ Ω → Ω−1/2 → Ω → Ω−1 → Ω
as Ω increases. This non-monotonic behavior is more pronounced for larger sphere sizes such as a/ξ = 100 (green).
On the other hand, the frequency dependence of the elastic
contribution, V e , crosses over as V e ∼ Ω2 → Ω0 .

The above results can be reproduced by eq. (2) when we
use the full expressions of the two-ﬂuid mobilities μ andM (most of the dotted lines in fig. 2 are invisible because
they almost coincide with the solid lines). Further, eq. (2)provides us with various asymptotic expressions. For example, we first discuss the limit of Ωτ → 0 in V v . Using
the first term of eq. (5) and the second term of eq. (8), we
obtain31d1 d2 a3 Ω
sin φ,
(9)
Vv ≈
144ℓ4
which is proportional to Ω. The complex non-monotonic
behaviors in the intermediate frequencies are separately
discussed in the SM. In the limit of Ωτ → ∞, we use thefirst term of eq. (6) and the second term of eq. (8) to obtain
31d1 d2 ξ 2 aΩsin φ.
(10)
16ℓ4
By evaluating the elastic term in eq. (2), one can also
obtain the asymptotic expressions for V e . In the limit of34002-p4

Vv ≈A three-sphere microswimmer in a structured ﬂuid
Ωτ → 0, we use the second terms of eqs. (5) and (8) toobtainVe ≈17(d21 − d22 )a5 4 + 3ǫ2(Ωτ )2 ,5184ℓ4ξ 2 τ
ǫ2

(11)

which is proportional to Ω2 . In the limit of Ωτ → ∞, thesecond terms of eqs. (6) and (8) yieldVe ≈17(d21 − d22 )ξ 2 a
,
32ℓ4τ

(12)

which is independent of Ω. Notice that the crossover frequency separating the diﬀerent scaling regimes is strongly
dependent on a/ξ. For example, the crossover frequency between eqs. (11) and (12) roughly scales as(Ωτ )∗ ∼ (a/ξ)−2 . This means that one can extract information concerning the internal structure of the surrounding medium by observing the average swimming velocity.
Moreover, the non-linear dependence on the sphere size,
such as V v ∼ a3 in eq. (9) or V e ∼ a5 in eq. (11), is alsoa unique feature of a two-ﬂuid gel.
For a medium swimmer (a ≪ ξ and ℓ ≫ ξ), we plot
the numerical results of V v and V e in figs. 2(c) and (d),respectively. In fig. 2(c), we find V v ∼ Ω over the entire
frequency range. This behavior is essentially explained byeq. (10), which was obtained for the large swimmer case.
The elastic contribution in fig. 2(d), conversely, crosses
over as V e ∼ Ω2 → Ω1/2 → Ω0 , where the Ω-independent
behavior can be explained by eq. (12) as before. In the SM,
we show the asymptotic expressions for smaller frequencies(see eqs. (S40) and (S41)).
Finally, we numerically plot V v and V e for a small
swimmer (a ≪ ξ and ℓ ≪ ξ) in figs. 2(e) and (f),respectively. The viscous contribution shows a linear
dependence, V v ∼ Ω. This is reasonable because the combination of the first terms in eqs. (6) and (7) simply represents a purely viscous ﬂuid. The elastic contribution
plotted in fig. 2(f) crosses over as V e ∼ Ω2 → Ω3/2 →Ω1/2 → Ω0 . Even though the first two scaling behaviors
cannot be obtained analytically, the last two behaviors are
given in the SM (see eqs. (S42) and (S43)). It should be
mentioned here that eq. (2) does not reproduce the numerical result in fig. 2(f) because the lowest expansion interms of a/ℓ is inappropriate for this limit. Hence the dotted lines deviate from the solid lines fig. 2(f) especially for
relatively smaller Ωτ values.
Discussion. – So far, we have primarily discussed the
motion of a three-sphere microswimmer in a polymer gel
described by the two-ﬂuid model. However, the importance of our work is not restricted to these specific models.
The prediction of the average velocity in eq. (2) is applicable to any structured viscoelastic ﬂuid that has an intermediate length scale and a characteristic time scale. For
example, one can also discuss the motion of a microswimmer in a polymer solution that is described by a diﬀerent“two-ﬂuid” model [24,25]. In addition, it is interesting
to discuss the dynamics of three-sphere microswimmers inliquid crystals [26], which typically exhibit complex rheological behavior depending on their diﬀerent phases [27].
Furthermore, we encounter a similar situation when we
consider the motion of a three-disk microswimmer immersed in a quasi-2D ﬂuid membrane [28]. Owing to
the presence of the hydrodynamic screening length in thequasi-2D ﬂuid, the geometric factor appearing in the average velocity exhibits various asymptotic behaviors whenchanging the ratio between the swimmer size and thescreening length. The result in ref. [28] can be obtained
from eq. (2) using the mobility of the disk and the coupling
mobility in a quasi-2D ﬂuid.
At this point, it is useful to give some numbers related to realistic microswimmers and systems describedby a two-ﬂuid model. Let us first consider a homogeneous sample of entangled F-actin network whose meshsize ξ can be controlled
by the actin monomer concentra√tion c as ξ = 0.3/ c (ξ in μm and c in mg/ml) [19,20].
Hence ξ ≈ 1 μm for c ≈ 0.1 mg/ml. On the other hand,a typical size of colloidal particles used in microrheology
experiments is a ≈ 0.5 μm [29]. If we assume that one
can construct a three-sphere microswimmer by connecting
these colloidal particles, its whole size (corresponding to ℓ)would amount to several microns, which is comparable toa typical size of living microorganisms such as bacteria.
Since a magnitude relation a < ξ < ℓ holds in this case, aliving microorganism swimming in an actin network may
correspond to a medium swimmer (see fig. 2(c) and (d)).
On the other hand, a three-sphere swimmer has been
experimentally realized by using ferromagnetic particles
at an air-water interface and by applying an oscillating magnetic field [5,6]. In this experiment, the size of
spheres is about a ≈ 200 μm and the rest length of thebonds is roughly ℓ ≈ 1 mm. Hence such a model swimmer behaves as a large swimmer in an actin network (seefig. 2(a) and (b)), since the mesh size is much smaller,i.e., ξ ≪ a < ℓ. Such a comparison shows that the swimming behaviors of a microorganism and a ferromagnetic
swimmer are essentially diﬀerent in a structured ﬂuid suchas F-actin networks due to the presence of the characteristic length scale ξ. This is the main message of thepresent work.
Concerning the characteristic time scale τ of a twoﬂuid model, we shall refer to a number that was measured
for F-actin networks [19,20] by using microrheology techniques [29]. With the water viscosity η ≈ 10−3 Pa · s andthe measured shear modulus G ≈ 10−1 Pa for a F-actin
network, we obtain the viscoelastic time scale as τ =η/G ≈ 10−2 s. For a typical microorganism and a modelswimmer mentioned above, characteristic frequencies areabout Ω ≈ 102 Hz [1] and 1 Hz [5,6], respectively. Hencethe corresponding Deborah numbers De = Ωτ for thesetwo cases are De ≈ 1 (microorganism) and 10−2 (modelswimmer) which are both considered in fig. 2.
Finally, we note that Fu et al. analyzed the swimming behavior of an infinite sheet undergoing transverse traveling-wave deformations in a two-ﬂuid gel [30].

34002-p5

Kento Yasuda et al.
They demonstrated that the boundary conditions between
the sheet and the network significantly aﬀect the swimming speed. In our study, we considered only the caseof “a sticking ﬂuid and a free network” for the boundary
condition between the sphere and the gel. Diﬀerent situations such as “a sticking ﬂuid and a sticking network” caseor “a sticking ﬂuid and a slipping network” case, as discussed in detail in ref. [18], will lead to diﬀerent swimming
behaviors because the self-mobility of the spheres is modified. Although several deficiencies of the two-ﬂuid model
have been explicitly pointed out [18], such investigations
are left to future studies.
Summary. – We discussed the locomotion of a threesphere microswimmer in a viscoelastic structured ﬂuid
with typical length and time scales. We derived a general expression for the average swimming velocity, eq. (2),which includes both viscous and elastic contributions. To
illustrate our result, we used the two-ﬂuid model for apolymer gel and demonstrated that the average velocity
exhibits various asymptotic behaviors depending on theswimmer size. Because one can extract information concerning the internal structure of the surrounding ﬂuid fromobservations of the motion of a microswimmer, the present
theory oﬀers a new approach to active microrheology [29].
∗∗∗KY acknowledges support by a Grant-in-Aid for JSPS
Fellows (Grant No. 18J21231) from the Japan Society for
the Promotion of Science (JSPS). SK acknowledges support by a Grant-in-Aid for Scientific Research (C) (Grant
No. 18K03567) from the JSPS.
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