PHYSICAL REVIEW E 97, 052612 (2018)

Three-disk microswimmer in a supported fluid membrane
Yui Ota, Yuto Hosaka, Kento Yasuda, and Shigeyuki Komura*
Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan(Received 25 February 2018; published 18 May 2018)
A model of three-disk micromachine swimming in a quasi-two-dimensional supported membrane is proposed.
We calculate the average swimming velocity as a function of the disk size and the arm length. Due to the presenceof the hydrodynamic screening length in the quasi-two-dimensional fluid, the geometric factor appearing in the
average velocity exhibits three different asymptotic behaviors depending on the microswimmer size and thehydrodynamic screening length. This is in sharp contrast with a microswimmer in a three-dimensional bulk fluidthat shows only a single scaling behavior. We also find that the maximum velocity is obtained when the disks areequal-sized, whereas it is minimized when the average arm lengths are identical. The intrinsic drag of the disks
on the substrate does not alter the scaling behaviors of the geometric factor.
DOI: 10.1103/PhysRevE.97.052612
I. INTRODUCTION

Biological membranes are composed of lipid molecules andvarious types of proteins which can move laterally due to themembrane fluidity [1]. Hence biomembranes play importantroles in various life processes, such as the transportation ofmaterials or the reaction between chemical species [2]. Whilesome proteins are subjected to thermal agitations of lipid
molecules and undergo passive Brownian motions [3,4], there
is also a large number of active proteins which cyclicallychange their conformations [5]. For instance, with a supplyof adenosine triphosphate, some proteins act as ion pumps bychanging their structural conformations in order to allow materials to pass through the membranes [6–8]. In general, such
cyclic motions of proteins can lead to their active locomotion
under certain conditions rather than just a passive motion.
By transforming chemical energy into mechanical work,
microswimmers change their shape and move in viscousenvironments [9]. Over the length scale of microswimmers, thefluid forces acting on them are governed by the effect of viscous
dissipation. According to Purcell’s scallop theorem [10], timereversal body motion cannot be used for locomotion in a
Newtonian fluid [11]. As one of the simplest models exhibiting
broken time-reversal symmetry in a three-dimensional (3D)
fluid, Najafi and Golestanian proposed a three-sphere swimmer[12,13], in which three in-line spheres are linked by two arms ofvarying length. This model is suitable for analytical treatment
because it is sufficient to consider only the translationalmotion, and the tensorial structure of the fluid motion canbe neglected. Recently, such a three-sphere swimmer has
been experimentally realized [14,15]. Moreover, some authors
proposed a generalized three-sphere microswimmer in which
the spheres are connected by two elastic springs with varying
natural lengths [16,17].
Compared to microswimmers in 3D bulk fluids, those in 2Dor quasi-2D fluids such as biomembranes have been less inves-*komura@tmu.ac.jp

2470-0045/2018/97(5)/052612(7)tigated in spite of their importance. Huang et al. considered amodel of an active inclusion in a membrane with three particles(domains) connected by variable elastic springs [18]. In theirmodel, the natural lengths of the springs depend on the discrete
states that are cyclically switched. They also performed a
microscopic dynamical simulation, where the lipid bilayer
structure of the membrane is resolved and the solvent effects
are included by multiparticle collision dynamics. For quasi-2D
fluids, there exists a hydrodynamic screening length which
distinguishes 2D and 3D hydrodynamic interactions [19,20]. In
the model by Huang et al., the longitudinal coupling mobility
has a logarithmic dependence on the distance between two
particles, which is valid only when the distance is much smaller
than the hydrodynamic screening length. As for the mobilityof a particle, they employed the 3D Stokes law even in a 2D
fluid membrane, which is justified only when the particle size
is much larger than the hydrodynamic screening length.
In this paper, we present a systematic and also analyticalinvestigation on the locomotion of a 2D microswimmer immersed in a supported fluid membrane, i.e., a lipid bilayer
membrane located on a solid substrate [21,22]. For supported
membranes, the membrane-substrate distance is usually notlarge, and such a direct contact leads to a frictional coupling
between the membrane and the solid support. Our swimmer
consists of three thin rigid disks (rather than spheres) connectedby two arms or springs which can undergo prescribed cyclic
motions. We employ the 2D mobility and the coupling mobility
that take into account the hydrodynamic interactions mediatedby the quasi-2D fluid in the presence of the substrate [19]. We
analytically obtain the average velocity of such a three-disk
micromachine as a function of the disk and arm sizes. Due to thepresence of the hydrodynamic screening length associated with
the quasi-2D fluid model, the geometric factor in the averagevelocity exhibits various asymptotic size dependencies, whichis in sharp contrast to a microswimmer in 3D bulk fluids
because they do not have any characteristic length scale.
In the next section, we briefly review the mobilities in aquasi-2D fluid describing a supported membrane. In Sec. III,
we discuss the motion of a 2D three-disk microswimmer in a

052612-1©2018 American Physical Society

OTA, HOSAKA, YASUDA, AND KOMURA

12

PHYSICAL REVIEW E 97, 052612 (2018)3

substrate
FIG. 1. A 2D three-disk micromachine swimming in a fluid
membrane supported by a solid substrate. The flat fluid membrane
located at z = 0 is infinitely large and its 2D viscosity is η. The 2D
momentum within the membrane can leak away due to the friction
between the membrane and the substrate [see Eq. (1)], and hence themembrane corresponds to a quasi-2D fluid. The 2D microswimmer
consists of three rigid disks of radii ai (i = 1,2,3) that are connectedby two arms of variable lengths Lj (j = 1,2). Without loss ofgenerality, the microswimmer is assumed to move along the x axis.
In the present model, the two arms can open and close in a prescribedform.

supported membrane. In Sec. IV, we argue various asymptotic
behaviors of the geometric factor appearing in the averagevelocity. We also examine the effects of structural asymmetryof 2D and 3D microswimmers in Sec. V. Finally, the summaryof our work and some discussions are given in Sec. VI.
II. MOBILITIES IN A QUASI-2D FLUID

Eq. (1) can be identified as λ = ηs / h, provided that h is smallenough [25].
Solving the above quasi-2D hydrodynamic equations, one
can obtain the translational mobility coefficient μ of a rigid diskthat is defined by V = μF , where V is the disk velocity andF is a driving force. By using the no-slip boundary condition,
the resulting expression becomes [23–25]−1
1 (κa)2
κaK1 (κa)μ(a) =
,
(3)
+
4π η
4
K0 (κa)√where a is the disk radius, κ = λ/η, and K0 and K1 aremodified Bessel functions of the second kind, order zero andone, respectively. Physically, κ −1 represents the hydrodynamic
screening length beyond which the 2D hydrodynamic interaction becomes irrelevant. Notice that κ −1 diverges as λ → 0,
which alludes the Stokes paradox in a pure 2D fluid [3,4].
The effect of intrinsic drag of the disk on the substrate will be
discussed later in the final section.
For κa  1, the above disk mobility asymptotically behaves as
  

2
1
ln
−γ ,
(4)
μ(a) ≈
4π η
κawhere γ = 0.5772 · · · is Euler’s constant. Here the mobility
is only weakly (logarithmically) dependent on the disk size a,which is characteristic for 2D fluids. For κa  1, on the other
hand, we haveμ(a) ≈1,π η(κa)2(5)(2)

which shows a stronger algebraic size dependence whencompared with the Stokes law for 3D fluids. Such a size
dependence can be understood in terms of mass conservation
principle because 2D momentum is not conserved any more in
the quasi-2D hydrodynamics [26].
Next we explain the hydrodynamic interaction between thetwo disks immersed in the membrane by using the velocity
Green’s function Gαβ (r). This tensor gives the flow velocityv(r) of the membrane at r due to a point force F exerted
on the membrane at the origin in the xy plane, according
to vα (r) = Gαβ (r)Fβ with α,β = x,y. The velocity Green’s
function can generally be expressed as Gαβ (r) = C1 (r)δαβ +C2 (r)(rα rβ /r 2 ), where δαβ is the Kronecker δ and r = |r|.
The longitudinal coupling mobility between the two disks inthe membrane can be obtained from M(r) = C1 (r) + C2 (r),where r here denotes the distance between the two disks andshould satisfy the condition r  a. Hence M does not depend
on a up to the lowest-order contribution.
Using the quasi-2D hydrodynamic equations, the longitudinal coupling mobility M is given by [27–31]
11K1 (κr).
(6)
M(r) =−
2π η (κr)2κr

It is worthwhile to briefly mention here the physical meaning of the friction parameter λ. For a supported membrane,
a thin lubricating layer of bulk solvent [thickness h and 3D
viscosity ηs (N s/m2 )] exists between the membrane and thesubstrate [23]. In such a situation, the friction parameter inFor κr  1, the above coupling mobility asymptotically behaves as
  

1
2
1
M(r) ≈
ln
−γ +
.
(7)
4π η
κr
2

We first describe the quasi-2D hydrodynamic model fora supported fluid membrane as shown in Fig. 1. Although alipid bilayer membrane itself can be treated as a 2D fluid, it
is not an isolated system because the membrane is supportedby the outer solid substrate (hence a quasi-2D fluid). Dueto the friction between the 2D membrane and the substrate,
the momentum within the membrane can leak away from themembrane. Such an effect can be taken into account through amomentum decay term in the hydrodynamic equation. Within
the Stokes approximation and by assuming the steady state, we
consider the following 2D equation for a supported membrane[23–25]:η∇ 2 v − ∇p − λv = 0.

(1)

In the above, ∇ = (∂x ,∂y ) is a 2D differential operator, v (m/s)
and p (N/m) are the 2D velocity and pressure, respectively,
η (N s/m) is the membrane 2D viscosity, and λ (N s/m3 ) is
the momentum decay parameter (or the friction coefficient).
In addition, we employ the 2D incompressibility condition as
expressed by∇ · v = 0.

052612-2

THREE-DISK MICROSWIMMER IN A SUPPORTED FLUID …

For κr  1, on the other hand, we haveM(r) ≈

1.
2π η(κr)2(8)Equations (7) and (8) are analogous to Eqs. (4) and (5),respectively, and the physical origins are exactly the same as
those for the disk mobility μ.
III. THREE-DISK MICROSWIMMER

Having explained the quasi-2D hydrodynamic model fora supported membrane and the resulting mobilities for inclusions, we now investigate the locomotion of a microswimmer
in a membrane. To calculate the swimming velocity, we follow
the procedure in Ref. [13] for a three-sphere swimmer in a 3D
bulk fluid. As shown in Fig. 1, we consider a 2D micromachineconsisting of three rigid disks of radii ai (i = 1,2,3) that areconnected by two arms of variable lengths Lj (j = 1,2). Sucha three-disk microswimmer is immersed in an infinitely large
and flat supported membrane having 2D viscosity η and thefriction coefficient λ, as described before. Each disk exerts aforce Fi on the quasi-2D fluid that we assume to be along theswimmer axis. Without loss of generality, the microswimmer
is assumed to move along the x axis. In the limit ai /Lj  1,
we can use Eqs. (3) and (6) to relate the forces Fi and thevelocities Vi as
V1 = μ(a1 )F1 + M(L1 )F2 + M(L1 + L2 )F3 ,(9)

V2 = M(L1 )F1 + μ(a2 )F2 + M(L2 )F3 ,(10)

V3 = M(L1 + L2 )F1 + M(L2 )F2 + μ(a3 )F3 .

(11)

The swimming velocity of the whole object is obtained byaveraging the velocities of the three disks, i.e., V = (V1 +V2 + V3 )/3. Since we are interested in the autonomous net
locomotion of the swimmer, there are no external forces acting
on the disks. This leads to the following force-free condition:
F1 + F2 + F3 = 0.

(12)

As assumed in Ref. [13], the motion of the arms connecting
the three disks is prescribed by the two given functions Lj (t).
In this situation, the arm motions are related to the velocitiesasL̇1 = V2 − V1 ,L̇2 = V3 − V2 ,(13)

PHYSICAL REVIEW E 97, 052612 (2018)

where the dot indicates the time derivative. The set of six
equations in Eqs. (9)–(13) is sufficient to solve for the six
unknown quantities Vi and Fi (i = 1,2,3).
We further assume that arm deformations are relatively
small as given by
L1 (t) =1 + u1 (t),L2 (t) =2 + u2 (t),(14)

where j are constants and uj / j  1. With these prescribedarm motions, we perform an expansion of the swimming
velocity to the leading order in both ai / j and uj / j . Aftersome calculations, we finally obtain the average swimming
velocity as [13]G(15)
u1 u̇2 − u̇1 u2 ,2where G is the geometric factor to be presented later in
Eq. (18), and the averaging · · · should be performedby time integration in a full cycle. In the above calculation, the terms proportional to u1 u̇1 , u2 u̇2 , and u1 u̇2 +u̇1 u2 are omitted because they average out to zero in acycle.
As studied for a three-sphere swimmer [13], one canassume, for example, that the two arms undergo the following
periodic motions:V =u1 (t) = d1 cos( t), u2 (t) = d2 cos( t − φ).

(16)Here d1 and d2 are the amplitudes of the oscillatory motions,is a common arm frequency, and φ is a mismatch in phases
between the two arms. Then the average swimming velocity in
Eq. (15) further reads
V =G
d1 d2 sin φ,2(17)which is maximized when φ = π/2. When the disks are connected by elastic springs with time-dependent natural lengths,a more general expression for V can be obtained [17].
IV. GEOMETRIC FACTOR

The geometric factor G (having the dimension of inverse
length) in Eq. (15) or Eq. (17) for a three-disk swimmer in aquasi-2D fluid turns out to beG(i ,δj )κ1 2 3 K0 (1 )K0 (2 )K0 (3 )[1 K0 (1 ) + 4K1 (1 )][2 K0 (2 ) + 4K1 (2 )][3 K0 (3 ) + 4K1 (3 )]=
2
41 K1 (1 )K0 (2 )K0 (3 ) + 42 K0 (1 )K1 (2 )K0 (3 ) + 43 K0 (1 )K0 (2 )K1 (3 ) + 12 + 22 + 32 K0 (1 )K0 (2 )K0 (3 ) K2 (δ1 ) K2 (δ2 ) K2 (δ1 + δ2 )
1
1
1
−
,
(18)
× 2 3+ 3−+−
δ1
δ2
δ1 + δ2
δ1
δ2(δ1 + δ2 )3
where i = κai and δj = κ j and K2 is modified Bessel function of the second kind, order two. We note that Eq. (18) is invariant
under the exchange of not only the three disks ai but also under the exchange of the two arms j .

052612-3

OTA, HOSAKA, YASUDA, AND KOMURA

10

10

10

10

10

PHYSICAL REVIEW E 97, 052612 (2018)

mobility coefficient μ and the longitudinal coupling mobility
M by0G(a, ) ∼

-2

-4

-6

-8

10

-1

10

0

10

1

10

2

10

3

10

4

FIG. 2. Plot of the scaled geometric factor G/κ [see Eq. (19)] fora fully symmetric 2D microswimmer as a function of the scaled arm
length δ = κ when the disk-to-arm ratio is a/ = 10−2 . The three
asymptotic expressions in Eqs. (20)–(22) are plotted by the dashedlines.

For the fully symmetric case with a1 = a2 = a3 = a and
2 = , the geometric factor in Eq. (18) reduces toG(,δ)
1 2 4K1 ()
 +
=
κ
36
K0 ()

15 8K2 (δ) 2K2 (2δ)
+
,
(19)
−
×
δ3
δ
δ

1 =

where  = κa and δ = κ . Equations (18) and (19) are themain results of this paper. In Fig. 2, we plot G/κ in Eq. (19) asa function of δ while keeping the ratio to a/ = 10−2 . In fact,
Eq. (19) has three asymptotic expressions
G(,δ)1≈κ3δ[ln(2/) − γ ](20)for   1 and δ  1,
G(,δ)
5≈ 3
κ3δ [ln(2/) − γ ](21)for   1 and δ  1, andG(,δ)
5 2
≈
κ
12δ 3

(23)

This relation holds irrespective of the dimensionality of the
micromachine and the surrounding fluid [32]. Due to thepresence of the hydrodynamic screening length κ −1 in a
supported membrane, both μ and M exhibit different asymptotic behaviors as shown in Eqs. (4) and (5) and Eqs. (7)and (8), respectively. Various limiting expressions of G in
Eqs. (20)–(22) can be understood as different combinationsof the asymptotic forms of μ and M. For example, Eq. (22)showing the scaling G ∼ (a 2 / 2 )/ is a direct consequence of
Eqs. (5) and (8). We also note that, because of the explicit 1/
dependence in Eq. (23), the logarithmic dependence of M( )on , as in Eq. (7), does not show up in Eq. (20) within thelowest-order expansion.
In order to compare our result with that of a three-sphereswimmer in a 3D fluid, we show here its corresponding
geometric factor as obtained in Ref. [13]:1
3a1 a2 a3
1
1
G3D =
,
(24)
+ 2−
(a1 + a2 + a3 )2 21( 1 + 2 )22
where ai here denote the radii of the three spheres (rather
than disks). For the symmetric case with a1 = a2 = a3 = a
and 1 = 2 = , the above expression reduces to
7a
.
(25)
12 2
First, we note that Eq. (24) or Eq. (25) does not depend on the
3D fluid viscosity, while Eq. (18) or Eq. (19) is dependent on
the membrane viscosity η through the inverse screening length
κ. Second, the essential size dependence in Eq. (25) is G3D ∼(a/ )/ which does not appear in the previous quasi-2D case.
On the other hand, such a dependence is in accordance withthe scaling relation Eq. (23) and the Stokes law in a 3D fluid
without any characteristic length scale. Hence the existenceof the characteristic length scale, κ −1 , for a quasi-2D fluid
completely changes the asymptotic size dependencies of theaverage velocity. This is an important finding of this paperand highlights the essential difference between 2D and 3D
microswimmers.
G3D =V. ASYMMETRIC MICROSWIMMERS(22)

for   1 and δ  1.
We note here that Eq. (20) decays as δ −1 , whereas Eqs. (21)and (22) decay as δ −3 . The dependence on the disk size  is only
logarithmic in Eqs. (20) and (21), while it is proportional to  2
in Eq. (22). In Fig. 2, we also plot the above three asymptotic
expressions by the dashed lines when a/ = 10−2 . They are allin good agreement with the solid line that corresponds to thefull expression of Eq. (19). Notice that the apparent behavior of
Eq. (22) is δ −1 because we have fixed the ratio to a/ = 10−2in this plot.
It is important to mention the physical meaning of thegeometric factor G in the average velocity. Within the scalingargument, the geometric factor is generally related to the

M( )
.
μ(a)

Since we have obtained the general expression of thegeometric factor G for a three-disk microswimmer, as shown in
Eq. (18), we discuss now the effects of structural asymmetryof a microswimmer on its geometric factor. We first set as
1 = 2 = and vary the two ratios between the disk sizes as
defined bya1a3
1 = , 3 = ,
(26)
a2
a2
whereas we keep, for instance, the sum of the three radii
being fixed to a1 + a2 + a3 = 3a. In Fig. 3, we plot the scaled
geometric factor G/κ in Eq. (18) as a function of the two
ratios 1 and 3 when κa = 0.1 and κ = 10. Notice that a/should be small within our expansion scheme and the color

052612-4

THREE-DISK MICROSWIMMER IN A SUPPORTED FLUID …

PHYSICAL REVIEW E 97, 052612 (2018)

2

10

-3.20

0

-3.28

1

-1

10

-3.36
-3.44

0

10

-3.52

-1

-2

-3.60
-3

10

-3.68

-2

-2

-2

-1

0

1

10

2

FIG. 3. Color plot of the scaled geometric factor log(G/κ) [seeEq. (18)] of a 2D microswimmer as a function of the asymmetry
parameters 1 = a1 /a2 and 3 = a3 /a2 [see Eq. (26)] when κa = 0.1
and κ = 10. Notice that a = (a1 + a2 + a3 )/3 and = 1 = 2 . All
the quantities including the color scale are plotted in the logarithmic
scale. The maximum of G occurs at 1 = 3 = 1.

scale indicates the quantity log(G/κ). From this plot, we findthat the maximum of the geometric factor is realized when thedisk size is identical, i.e., 1 = 3 = 1. In other words, any
asymmetry in the disk size leads to a reduction of the averageswimming velocity.
In Fig. 4, we further consider the case when 1 = 3 = (or, equivalently, a1 = a3 ) and plot G/κ as a function of .
Such a plot corresponds to a cross section of Fig. 3 along the
diagonal line. To compare the quasi-2D case with the 3D case,
we also plot G3D /κ [see Eq. (24)] as a function of  under
the same condition. For G3D , the ratios 1 and 3 correspond
-3

10

-4

10

-1

10

0

10

10

1

2

10

FIG. 5. Plot of the scaled geometric factor G/κ (solid line) [seeEq. (18)] and G3D /κ (dashed line) [see Eq. (24)] as a function of
the asymmetry parameter  = 1 / 2 [see Eq. (27)] when κa = 0.1
and κ = 10. Notice that a = a1 = a2 = a3 and = ( 1 + 2 )/2. The
minima of G and G3D occur at  = 1.

to those between the sphere radii. It should be also noted thatEq. (24) does not contain κ and it is used only to compare with
the quasi-2D case. Although both G and G3D are maximizedat  = 1, the dependence on  is much weaker for G. This
weak dependence originates from the logarithmic dependenceof the disk mobility μ on the disk size a, as shown in Eq. (4).
Alternatively, one can set as a1 = a2 = a3 = a and vary theratio between the arm lengths defined by
=

1

,

(27)

2

whereas we keep the sum of the two arm lengths being fixed
to 1 + 2 = 2 . In Fig. 5, we plot the scaled geometric factors
G/κ and G3D /κ as a function of  when κa = 0.1 andκ = 10 as before. Here it is remarkable that both G and G3D
are minimized when  = 1. On the other hand, the overallbehaviors of G and G3D are rather similar because the chosenparameter value satisfies the condition κ  1 for which thecoupling mobility M exhibits an algebraic dependence on the
distance , as shown in Eq. (8).
VI. SUMMARY AND DISCUSSION

-5

10

-6

10

-2

10

-1

10

0

10

10

1

2

10

FIG. 4. Plot of the scaled geometric factor G/κ (solid line) [seeEq. (18)] and G3D /κ (dashed line) [see Eq. (24)] as a function of
the asymmetry parameter  = 1 = 3 when κa = 0.1 and κ = 10.
Notice that a = (a1 + a2 + a3 )/3 and = 1 = 2 . The maxima of G
and G3D occur at  = 1.

In summary, we have proposed a model of 2D three-disk
micromachine swimming in a quasi-2D supported membrane.
In particular, we have obtained the average swimming velocity
as a function of the disk size and the arm length. Due to thepresence of the hydrodynamic screening length in the quasi-2D
fluid, κ −1 , the geometric factor in the average velocity exhibits
various asymptotic behaviors depending on the microswimmer
size and the screening length. Our result has been confirmedby the scaling argument for the geometric factor. We have also
looked at the effects of structural asymmetry of a microswimmer and found that the geometric factor is maximized when
the disks are equal sized, whereas it is minimized when the
average arm lengths are identical.

052612-5

OTA, HOSAKA, YASUDA, AND KOMURAPHYSICAL REVIEW E 97, 052612 (2018)

At this point, a rough estimate of the characteristic lengthscales would be useful [25,29,31]. The membrane viscosity oflipid bilayers at physiological temperatures is approximatelyη ≈ 10−9 N s/m and the viscosity of surrounding water is ηs ≈10−3 N s/m2 . For supported membranes, we can approximate−9
the height of the intervening√ solvent region as h ≈ 10 m.
Hence we obtain κ −1 = ηh/ηs ≈ 3 × 10−8 m. We note that
this length scale is relatively small and the large-scale behavior
is expected for micron-sized swimmers.
In our model of a three-disk microswimmer, we haveassumed that the three disks are connected by two arms
and their time-dependent motions are given by Eq. (14)[12,13]. Alternatively, one can also consider a three-disk
microswimmer in which the disks are connected by two elastic
springs, while the natural length of each spring is assumed
to undergo a prescribed cyclic change. Using the results in
Ref. [17], one can immediately estimate the average quantity
u1 u̇2 − u̇1 u2 in Eq. (15) and obtain the average velocity Vof an elastic 2D microswimmer. It can be generally shownthat the swimming velocity increases with frequency in the
low-frequency region, whereas in the high-frequency region,
the average velocity decreases when the frequency is increased[17]. Such a behavior originates from the intrinsic spring
relaxation dynamics of an elastic swimmer.
Although we have taken into account the hydrodynamic
friction between the fluid membrane and the substrate throughthe friction coefficient λ in Eq. (1), the effect of intrinsic dragof the disks on the substrate was not considered in Eq. (3).
One can naturally assume that the drag coefficient of a disk on
the substrate is proportional to its area and is given by λd π a 2 ,where λd is the disk friction coefficient. In this case, the translational mobility coefficient in Eq. (3) will be modified [23]:



−1
λd
κaK1 (κa)
1 (κa)2
1++μ(a) =
.
(28)
4π η
4
λ
K0 (κa)Notice that the correction term due to λd /λ gives rise to acontribution that is independent of λ. Since only the coefficientof (κa)2 is altered when compared with Eq. (3), the drag acting
on the disks modifies our result only up to a numerical factor
when κa  1. This means that the asymptotic scaling
behaviors in Sec. IV are not affected by the drag forces on thedisks. In principle, the disk friction coefficient can be different
for different disks. Such an effect can be effectively taken into
account by considering different disk radii as long as they aremuch larger than the screening length κ −1 .
In this paper, we have discussed the behavior of a 2D
microswimmer in a quasi-2D fluid that is characterized bya hydrodynamic screening length, κ −1 . It should be noted,
however, that we encounter a similar situation in which a 3D
micromachine swims in a structured viscoelastic fluid having
a characteristic length scale. According to our preliminary
result, we find that the frequency dependence of the averagevelocity exhibits fairly complex behaviors depending on themachine size relative to the characteristic length scale of thesurrounding structured fluid. Details of such an investigation
will be reported elsewhere [32].
In the future, we shall also investigate the case when thesurrounding bulk fluid is viscoelastic [30]. Such a study will
enable us to obtain the frequency-dependent complex viscosityof the surrounding 3D fluid by measuring the velocity of a 2D
microswimmer in a membrane [33]. Such a method will provide us with a new type of noncontact surface microrheology.

[1] S. J. Singer and G. L. Nicolson, Science 175, 720 (1972).
[2] B. Alberts, A. Johnson, P. Walter, J. Lewis, and M. Raff,
Molecular Biology of the Cell (Garland Science, New York,2015).
[3] P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. USA 72,
3111 (1975).
[4] P. G. Saffman, J. Fluid Mech. 73, 593 (1976).
[5] Y. Togashi and A. S. Mikhailov, Proc. Natl. Acad. Sci. USA 104,8697 (2007).
[6] J.-B. Manneville, P. Bassereau, D. Lévy, and J. Prost, Phys. Rev.
Lett. 82, 4356 (1999).
[7] J.-B. Manneville, P. Bassereau, S. Ramaswamy, and J. Prost,
Phys. Rev. E 64, 021908 (2001).
[8] M. D. El Alaoui Faris, D. Lacoste, J. Pécréaux, J.-F. Joanny, J.
Prost, and P. Bassereau, Phys. Rev. Lett. 102, 038102 (2009).
[9] E. Lauga and T. R. Powers, Rep. Prog. Phys. 72, 096601 (2009).
[10] E. M. Purcell, Am. J. Phys. 45, 3 (1977).
[11] E. Lauga, Soft Matter 7, 3060 (2011).
[12] A. Najafi and R. Golestanian, Phys. Rev. E 69, 062901 (2004).

[13] R. Golestanian and A. Ajdari, Phys. Rev. E 77, 036308(2008).
[14] G. Grosjean, M. Hubert, G. Lagubeau, and N. Vandewalle, Phys.
Rev. E 94, 021101 (2016).
[15] G. Grosjean, M. Hubert, and N. Vandewalle, Adv. Colloid
Interface Sci. 255, 84 (2018).
[16] J. Pande, L. Merchant, T. Krüger, J. Harting, and A.-S. Smith,
New J. Phys. 19, 053024 (2017).
[17] K. Yasuda, Y. Hosaka, M. Kuroda, R. Okamoto, and S. Komura,
J. Phys. Soc. Jpn. 86, 093801 (2017).
[18] M.-J. Huang, A. S. Mikhailov, and H.-Y. Chen, Eur. Phys. J. E
35, 119 (2012).
[19] S. Komura, S. Ramachandran, and M. Imai, in Non-Equilibrium
Soft Matter Physics, edited by S. Komura and T. Ohta (World
Scientific, Singapore, 2012), p. 197.
[20] S. Komura and D. Andelman, Adv. Colloid Interface Sci. 208,
34 (2014).
[21] E. Sackmann, Science 271, 43 (1996).
[22] M. Tanaka and E. Sackmann, Nature 437, 656 (2005).

ACKNOWLEDGMENTS

We thank R. Okamoto for helpful discussions. S.K. acknowledges support by a Grant-in-Aid for Scientific Researchon Innovative Areas “Fluctuation and Structure” (Grant No.
25103010) from the Ministry of Education, Culture, Sports,
Science, and Technology (MEXT) of Japan and by a Grantin-Aid for Scientific Research (C) (Grants No. 15K05250 and
No. 18K03567) from the Japan Society for the Promotion of
Science (JSPS).

052612-6

THREE-DISK MICROSWIMMER IN A SUPPORTED FLUID …

PHYSICAL REVIEW E 97, 052612 (2018)[23] E. Evans and E. Sackmann, J. Fluid Mech. 194, 553 (1988).
[24] K. Seki and S. Komura, Phys. Rev. E 47, 2377 (1993).
[25] S. Ramachandran, S. Komura, M. Imai, and K. Seki, Eur. Phys.
J. E 31, 303 (2010).
[26] H. Diamant, J. Phys. Soc. Jpn. 78, 041002 (2009).
[27] S. Ramachandran, S. Komura, and G. Gompper, Europhys. Lett.
89, 56001 (2010).
[28] N. Oppenheimer and H. Diamant, Phys. Rev. E 82, 041912(2010).

[29] S. Ramachandran, S. Komura, K. Seki, and G. Gompper, Eur.
Phys. J. E 34, 46 (2011).
[30] S. Komura, S. Ramachandran, and K. Seki, Europhys. Lett. 97,
68007 (2012).
[31] Y. Hosaka, K. Yasuda, R. Okamoto, and S. Komura, Phys. Rev.
E 95, 052407 (2017).
[32] K. Yasuda, R. Okamoto, and S. Komura (unpublished).
[33] K. Yasuda, R. Okamoto, and S. Komura, J. Phys. Soc. Jpn. 86,
043801 (2017).

052612-7