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Two-sphere low-Reynolds-number propeller

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Two-sphere low-Reynolds-number propeller
Ali Najaﬁ
1,
*
and Rojman Zargar
2
1
Department of Physics, Zanjan University, Zanjan 313, Iran
2
Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan 45195, Iran
Received 7 January 2010; revised manuscript received 12 May 2010; published 4 June 2010
A three-dimensional model of a low-Reynolds-number swimmer is introduced and analyzed in this Brief Report. This model consists of two large and small spheres connected by two perpendicular thin rods. Thegeometry of this system is motivated by the microorganisms that use a single tail to swim; the large sphererepresents the head of microorganism and the small sphere resembles its tail. Each rod changes its length andorientation in a nonreciprocal manner that effectively propels the system. Translational and rotational velocitiesof the swimmer are studied for different values of parameters. Our ﬁndings show that by changing theparameters we can adjust both the velocity and the direction of motion of the swimmer.DOI:10.1103/PhysRevE.81.067301PACS number
s
: 47.15.G
, 87.19.ru, 45.40.Ln
I. INTRODUCTION
The propulsive motion of artiﬁcial and biological micron-scale objects is an interesting problem at low-Reynolds-number hydrodynamics. In this condition the dynamics isdominated by viscous forces. Examples of these micron-scale objects include biological microorganisms such as bac-teria and also man made microswimmers, useful to operate atmicroﬂuidic investigations
1
.Propulsive motion at low Reynolds number is subject tothe
scallop theorem
2
. At small scales, where the Reynoldsnumber is very low, the governing hydrodynamic equations,i.e., the Stokes and continuity equations, are linear and in-variant under time reversal
3
. Any reciprocal shape defor-mation retraces its trajectory and the system stays back at thepoint where it started. In order to achieve a net translationaldisplacement, the system should perform the body deforma-tions in a nonreciprocal manner. As mentioned by Purcell alow-Reynolds-number propeller must have at least two inter-nal degrees of freedom and he proposed a three-link swim-mer. The detailed motion of Purcell’s swimmer was exam-ined by Becker
et al.
where it was shown that Purcell’ssystem could swim and its dynamical properties were calcu-lated
4
. Inspired by Purcell’s system, a low-Reynolds-number swimmer constructed by three linked spheres wasintroduced and analyzed by Najaﬁ
et al.
5
and experimen-tally realized by Leoni
et al.
6
. After Purcell’s proposalthere have been considerable scientiﬁc efforts in designingartiﬁcial swimmers. Such swimmers would be useful in de-veloping microﬂuidic experiments. Furthermore, progressesin assembling microswimmers show the possibility of usingmicromachines inside the biological cells for noninvasivetherapeutic treatments
7
. On the other hand, there are manytheoretical works devoted to the study of different aspects inthe motion of biological microorganisms at low-Reynolds-number condition
8–12
. Such interests include spermswimming, metachronal waves in cilia,
E. Coli
chemotaxis,and coupling mediated by hydrodynamic interaction betweennearby microorganisms
13,14
. For a review of recentprogress on low-Reynolds-number hydrodynamics of micro-organisms, see, for example, the review paper by Lauga andPowers
15
.Our ﬁrst aim in this Brief Report is to present a simpliﬁedmodel that captures the characteristics of a swimming bio-logical organism like a bacterium. Dipolar far velocity ﬁeldand asymmetric shape, corresponding to the head-tail geom-etry of the organisms, are two important features of mi-croswimmers. We model these systems by considering twospheres with different radii that are changing their separa-tion. We will study the translational and angular motions of this system.
II. TWO-SPHERE MODEL
Figure1shows the schematic geometry of a model swim-mer composed of two spheres. As shown in this ﬁgure twosmall and large spheres with radii
a
and
R
are connected bytwo perpendicular and negligible diameter rods. Let denotethe lengths of long and short rods by
L
and
l
, respectively.The connection is established in a way that the angle be-tween two rods is ﬁxed to
2
while the relative angular posi-tion of small rod with respect to the large sphere can bevaried. Additionally, we assume that the length of the longrod can be dynamically changed. In this case, the system willhave two internal degrees of freedom: the length of the longrod
L
t
and the rotational angle of the short rod
t
.The geometry which we are introducing here resemblesthe body shape of a bacterium with a single ﬂagellum orcilium. Bacteria use beating patterns in their tails to move.The small sphere in our two-sphere model acts as a beatingtail and the large sphere resembles the head of animal. Theminimum condition for swimming at low Reynolds numbercan be achieved in our three-dimensional model. By chang-ing the length of long rod and the angle of small one in aprescribed form, we are able to choose the motion whichbreaks the time-reversal symmetry, the necessary conditionfor translational motion, and consequently propel the system.As an example for the internal motion of the system, welet the angle
t
increase with constant angular velocity andthe length of long rod change periodically around an averagelength. The explicit form of this motion is given by
L
t
*
najaﬁ@znu.ac.irPHYSICAL REVIEW E
81
, 067301
2010
1539-3755/2010/81
6
/067301
4
©2010 The American Physical Society067301-1
=
L
0
+
h
0
cos
L
t
−
0
and
t
=
t
. In this case, the posi-tion vector of the small sphere, seen in the reference framethat is comoving with the large sphere, is given by
X
0
=
„
l
cos
t
,
l
sin
t
,
L
0
+
h
t
…
,
1
where
l
is the length of the small rod and
L
0
represents theaverage length of the long rod. Figure2shows a typicalreal-space trajectory of the small sphere that is seen in thereference frame comoving with the large sphere. Differentchoices for
L
and
correspond to different forms of theinternal motion. For
/
L
=
m
=
p
/
q
with
p
and
q
as twopositive integer numbers, we see that the paths in the
h
,
space are closed loops. One should note that the phase spaceof internal motion
h
,
space
is the surface of a cylinder.The axial direction on the cylinder represents the
h
directionand the azimuthal angle is shown by the transverse directionon the cylinder. For
m
1, the phase-space trajectory is aclosed curve which traces exactly one complete turn aroundthe cylinder, while for
m
1 the trajectory is a closed loopthat turns many times around the cylinder. In both cases, thegeometry of the closed curves in the cylindrical-shape phasespace is an example of nonreciprocal motion that can gener-ate a net translational motion.
III. POINT FORCE NEAR A RIGID SPHERE
At zero Reynolds number the Stokes equations govern thedynamics of ﬂuid. The solution of the Stokes equation for apoint force singularity is formulated in terms of the Green’sfunction and is called Stokeslet. For a point force singularitywith strength
f
located at point
X
0
, the velocity ﬁeld gener-ated in the ﬂuid is given by
u
X
=
1
8
G
X
,
X
0
·
f
, where
is the viscosity of the ﬂuid. Oseen derived the explicit formof the Green’s function
G
for an inﬁnite ﬂow that is boundedinternally by a solid sphere with radius
R
16
. For the ex-plicit form of this solution, we refer to the srcinal paper byOseen. As it is manifested by Oseen’s solution, the ﬂow ﬁelddue to a point force in the presence of a solid sphere can beregarded as the ﬂow of the srcinal point force and the ﬂowdue to the singular parts that are located at an image positioninside the sphere. The location of the image point inside thesphere is given by its position vector,
X
0
=
R
2
/
X
02
X
0
, rela-tive to the sphere’s center.As argued by Higdon the total force acting by a pointforce on ﬂuid bounded by a no-slip sphere is equivalent tothe total Stokeslet strength
17
. The total Stokeslet strengthincludes the image point force inside the sphere. For a pointforce
f
, the image point force is deﬁned by
f
I
=
c
r
f
r
+
c
t
f
t
where the radial and tangential components of this imageforce are given by
f
r
=
f
·
X
0
/
X
0
,
f
t
=
f
−
f
r
. Here, two coefﬁ-cients
c
r
and
c
t
are given by
c
r
= −32
R X
0
+12
R
3
X
03
,
c
t
= −34
R X
0
−14
R
3
X
03
.
2
In the next section we will use these results for analyzing themotion of two-sphere system.
IV. TWO-SPHERE DYNAMICS
In this section we will develop the dynamical equationsfor two-sphere system. To simplify the equations We willassume that the radius of small sphere is much smaller thanthe radius of large sphere
a
R
. We further assume that
a
is smaller than the characteristic distance between thespheres. With this approximation the velocity ﬁeld of thesystem is described by the velocity ﬁeld of a point force thatis moving near a rigid sphere. This simpliﬁcation allows usto use the results of the preceding section and derive simplerdynamical equations of the system. However, one shouldnote that the ﬁnite-size effect of the small sphere can besystematically considered by Faxen’s theorem
18
.To obtain the swimming velocity of the system we work in the reference frame that is comoving and rotating with thelarge sphere. In this coordinate system the velocity of theﬂuid at inﬁnity is the swimming velocity. Denoting theswimming velocity of the system by
V
and its angular veloc-ity by
, we can express the velocity ﬁeld of the ﬂuid at ageneral point
X
as
u
X
= −
V
−
X
+
M
·
V
+
m
+
G
·
f
,
3
where the tensor
M
and vector
m
give the ﬂow ﬁeld due tothe translational and rotational motions of a moving sphere.These quantities are given by
M
=34
R X
I
+
XX
X
2
+14
R
3
X
3
I
− 3
XX
X
2
,
m
=
R
3
X
3
X
.
4
In the absence of external force and torque, the swimmeris force and torque free. Therefore, we require the total forceand torque acting on the ﬂuid by the system to be zero.Including the point force and its image counterpart and add-
h(t) L Rl
t
φ( )
0
a
x z y
FIG. 1. Schematic showing the geometry of a two-sphere swim-mer. Two large and small spheres are connected through two per-pendicular rods, one with ﬁxed but the other with variable length.The short rod is rotating around the long rod. This model systemresembles the motion of a bacterium that has a single tail.
x z y
0.80.40−0.4−0.8−0.8 −0.40.4 0.8−0.80.800
FIG. 2. Trajectory of the small sphere seen in the frame of reference that is comoving with large sphere. Here, we chose
h
0
=1 and
/
L
=2.BRIEF REPORTS PHYSICAL REVIEW E
81
, 067301
2010
067301-2
ing the contributions due to the translational and rotationalmotions of large sphere, we arrive at the following force andtorque balance equations:
f
+
f
I
+ 6
R
V
= 0,
X
0
f
t
−
R
3
X
03
X
0
f
t
+ 8
R
3
= 0.
5
The ﬂuid velocity ﬁeld at the location of small sphere issubject to the boundary condition
u
X
0
=
X
˙
0
. Together withthis boundary condition, the above equations make a com-plete set of dynamical governing equations for the swimmer.To solve the dynamical equations for the system, we canuse the force and torque balance conditions and obtain a setof equations which relate the different components of thetranslational or angular velocities of the system to the com-ponent of the vector
X
˙
0
in the following matrix form:
V
=
AC
−1
X
˙
0
,
=
BC
−1
X
˙
0
,
6
where the details for of the matrix elements
A
ij
=
a
ij
,
B
ij
=
b
ij
, and
C
ij
=
c
ij
are given in the Appendix.
V. RESULTS AND DISCUSSION
In this section we will present the numerical solution forthe governing equations and obtain the real-space trajectoryof the swimmer. For this purpose we plot the trajectory of large sphere. For the prescribed internal motion given by Eq.
1
and a special choice of parameters, we have plotted inFig.3the space trajectory of the large sphere. As one candistinguish, the trajectory is a helical-shaped path with anoverall translational movement in each turn. The differentcharacteristics of the trajectory, preferred direction, averageswimming velocity, and the effective radius of the helix canbe controlled by the geometrical as well as dynamical param-eters of the swimmer.The average orientation of the long rod, which is notshown in the ﬁgure, is in the direction of the longitudinalaxis of the helix. This is achieved by numerically solving forthe rotational velocity. Controlling and adjusting the dynami-cal behavior of the swimmer are of prime importance in de-signing artiﬁcial micromachines. Here, we see that by chang-ing the parameters of the system we can do this favor. In Fig.3, we have shown that the overall swimming direction issensitive to the initial phase
0
. Additionally and as anotherexample, in Fig.4,we have shown that by changing
L
0
, thelength of long rod, the average swimming velocity can bechanged.As the geometry of the two-sphere swimmer is not sym-metric, the far-ﬁeld distribution of ﬂuid velocity at the lead-ing order of approximation resembles a velocity ﬁeld of asingle force dipole. This is the main characteristic of mostswimming microorganisms with a dipolelike velocitypattern.In summary, inspired by bacterium swimming, we pro-posed and analyzed a swimmer, constructed by two jointspheres. We have shown that this simple three-dimensionalswimmer is a model for a low-Reynolds-number propellerthat captures a number of dynamical features in microorgan-isms. It will be interesting to use this model swimmer andstudy many interesting problems such as the hydrodynamicinteraction between such swimmers, the effects due to theconﬁnement in the bounded ﬂuids, and also chemotaxis phe-nomena. Inspired by the colonies of microorganisms, we areextending our model to investigate the hydrodynamic inter-action in an ensemble of two-sphere swimmers.
APPENDIX: MATHEMATICAL DETAILS
Here, we present the explicit form of the matrix elements
a
ij
,
b
ij
, and
c
ij
which were introduced in the text,
a
ii
= −16
R
1 +
c
t
+
c
r
−
c
t
X
0
i
2
X
02
,
a
ij
=
a
ji
= −16
R
c
r
−
c
t
X
0
i
X
0
j
X
02
for
i
j
,
φ =π
0
φ =0.8π
0
z x y
0.0050.0040.0030.0020.0010−0.04−0.0200.02−0.02−0.01 00.020.01 0.030.04 0.05
FIG. 3. The trajectory of the swimmer in the
x
,
y
,
z
space isplotted for two different values of
0
. Other parameters are
R
=1,
a
=0.5,
L
0
=4,
h
0
=0.1,
l
=0.3, and
=2
l
=1. Line shows the realpath of the large sphere. The swimmer starts its motion from theinitial state where the large sphere is located at the srcin and thelong rod is orientated along the −
zˆ
direction. As one can see, theoverall swimming direction can be varied by changing the param-eters of the system. Average orientation of the long rod which is notshown here is along the average swimming direction.
V
0
L
0.0040.0020.0030.0050.0062.6 2.8 3 3.2 3.4 3.6 3.8 4
FIG. 4. Average swimming velocity is plotted in terms of thelength of long rod. Other parameters are set to
R
=1,
a
=0.5,
0
=0,
h
0
=0.1,
l
=0.3, and
=
l
=1.BRIEF REPORTS PHYSICAL REVIEW E
81
, 067301
2010
067301-3
b
ii
= 0,
b
ij
= −
b
ji
=18
R
3
1 −
R
3
X
03
X
0
k
for
i
j
k
,
c
11
=
M
xx
− 1
a
11
+
M
xy
a
21
+
M
xz
a
31
+
m
z
−
z
0
b
21
−
m
y
−
y
0
b
31
+
G
xx
,
c
12
=
M
xx
− 1
a
12
+
M
xy
a
22
+
M
xz
a
32
−
m
y
−
y
0
b
32
+
G
xy
,
c
13
=
M
xx
− 1
a
13
+
M
xy
a
23
+
M
xz
a
33
+
m
z
−
z
0
b
23
+
G
xz
,
c
21
=
M
yx
a
11
+
M
yy
− 1
a
21
+
M
yz
a
31
+
m
x
−
x
0
b
31
+
G
yx
,
c
22
=
M
yx
a
12
+
M
yy
− 1
a
22
+
M
yz
a
32
+
m
x
−
x
0
b
32
−
m
z
−
z
0
b
12
+
G
yy
,
c
23
=
M
yx
a
13
+
M
yy
− 1
a
23
+
M
yz
a
33
−
m
z
−
z
0
b
13
+
G
yz
,
c
31
=
M
zx
a
11
+
M
zy
a
21
+
M
zz
− 1
a
31
−
m
x
−
x
0
b
21
+
G
zx
,
c
32
=
M
zx
a
12
+
M
zy
a
22
+
M
zz
− 1
a
32
+
m
y
−
y
0
b
12
+
G
zy
,
c
33
=
M
zx
a
13
+
M
zy
a
23
+
M
zz
− 1
a
33
+
m
y
−
y
0
b
13
−
m
x
−
x
0
b
23
+
G
zz
.
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81
, 067301
2010
067301-4

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