Nonlinear dynamics of a microswimmer in Poiseuille flow

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by  Andreas Zöttl

Nonlinear dynamics of a microswimmer in Poiseuille flow
Andreas Zo¨ttl and Holger Stark
Institut fur Theoretische Physik, Technische Universita¨t Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
(Dated: January 4, 2012)
We study the three-dimensional dynamics of a spherical microswimmer in cylindrical Poiseuille
flowwhichcanbemappedontoaHamiltoniansystem. Swingingandtumblingtrajectoriesareiden-
tified. In 2D they are equivalent to oscillating and circling solutions of a mathematical pendulum.
Hydrodynamic interactions between the swimmer and confining channel walls lead to dissipative
dynamics and result in stable trajectories, different for pullers and pushers. We demonstrate this
behavior in the dipole approximation of the swimmer and with simulations using the method of
2 multi-particle collision dynamics.
1
0 PACSnumbers: 47.63.Gd,47.63.mf,47.61.-k
2

n
Microswimmersoftenhavetorespondtofluidflowand
a
J confining boundaries, like sperm cells in the Fallopian

3 tubes [1] or pathogens in blood vessels [2]. Artificial mi-
croswimmers constructed with the vision to act as drug-

] deliverers in the human body [3] would have to swim in
t
f narrow channels like arteries. Two properties influence
o
s the swimming in microchannels. On the one hand, vor-
t. tices in flow reorientthe swimming direction of microor- Figure 1. Swimmer in Poiseuille flow. (a) Flow profile
a ganisms. In simple shear flow, for example, microswim- v (ρ), cylindrical coordinate system with coordinate basis
m f
merstumbleduetoaconstantflowvorticity[4]. Vortices (ρˆ,ϕˆ,zˆ) and orientation angle Ψ for the projected orienta-
- in Poiseuille flow in combination with bottom-heaviness tion into the ρ-z-plane. When eϕ=sinΘ=0, the motion is
d two-dimensional. Note that the sign of vorticity Ω changes
n due to gravitation lead to stable orientations of swim- f
when crossing the centerline. (b) Cross section of the mi-
o ming algae cells [5]. On the other hand, microorganisms
crochannel. The orientation in ϕ-direction defines the angle
c swimming near surfaces are trapped by hydrodynamic
Θ.
[
interactions [6] and ultimately escape with the help of

1 rotational diffusion [10]. Finally, bacteria in Poiseuille
v flow show a net-upstream flux at the walls due to the tion for the swimmer position r and orientation eˆ are
9 interplay of confinement and flow vorticity [7, 8]. All given by
2
these examples show there is genuine interest in under-
6 d d 1
0 standinggenericfeaturesofmicroorganismsandartificial r=v0eˆ+vf, eˆ= Ωf eˆ (1)
dt dt 2 ×
. swimmers in Poiseuille flow.
1
0 In this letter we demonstrate that the dynamics of a where Ωf = vf = vfρ/RC2hϕˆ is the flow vorticity.
2 simplesphericalmicroswimmerinacylindricalPoiseuille The swimmer∇o×rientation eˆ = eρρˆ+eϕϕˆ+ezˆz has the
1 flow can be mapped onto a conservative dynamical sys- components
:
v tem with the Hamiltonian as a constant of motion. In
e = cosΘsinΨ,e =sinΘ,e = cosΘcosΨ, (2)
Xi analogytotheoscillatingandcirclingsolutionsofamath- ρ − ϕ z −
ematicalpendulum,wediscussindetailtheswingingand
r whereΨ π,π istheangleintheρ-z-plane[Fig.1(a)]
a tumbling motion of the microswimmer in 2D and gener- and Θ ∈{π−/2,π}/2 measures the orientationin the az-
alize them to three dimensions. Hydrodynamic interac- ∈{− }
imuthal ϕ-direction [Fig. 1(b)]. We note that Ψ <π/2
tions with the channel walltreatedin the dipole approx- | |
means upstream and Ψ >π/2 downstream orientation,
imation introduce dissipation andthe microswimmeras- | |
respectively. In the following we use rescaled units,
sumesspecificstableswimmingtrajectoriesdependingon
ρ/RCh ρ 0,1 , z/RCh z and t/t0 t with
its type as puller or pusher. → ∈ { } → →
t0=RCh/v0. We also introduce the dimensionless flow
We first introduce the geometry. We consider a point- speed v¯f =vf/v0, which is the only essential parameter
like microswimmer that moves with a constant intrinsic in our problem.
swimming speed v0 in a cylindrical microchannel where First, we discuss 2D solutions of Eqs. (1) since they
a Poiseuille flow is imposed. Using a cylindrical coordi- already capture many aspects of the swimmer dynam-
nate system (ρ,ϕ,z) with the coordinate basis (ρˆ,ϕˆ,ˆz), ics. When Θ =0, the trajectories of the swimmer are
the flow is given by v =v (1 ρ2/R2 )ˆz, where v is restricted to two dimensions, for example, to the x-z-
f f Ch f
−
the maximum flow speed in the center of the channel plane, x 1,1 . Due to the translationalsymmetry in
∈{− }
[Fig. 1(a)]. In the absence of noise the equations of mo- z-direction, only the equations for x and Ψ are coupled,
2
and Eqs. (1) give x˙ = sinΨ, Ψ˙ =v¯ x. Eliminating x
f
−
results in
Ψ¨ =v¯ sinΨ, (3)
f
whichistheequationofmotionofthemathematicalpen-
dulum. Since in this analogyx plays the role of velocity,
we can immediately write down the 2D Hamiltonian
1
2
H2D = v¯fx +1 cosΨ (4)
2 −
asaconservedquantity. Fig.2showsthex-Ψphasespace
and typical trajectories z(x) for several flow strengths
v¯ . In analogy to the pendulum two swimming states
f
exist. The flow vorticity rotates the upstream oriented
microswimmer always towards the center. Hence, the
swimmer performs a swinging motion around the cen-
terline of the channel for H2D<2 which corresponds to
the oscillating solution of the pendulum [e. g. blue tra- Figure 2. Phase spaces x-Ψ (left) and typical trajectories
z(x)(right)forseveralflowstrengthsv¯ . Alltrajectoriesstart
jectory of Fig. 2(a)]. For small amplitudes (Ψ 1) the f
≪ at z=0. (a) upstream motion, (b) intermediate motion and
swingingfrequencyis ω0=√v¯f. Whenthe upstreamori- (c)downstreammotion. Notethevariousscalesforthez-axis.
ented swimmer moves exactly in the center of the chan- Thearrows indicatetheorientation vectoreˆof theswimmer.
nel (stable fixed point), the Hamiltonian is zero. Down-
stream swimming along the centerline (Ψ=π) is an un-
stable fixed point. After a slight disturbance of x=0, exist but a net upstream motion only occurs for v¯f .
vorticity rotates the swimmer away from the centerline. 1+H2D/2 [blue line in Fig. 2(b)].
The swimmer performs a tumbling motion (H2D > 2) For a non-zero azimuthal component, eϕ = sinΘ =
6
which corresponds to the circling solution of the pen- 0, the swimmer trajectory is three-dimensional. Using
dulum [green trajectory of Fig. 2(c)]. At H2D =2, the Eqs.(1) andEqs.(2), we obtainthree coupledequations
separatrix x2=2(cosΨ+1)/v¯ divides the swinging and for Ψ, Θ and ρ,
f
tumbling region in phase space [red curves in the phase
ρ˙ = cosΘsinΨ
portraits of Fig. 2]. Since the Poiseuille flow is bounded
−
by the channel walls, tumbling motion only occurs for Ψ˙ =v¯ ρ sinΘtanΘcosΨ/ρ (6)
f
−
v¯ >4[Fig.2(c)]. Meaningsufficientlystrongvorticityto
f Θ˙ =sinΘsinΨ/ρ.
prevent the swimmer from crossing the centerline.
If we only consider steric interactions of the swimmer Due to translational symmetry in z-direction and rota-
withthe channelwall,the swimmercrashesintothe wall tional symmetry about the channel axis, Eqs. (6) do not
at x =1forH2D>v¯f/2,reorientsduetotheflowvortic- dependonz andϕ. Weareabletoidentifytwoconstants
| |
itytowardstheupstreamorientation,andleavesthewall of motion,
at Ψ=0 with Hmax=v¯ /2. The swimmer then performs
2D f
aswingingmotionbetweenthewallswithmaximumam- L =ρsinΘ
z
pSloitfuodrev¯f|x<|=41thfeorswv¯fim<m4er[garleweanytsreanjetcetrosraysiwniFngigin.g2(ma)o]-. H = 21v¯fρ2+1−cosΨcosΘ, (7)
tion oriented upstream, at the latest after contact with
the wall, whereas it tumbles close to the wall for v¯f>4. where Lz is proportional to the angular momentum of
To determine the full 2D trajectory in the microchan- the swimmer in z-direction. Due to this constant the
nel, we solve the dynamic equation for z(t), sign of Θ along a swimmer trajectory does not change.
Eliminating Θ from Eqs. (6), reduces the equations of
z˙ =v¯f[1−x(t)2]−cosΨ(t). (5) motions to ∂∂HΨ=−ρ˙, ∂∂Hρ =Ψ˙. So, H againplays the role
of a Hamiltonian for the conjugate variables Ψ and ρ.
Acarefulanalysisrevealsthefollowing. Theswimmeral- The intersection of the two constants of motion
waysmovesupstream(z˙<0),whenv¯f<1 H2D,asshown gives the orbit of the swimmer in ρ-Ψ-Θ phase space
−
inFig.2(a). Whenthe flowisstrong(v¯f>1+2H2D), the [Fig.3(a,b)]. ThestablefixedpointsofEqs.(6)lieonthe
swimmer always drifts downstream (z˙>0), while swing- fixed-point line (ρ∗= sinΘ∗tanΘ∗/v¯ , Ψ∗=0), drawn
f
ing or tumbling [Fig. 2(c)]. In between, mixed up- and in Fig. 3(a). Swimmipng at a fixed point corresponds to
downstream segments within one trajectory [Fig. 2(b)] a helical trajectory [Fig. 3(c)]. The swimmer moves up-
3
drodynamic interactions with the wall all the time. To
capturethebasicidea,weconcentrateon2D-trajectories
and consider instead of the cylindrical channel wall, two
parallel plates located at x=1 and x= 1. We calculate
−
the wall-induced translational and angular velocities us-
ing the force-dipole approximationof Ref. [6] and obtain
the equations of motion,
2
3p¯(3sin Ψ 1) 1 1
x˙ = sinΨ −
− − 64π (cid:18)(1 x)2 − (1+x)2(cid:19)
−
3p¯sinΨcosΨ 1 1
Ψ˙ =v¯ x +
f − 64π (cid:18)(1 x)3 (1+x)3(cid:19)
−
(8)
Figure 3. ρ-Ψ-Θphase space. The intersection between L =
z
const. (orange) and H=const. (green) gives the phase space
trajectory. (a) helical-like swinging motion (blueintersection where p¯=p/(ηv0) is the reduced dipole strength. Fig. 4
curve) for L = 0.2, H = 1. Black curve: fixed-point line shows typical phase space plots generated from Eqs. (8)
z
corresponds to helical trajectories. (b) helical-like tumbling for a puller (a) and pusher (b). The swinging motion
motion (blue intersection curve) for Lz =0.2, H =3. (c-e) of an upstream oriented puller becomes damped and an
sketch of trajectories in the channel for helical motion (c),
attractive fixed point in the center exists. Repelled by
helical-like motion (d) and tumbling motion (e).
both walls, the puller swims upstream along the center-
line. On the other hand, a puller tumbling near the wall
is attracted,onaverage,bythe wallandstaysnear to it.
∗
stream for v¯ < 1/cosΘ , either on a left-handed he-
f This is indicated by the green stable trajectory. All tra-
lix (Θ > 0) or a right-handed helix (Θ < 0). Closed
jectoriesoutsidetheunstableredlimitcycleorseparatrix
orbits around the fixed-point line correspond to swing-
convergeto it. However,due to thermal fluctuations the
ing motion around a helical path [Fig. 3(d)] and open
pullermaycrosstheseparatrix. Apusherbehavesdiffer-
orbits [Fig. 3(b)] are complicated tumbling trajectories
ently, it is attractedby the wallwhen orientedupstream
[Fig. 3(e)].
in the center of the channel, but it is pushed away from
Nowweconsiderhydrodynamicinteractionsofthemi- thewallwhentumblingnearthewall. So,alltrajectories
croswimmer with the bounding channel wall. The flow convergetowardsaswingingmotionaboutthecenterline,
field for neutrally buoyant swimmers in a bulk fluid is characterizedby a stable limit cycle in the x-Ψ-plane.
in leading order a force dipole, v(r)=8πpηr2[3(ˆr·eˆ)2−1]ˆr Totestourfindingswesimulatethemotionofaspher-
where ˆr = r/r and η is the viscosity of the fluid. For icalmicroswimmerinPoiseuilleflowusingthemethodof
positive dipole strength, p>0, the propelling apparatus multi-particle collision dynamics (MPCD) [12]. It solves
ofthe swimmeris typicallyatthe back(pusher), andfor theNavier-Stokesequationsonacoarse-grainedleveland
p <0 in the front (puller). Ref. [6] treated the swim- calculatestheflowfieldaroundtheswimmerinthecylin-
mer close to a plane wall. The authors showed that hy- drical microchannel taking into account both hydrody-
drodynamic interactions between the swimmer and the namic interactions and thermal noise. In every simula-
wall lead to re-orientation due to the wall-induced vor- tionsteprandomlydistributedpoint-particlesofmassm
ticity ΩW and to attraction/repulsion. For example, a at temperature kBT first move ballistically for a time
pusherswimmingparalleltothewallisattractedandre- ∆t and then they are sorted into cubic cells of length a.
mainstrappedatthewall,whileapullerisrepelledfrom They interact with all other particles in the cell with a
the wall. We note that close to a wall the force-dipole specific collision rule such that momentum is conserved
approximation for ΩW is no longer valid. Instead, ge- locally. The density of the fluid is mNc/a3 where Nc is
ometric details of the swimmer and thereby near fields the average number of particles per cell. Depending on
become important for ΩW. Combining flow and wall ef- the parametersand the specific collisionrule, the viscos-
fects,the totalangularvelocityofaswimmernearawall ity η of the fluid can be calculated [12, 18].
isΩ(x,Ψ)=ΩW(x,Ψ)+Ωf(x). Recentworkshowedthat Asamodelmicroswimmerweuseasphericalsquirmer
hydrodynamicre-orientationisalmostnegligibleforbac- of radius R [13, 14]. It propels itself by a static,
S
teria swimming near walls [9, 10] but rotates the swim- axisymmetric and tangential velocity field on its sur-
mer at the wall. For a sufficiently weak Poiseuille flow face, vs(ˆrs,eˆ)=(B1+(eˆ ˆrs)B2)[(eˆ ˆrs)ˆrs eˆ] where ˆrs is
· · −
field, stable orientations ΨW for swimming at the wall the radial unit vector pointing from the center of the
exist when ΩW(RCh,ΨW)+Ωf(RCh)=0 [11]. Thermal squirmer to the surface. The first mode B1 determines
fluctuations will, however, reorient the swimmer so that theswimmingspeedv0=2B1/3andthesecondmodethe
it leaves the wall [10]. strength of the force dipole p/η=−4πB2RS2, so B2>0
In narrow channels a microswimmer experiences hy- models a puller and B2<0 a pusher. The squirmer has
4
stable trajectories within seconds in sufficiently narrow
channels. Similar estimates apply to the E. coli bac-
terium where p 0.8pNµm was measured recently [10].
≈
Althoughhydrodynamicinteractionsbetweenasmallmi-
croswimmerandasinglewallmayplaynosignificantrole,
theybecome importantinchannelswhenthe channeldi-
ameter is only few times the size of the swimmer.
Inconclusion,throughaformalmappingontoaHamil-
tonian dynamical system we have shown that spheri-
cal microswimmers perform either an upstream oriented
swingingoratumblingmotionwhenmovinginPoiseuille
flow. Hydrodynamic interactions of the swimmer with
the wall stabilizes the upstream orientation of pullers in
the centerof the channelwhereasa pusher performs sta-
ble oscillations around the centerline with a specific am-
plitude.
Sphericalartificialswimmerswithdifferentlocomotion
mechanisms have been constructed and studied recently
[21]. Investigating them in microfluidic channels under
Poiseuille flow, the generic features presented in this ar-
ticle should be accessible in experiments.
Figure 4. Phase space trajectories for a microswimmer in a
We thank R. Goldstein, I. Pagonabarraga, T. Pfohl,
narrow channel for v¯ =10 for a puller (left) and a pusher
f
(right). Green and red indicate, respectively, stable and un- S. Uppaluri and R. Vogel for helpful discussions and the
stable trajectories. (a) and (b) are obtained from Eqs. (8) Deutsche Forschungsgemeinschaft for financial support
and (c) and (d) from MPCD simulations where we used the through the researchtraining group GRK1558.
parameters B1=0.045, β=B2/B1=±5, RS=6, RCh=18,
N =30, ∆t=0.02 setting a=m=k T =1 and the colli-
c B
sion rule MPC-AT-a [18]. Each trajectory was obtained by
averaging over10 individual runs.
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5
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