A frictionless microswimmer

0.2 MB·English

by  Alexander M. Leshansky

A frictionless microswimmer
AlexanderM.Leshansky∗,OdedKenneth†,OmriGat‡, andJosephE.Avron†§
∗DepartmentofChemicalEngineering,and†DepartmentofPhysics,Technion-IIT,Haifa32000,Israel
SubmittedtoProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica
7
0
We investigate the self-locomotion of an elongated microswim- swimmeroflengthℓ,thicknessdwithroundedcapsundergoingtread-
0 merbyvirtueoftheunidirectionaltangentialsurfacetreadmilling. millingatvelocityU. Itisreasonabletoassume(andtheanalysisin
2 Weshowthatthepropulsioncouldbealmostfrictionless,asthe
thefollowingsectioncanbeusedtojustify)thatallthedissipationis
microswimmer is propelled forwardwith the speed ofthe back-
n associatedwiththerounded ends. Hence, bydimensional analysis,
wardsurfacemotion, i.e. itmovesthroughoutanalmostquies-
a centfluid.Weinvestigatethisswimmingtechniqueusingthespe- thepowerdissipatedintreadmillingisoftheorderofµdU2whereµ
J
cialspheroidalcoordinatesandalsofindanexplicitclosed-form istheviscositycoefficient. Letuscomparethiswiththepowerneeded

7 optimal solution for a two-dimensional treadmiler via complex- todragthe“frozen"treadmiler. ByCoxslenderbodytheory[12]the
variabletechniques. powerneededtodragthetubewithvelocityUis µℓU2/log(2ℓ/d).
] ∼
n Hencetheratioofpowerinvestedindraggingandswimmingscales
y self-locomotion | creepingflow | motility | propulsionefficiency like(εlog1/ε)−1 andcanbemadearbitrarilylarge. Hereε = d/ℓ
d istheaspectratiooftheswimmer.
Tinyswimmers,betheymicro-organismsormicrobots, liveina
- One can now ask if there are slender treadmilers that are arbi-
u worlddominatedbyfriction[1]. Inthisworld, technically,the
trarily better than the slender rod-like treadmiler above? Consider
l world of low Reynolds numbers, motion is associated with energy
f nowanelongatedellipsoidalmicroswimmerwhosesurfaceisgiven
s. dissipation. Intheabsenceofexternalenergysupplyobjectsrapidly byz2/b2+r2/a2 =1wherer2 x2+y2. Letusassumeagain,that
c cometorest[2]. Itisbothconceptuallyinteresting,andtechnologi- theviscousdissipationisaresult≡ofatippropulsion,andestimatethe
i callyimportant,totryandunderstandwhatclassesofstrategieslead
s positionofthetipfromthecondition dr/dz =1. Itcanbereadily
y toeffectiveswimminginasettingdominatedbydissipation. Apar- demonstratedthatinthecaseofε = a|/b |1thetipislocatedata
h ticularly promising class of strategies is where the motion is, in a distanceofb(1 1ε2)fromthecenterand≪itstypicalwidthscalesas
p sense,onlyapparent;whereashapemoveswithlittleornomotionof −2
aε. Therefore,applyingthesameargumentsasbefore,thedissipation
[ materialparticles.
rateshouldthenscaleas µaεU2andtheratioofpowerexpanded
1 Thewheelisthemechanicalapplicationofthisstrategyanditis indraggingandswimmiPng∼becomes bµU2/log(2b/a) 1 .
v instructivetoexamineitfromthispointofview. The(unholonomic) Weshallsee,byamoreaccurateanalysµias,εUth2atforp∼rolεa2telosgp1h/eεroid
constraintofrollingwithoutslippingcomesaboutbecauseofthelarge
0 the ratio of power in treadmilling to dragging is of the order of
frictionbetweenthewheelandthesurfacesupportingit. Neverthe-
8 (εlogε)−2. Inthefollowingsectionsweshallanalyzetwomodelsof
0 less,awheelcanrollwithlittleornodissipationofenergy. Oneway
treadmillersin3and2dimensions,respectively.
1 toviewthisistonotethatthemotionofthepointofcontactisonly
0 apparent. The point of contact moves, even though the part of the The theoreticalframework.Wemodel themicro-swimmer as a
7 wheelincontactwiththesurface,ismomentarilyatrest. prolatespheroidswimminginanunboundedfluidbycontinuoustan-
0 AnexampleclosertotheworldoflowReynoldsnumbers,isthe gentialsurfacemotion. TheCartesian-coordinatesystem(x1,x2,x3)
/ actin-basedpropulsionoftheleadingedgeofmotilecells[3],intra- is fixed with the center O of the spheroid. A modified orthogo-
s
c cellularbacterialpathogens[4]andbiomimeticcargoes[5,6,7]. The nal prolatespheroidal coordinate system (τ,ζ,ϕ) withunit vectors
si actinfilamentsassemblethemselvesfromtheambientsolutionatthe (eτ,eζ,eϕ) is defined via the relations x1 = c{τ2 −1}1/2{1−
y frontendanddisassemblethemselvesattherearend. Hereagain,it ζ2 1/2cosϕ,x2 =c τ2 1 1/2 1 ζ2 1/2sinϕandx3 =cτζ,
} { − } { − }
h isonlyshapethatismovingandinprincipleatleasttheenergyin- where 1 ζ 1 τ < , 0 ϕ 2π and cis thesemi-
− ≤ ≤ ≤ ∞ ≤ ≤
p vestedatthefrontendcanberecoveredattherearend. (Thereare focal distance[9]. Thecoordinatesurfacesτ = τa = const area
: thermodynamicsandentropicissuesthatweshallnotconsiderhere.) familyofconfocalspheroids,x2/b2+(x2+x2)/a2 = 1,centered
v 3 1 2
i Apparentmotionsseemtobegoodatfightingdissipation[8]. attheoriginwithmajorandminorsemi-axisgivenbyb = cτa and
X Hereweshallfocusonacloselyrelatedmodeoflocomotion: sur- a=c τ2 1 1/2,respectively. Weassumethatasteadyaxisymmet-
{ a− }
r facetreadmilling. Insurfacetreadmillingtheswimmermoveswithout ricflowhasbeenestablishedaroundthemicro-swimmerasaresultof
a achangeofshape,byatangentialsurfacemotion. Surfaceisgener- thetangentialsurfacetreadmillingwithauniformfar-fieldvelocityU
atedatthefrontendandisconsumedattherearend1. Incontrastto (equaltothelaboratoryframepropulsionspeed)inthenegativex3-
actinandmicrotubules,thesurfacetreadmillingdoesnotrelyonthe direction. Thelow-ReincompressibleflowisgovernedbytheStokes
exchange ofmaterialwiththeambient fluid. (Theswimmerneeds, andcontinuityequations,
ofcourse,aninnermechanismtotransfermaterialfromitsreartoits
front). Itisintuitivelyclearthataneedleshapedswimmerundergoing ∆v=µgradp, divv =0, [1]
treadmillingcanmovewithverylittledissipationbecausetheambi-
respectively,accompaniedbytheboundaryconditionattheswimmer
ent fluid isalmost quiescent and thereisalmost no relativemotion
surfaceτ =τ
betweenthesurfaceoftheswimmerandthefluid. Onecannotmake a
treadmillingcompletelynon-dissipativebecausethereisalwayssome v =u(ζ)eζ. [2]
remanentdissipationassociatedwiththemotionofthefrontandrear
ends. Themainquestionthatweshalladdresshereishowcanone
§Towhomcorrespondenceshouldbeaddressed.E-mail:[email protected]
quantitativelyestimatethisremanentdissipation.
(cid:13)c2006byTheNationalAcademyofSciencesoftheUSA
Let us first consider simple qualitative estimates of the power 1Alternatively,wemaythinkofaaslendermicrobotthatistopologicallyequivalenttoatoroidal
dissipatedbyviscosityintreadmilling. Consider,arod-likeslender swimmerproposedbyPurcell[1],i.e.thesurfaceisnotcreatedordestroyed,butratherunder-
goesacontinuoustank-treadingmotion
www.pnas.org—— PNAS IssueDate Volume IssueNumber 1–7
Sincetheflowisaxisymmetricweintroducethescalarstream-function
Ψ(uniqueuptoanadditiveconstant)thatsatisfiesthecontinuityequa-
1
tion
0.95
1
v =v e +v e =curl Ψe . [3] 0.9
τ τ ζ ζ ϕ
Hϕ ! 0.85
us
Thevelocitycomponentsarereadilyobtainedfrom[3]as (cid:144) 0.8
b U
0.75
1 ∂Ψ 1 ∂Ψ
v = ,v = , 0.7
τ H H ∂ζ ζ −H H ∂τ
ζ ϕ ζ ϕ 0.65
wherethesymbolsHbstbandfortheappropbriabteLame´ metriccoeffi- 0 2 4 6 8 10 12
cientsHτ =c(τ2 ζ2)12(τ2 1)−21,Hζ =c(τ2 ζ2)12(1 ζ2)−21, c(cid:144)a
Hϕ=c(τ2 1)12−(1b ζ2)12.−Thevorticityfieldca−nbeobtai−nedfrom
[3]asb − − b Fig. 1.Thepropulsionvelocityofthe‘cigar-shaped’ microswimmervs. the
b ω =curlv = 1 E2Ψe , [4] elongation.
ϕ
−H
ϕ
wheretheoperatorE2isgivenby inadditiontothesurfacevelocity, u(ζ). Intheswimmingproblem
b
F =C =0,andu(ζ)determinesthepropulsionvelocityU.
2
E2 = 1 (τ2 1) ∂2 +(1 ζ2) ∂2 . Theproblemofthe“frozen"spheroid(i.e. u(ζ)=0)intheuni-
c2(τ2 ζ2) − ∂τ2 − ∂ζ2 formambientflow Ue correspondstosubstitutingb =0,m 2
− » – − 3 m ≥
in[7].InthiscasetheequationsforC canreadilybesolvedyielding
Following the standard procedure, the pressure is eliminated from m
thewell-knownresultforΨandthedragforce[11],
theStokesequationbyapplyingthecurloperatortobothsides,with
conjunctionwith[4]thisyieldstheequationE4Ψ=0forthestream- 8πcµU
F = . [9]
function. Theboundaryconditions[2]atthemicroswimmersurface (1+τ2)coth−1τ τ
a a− a
τ =τ intermsofthestream-functionbecome
a
Propulsionvelocity.Inordertodeterminethevelocityofpropul-
Ψ=0, ∂τΨ=−c2{τa2−ζ2}12{1−ζ2}12 u(ζ), [5] smiounstosfotlhveemEqicsr.o[s7w]imwimtherCfre=ely0sausstpheenpdaerdtiicnlethisefvoirscceou(asnfldutiodr,qounee)
2
and the conditions at infinity (τ ) are vτ Uζ, vζ free. Letusconsiderthefollowingvelocitydistributionatthebound-
U(1 ζ2)12. → ∞ ∼ − ∼ ary
− −
ThesolutionforΨthatisregularontheaxisandatinfinity,and
alsoeveninζ canbederivedfromageneralsemiseparablesolution u(ζ)=−2τaus τa2−ζ2 −12 1−ζ2 −21 G2(ζ), [10]
[9,10],
whereu isatypicalv`elocityof´surf`acetrea´dmillingandG (ζ) =
s 2
∞ 1(1 ζ2). One may verify that for a sphere (c/a 0) u(ζ) =
Ψ=−2c2UG2(τ)G2(ζ)+ {AmHm(τ)Gm(ζ)+ u2ssin−θ,whileforanelongatedswimmeru(ζ) ≃ us→almostevery-
m=X2,4,... whereexceptthenearvicinityofthepolesζ = 1. Moregenerally
±
CmΩm(τ,ζ) , [6] itcanbereadilyshownthatthesolutionsatisfying[5],[10]isgiven
}
by[6]with m,C =0;m 4, A =0and
whereΩ (τ,ζ)isasolutionofE4Ψ=0composedfromspheroidal ∀ m ≥ m
m
harmonicsthatdecayatinfinity,andG andH aretheGegenbauer A = 2c2u τ ( 1+τ2),
m m 2 − s a − a
functionsof thefirstandthesecondkind, respectively. Thecoeffi- U =u τ2 τ ( 1+τ2)coth−1τ . [11]
cientsA in[6]canbeexpressedintermsofC andU viatheuse s a − a − a a
m m
oftheboundary conditionΨ = 0atτ = τ . Substituting[6]into NotethatU canberela˘tedtothesurfacemotion2 viath¯euseofthe
a
[5]wearriveaftersomealgebraatthetridiagonalinfinitesystemof Lorentzreciprocaltheorem[12],
equationsforU andthecoefficientsC ,
m
F U = (σ n) udS, [12]
Em(−)Cm−2+Em(0)Cm+Em(+)Cm+2 =bm, m≥2 [7] · −ZS · ·
HereC = c2U, (0,±)areknownfunctionsofτ ,and where(u,σ)isthebvelocityandstrbessfieldcorrespondingtotransla-
0 − Em a tionofthesameshapedobjectwhenacteduponbyanexternalforce
bm = c22m(m−1)(2m−1) +1 τ1a2−ζζ22 12 u(ζ)Gm(ζ)dζ. [8] Fha.veF(oσrbp·unrbe)ly·utan=geσnτtiζauls(uζr)f,awcehemreotionconsideredinthisworkwe
−Z1  − ff b
H ∂ vˆ H ∂ vˆ
b σ = b τ τ + ζ ζ .
RegularityofΨimpliesthattheadmissiblesolutionof[7]shouldsat- τζ Hζ ∂ζ „Hτ« Hτ ∂τ Hζ!
isfyC H (τ ) 0asm whiletheexponentiallygrowing b b
m m a
solutTiohnevwiistchoCusmd→r∼agOfo(r(cτeae+x→er√te∞τda2o−nt1h)e2pmro)lsahteouslpdhebreodidis(cianrtdheedx. - Wluteiocnalccourlraetsepbtohnedlioncgabtlotastnrgeeanmtiibanlgsptraesstsabcroigmidpopnreonlabtteστspζhferroomidt,hwehsiole-
3
direction)issolelydeterminedbytheC2-termin[6]corresponding b
toamonopole (Stokeslet)velocitytermdecaying like1/r farfrom
2Inthelaboratoryframethevelocityatthesurfaceisasuperpositionofthetranslationalvelocity
theparticle,F = (4πµ/c)C2[11]. EitherF orUcanbespecified Uandpurelytangentialmotionsu
−
2 www.pnas.org—— FootlineAuthor
F isgivenby[9]. Substitutioninto[12]withdS = H H dϕdζ
ϕ ζ
yieldsaftersomealgebra
b b b
τ +1 1 ζ2 12 10 1
U = a − u(ζ)dζ, [13] ~€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
− 2 −Z1 „τa2−ζ2« 5 2 Ε2 Hlog €Ε1€€€€L2
∆
which holds for an arbitrary tangential boundary velocity u(ζ). In
1
thespecialcaseofu(ζ)givenby[10]itcanbereadilydemonstrated
that integrationyieldsthe propulsion speed [11]. Also, [13] actu- 0.5
ally solves the infinite tridiagonal system [7], since knowing U,F
(i.e. C ,C )onecaniterativelyobtainalltheotherC ’sbydirect 0 5 10 15 20
0 2 m
substitution. The scaled swimming speed of the microswimmer is c(cid:144)a
depictedinFigure1asafunctionofthescaledelongation. Theval-
uesofthepropulsionvelocitycorrespondingtoasphericalswimmer Fig. 2.Theswimmingefficiencyofthespheroidalmicroswimmer,δ,vs. the
(c=0)andaslenderswimmer(c a)canbedeterminedvia[12] scaledelongationc/a(linear-logplot). Thesolidlinecorrespondstotheexact
without invoking special spheroida≫l coordinates. For a sphere, the calculation,thedashedlineistheasymptoticresult.
localtractionforceσ n= 3µUˆ andFˆ = 6πµaUˆ andthusthe
· −2a −
self-propulsionvelocitycanbefoundas[13]
whereE istherate-of-straintensor,V isthefluidvolumesurround-
b 1 ing the swimmer and S its surface. Expressing the product E:E
U = u edS,
−4πa2 · as ωiωi+2(∂ivj)(∂jvi)allowsre-writing formicroswimmers
ZS self-propelledbypurelytangentialmotionsuaPs[13]
where e is the unit vector in the direction of locomotion. Sub- P
θstiitsuttihneg suph=ericuaslsainngθleemθeaansdureed=witehrrecsopseθct−toeeθ, swineθa,rrwivheeraet P =µZV ω2dV +2µZS u2κsdS, [14]
U = 12 0πussin3θdθ = 32 us inagreementwiththeresultshown whereκs = −(∂s/∂s)·nisthecurvaturemeasuredalongthepath
inFigure1. of the surface flow, expressible in terms of the unit tangential and
R
Thedragforceexertedontherod-likemicroswimmerupontrans- normalvectors,sandn,respectively. Letusnowestimateδofthe
lation along its major axis with velocity Uˆ|| is given by Fˆ spheroidaltreadmilerdescribedintheprevioussubsection. Forapro-
4πµaUˆ||/(εlog1/ε) and the local friction force is σ n ≈ latespheroids=eζ,n=eτ,respectively,andκscanbecalculated
− · ≈ as
µUˆ||/(alog1/ε), where ε = a/b = [1+(c/a)2]−1/2 1 is
t−heaspectratio. Thus,from[121]follows b ≪ κs = Hτ1Hζ ∂∂Hbτζ˛˛˛τ=τa = c(ττaa2√−τa2ζ2−)31/2 ,
U u edS. Sincethesolution[11]corresp˛ondstoirrotationalflow3,i.e. ω =0
≈−4πab · b b ˛
ZS thevolumeintegralin[14]dropsout. Substitutingtheexpressionfor
For the ‘needle-shaped’ microswimmer the surface velocity u = thesurfacevelocity[10]andκsintothesurfaceintegralin[14]we
u(ζ)e u e over almost the whole surface, it follows that find
ζ s
≃ −
U u . As seen in Figure1 the propulsion velocity U/u 1
asc≃/agsrowsandequalsto0.95alreadyatc 5.3a. Asintusit→ively P =4πµcu2s(τa2−1) (1+τa2)coth−1τa−τa . [15]
≃
expected,themicro-swimmerisself-propelledforwardwiththeve- Collecting the expressions fo˘r the drag force [9], velo¯city of self-
locityofthesurfacetreadmilling,whiletheboundaryvelocityinthe propulsion[11]andthedissipation[15]onecancomputetheswim-
laboratoryframeis(almost)zero. mingefficiency,
Swimming efficiency.Since the fluid around the elongated mi-
croswimmerpropelledbycontinuoussurfacetreadmillingisalmost δ= RU2 = 2 τa2−τa(τa2−1)coth−1τa 2 . [16]
quiescent, except for the near vicinity of the poles, it is natural to P (τa2−˘1) (1+τa2)coth−1τa−¯τa 2
expectlowviscousdissipationandhighhydrodynamicswimmingef-
δisplottedasafunctionofth˘eelongationc/ainFigure¯2. Evidently,
ficiency. Several definitions of hydrodynamic efficiency have been
δ growsunbounded asc/a does, corresponding, inthelimit
proposed [13,14,19] herewefollow thedefinitionδ = F·U/ , → ∞
where istheenergydissipatedinswimmingwithvelocityU,aPnd toafrictionlessswimmer. Forthesphericaltreadmilerδcanbecal-
theexpPressioninthenumeratoristheworkexpandedbydraggingthe culatedfrom[14]withu = us sinθeθ,us = 32U andκs = 1/a,
= 2µ 9 U2 sin2θ 1 dS = 12πµaU2. Dividing 6πµaU2
“frozen"swimmeratvelocityU uponactionofanexternalforceF P S 4 a
by wefindδ = 1 inagreementwith[16](seeFigure2)andthe
[13]. δisdimensionlessandcomparestheself-propulsionwithdrag- P R 2
theoreticalbound(i.e. δ 3)in[13].
ging(someauthorsusethereciprocalefficiency1/δ). Thehigherδ ≤ 4
Fortheslenderswimmertheasymptoticbehaviorofδcanbeesti-
the more efficient the swimmer is. For an axisymmetric swimmer
matedfrom[16]byexpandingδinaseriesaroundτ =1andusing
propelledalongthesymmetryaxis,F·U = U2,wherethescalar a
R τ =1/√1 ε2 1+ ε2 whereε= a 1,
istheappropriatehydrodynamicresistance. Theworkdonebyan a − ∼ 2 b ≪
R
arbitraryshapedswimmeranddissipatedbyviscosityinthefluidis 1 1
δ . [17]
givenby ≃ 2(εlogε)2
= (σ n) vdS =2µ E:EdV ,
P −ZS · · ZV 3SinceE2(HmGm)=0thevotricityωisdeterminedbytheCm, m≥4termsin[6]
FootlineAuthor PNAS IssueDate Volume IssueNumber 3
ThisresultisshowninFigure2asadashedline. Forcomparison,the
1
efficiencyof spherical squirmersself-propelled bypropagating sur-
facewavesalongtheirsurface(themathematicalmodelofcianobac- 0.8
teria [15]) has the upper bound δ 3, while numerically calcu-
latedvaluesofδdomuchworsethan≤dra4ggingandthecorresponding us 0.6
swimmingefficiencyisusuallylessthan2%[13]. Swimmingbysur- (cid:144)u
0.4
facetreadmillingisremarkablymoreefficientthantherotatinghelical
flagellum[16],beatingflexiblefilament[17],thePercell’s“three-link 0.2
swimmer"[18]orlocomotionbyvirtueofshapestrokes[14,19]. The
0
surfacetreadmillingisprobablysuperiortoanyinertialessswimming
0 0.2 0.4 0.6 0.8 1
techniquesproposedsofar.
Ζ
Also, the swimming efficiency of the ellipsoidal treadmiler is
superior by a factor of (εlog1/ε)−1 over the estimate of δ corre- Fig. 3.Theoptimalboundaryvelocityuvs. thespheroidalcoordinateζfor
spondingtotherod-liketreadmilerwithroundedendsderivedfrom elongationc = 2.5aatvarioustruncationlevels: L = 4(red)andL = 10
(blue).Theblacklinecorrespondtotheone-termboundaryvelocity[10]
purelyscalingargumentsintheintroduction. Therefore,thegeome-
try(viaκ )playsanimportantroleinminimizingthedissipationin
s
surfacetreadmilling, whichisrathersurprisingsincethedragforce
onslendernonmotileobjectdoesnotdependonitsshapetothefirst
15
approximation.
12.5
Optimal swimming.We can set an upper bound on δ for a
spheroidal microswimmer in terms of surface integrals of an arbi- 2us 10
a
trary velocity u(ζ) analogously to [13]. The power dissipated in Μ 7.5
(cid:144)
self-propulsionisboundedfrombelowaccordingto[14]by P
5
+1 2.5
u2(ζ)
2µ u2κ dS =4πµcτ τ2 1 dζ,
P ≥ s a a − τ2 ζ2 0
Z ` ´−Z1 a − 0.5 1 5 10 50
whereweusedthepreviouslyderivedresultforκ . Thepowerex- c(cid:144)a
s
panded in dragging at the same speed is found from [9] and [13] Fig. 4.Thedissipationintegral vs. thescaledelongationcorresponding to
as optimalswimmingatvarioustruncationlevels(linear-logplot): L=4(red)and
L=10(blue).Theblacklinerefersto[15].
+1 1 2
2πcµτ2 1 ζ2 2
U2 = a − u(ζ)dζ .
R (1+τa2)coth−1τa−τa 2−Z1 τa2−ζ2ff 3 calculatedas
4 5
Combiningthelasttworesultsweobtainanupperboundonδas 2µ u2κ dS =u2 (τ )+u u (τ ),
s 2I2 a 2 4I4 a
δ≤ (τa2−1) (1+τaτ2a)coth−1τa−τa 8><"−+R11„2−+τ1a121−−τζζa22u2−2«ζ212udζdζ#29>= τUwah.=erTeuh2IeF2v(2τe(alτo)ac)iiZts+ySguoivf4eFpnr4o(bpτyual)[s1ifor5on]mcaa[nn1d3bI]e,4wfoihsuensrdeomFine2tiohsteheeqcruloafsluetnocfottihromenexaos-f
` ´ R pressioninthefigurebracketsin[11]and issomeotherfunction
The term in the figure brackets can be>:shown to be bounde>;d ofτ . AsthepropulsionvelocitytobefixeFd,4werequireU =u .
from above by 2/3 while its maximum is obtained for u(ζ) = a sF2
Thisyieldsdissipation
u 1 ζ2 τ2 ζ2correspondingtothe2-termboundaryvelocity
s − a −
esexnpptainnsgioanro[t7pat]iowniathlflbo2w,)b.4T6=hus0,,fobrman=el0o,nmgat≥ed6sw(aimndm,ethru(τs,ar→epr1e-) P =u2sI2+u4us„I4−2FF24 I2«+O(u24).
wearriveat
1 wherethefunctioninthebracketscanbeshown tobepositiveand
δ
≤ 3ε2log1/ε bounded for allτa > 1, and vanishes only at τa 1. Therefore,
onecanalwayschoosesomeu <0suchthatP <→ u2leadingto
whichdoesbetterthan[17]byafactorof(log1/ε)−1 andalsosu- 4 I2 s
reductioninthedissipationin[15]. Theaboveperturbationanaly-
perioroverthescalingestimatefortherod-likeswimmerbyafactor
sisshowsthat,quitesurprisingly,vorticityproductioncouldbringa
of (1/ε). Note that the asymptotic behavior δ 1 was
O ∼ ε2log1/ε reductioninviscousdissipation,leadingtomoreefficientswimming.
derivedfromsimplescalingargumentsintheintroduction.
Toaddressthequestionofoptimalswimmingweconsideranar-
Itcanbedemonstratedthatthesurfacevelocity[10]isnotopti-
bitraryboundaryvelocityviatheexpansion,thatmeetsalltheabove
mal,i.e. itdoesnotminimize foraprescribedpropulsionspeedU.
P requirementsforregularityandevennessinζ,
Toseethisconsidertheslightlyperturbedboundaryvelocity
L
u(ζ)= (τa2−ζ2−)122τ(a1−ζ2)21 {u2G2(ζ)+u4G4(ζ)}, [18] u(ζ)= (τa2−ζ2−)212τ(a1−ζ2)12 m=X2,4,...umGm(ζ). [19]
suchthat u u andu u . Thesolutionofthelinearprob- whereitfollowsfrom [8] thatu = b /(2c2τ ). Tofindaset
4 2 2 s m m a
lem [7] y|ield|s≪ω| =| (u )∼and therefore, the volume integral in ofFouriercoefficientsu , m=2, 4, 6−,... Lcorrespondingtothe
4 m
[14]isµ ω2|dV| =O(u2). Thesurfaceintegralin[14]canbe optimalswimming,oneshouldminimizethedissipationintegral, ,
V O 4 P
R
4 www.pnas.org—— FootlineAuthor
while keeping the propulsion speed U fixed. The dissipation inte- In the swimmer frame of reference, the boundary condition at
gral = σ udS beingbilinearinu , canbeexpressedas infinity v( ) = U (where U is the laboratory-frame swimming
P − S τζ i ∞ −
= 1 u u . Notehoweverthatthetangentialstressσ at speed)impliesLaurentexpansions
P 2 iRPij j τζ
thesurfaceofthemicroswimmerrequirestheknowledgeoftheveloc- ∞ ∞
itygradiPentatthesurface(ratherthanvelocityalong). Alternatively, f = a ζ−n, g= U+ b ζ−n, [22]
n n
−
since the optimal velocity fieldisrotational, calculation of from n=1 n=1
P X X
[14]requirestheknowledgeofvorticityeverywhere. whereU isarbitrarilyappendedtog.
Thepropulsionvelocitygivenby[13]islinearinu ,i.e. U = The boundary condition on the swimmer surface is fulfilled
i
u . Theoptimalsetofcoefficientsu istobedeterminedfrom by matching v(ζ) to a prescribed boundary motion v =
jFj j i |ζ=eiθ
P∂∂ui(P −λU) = 0, orjustfrom jPij uj = λFi,whereλisa w(θ) = ∞n=−∞wneinθ. It is useful to express w(θ) as w =
Lagrangemultiplier. Wefoundtheclosedformoptimalsolutionfor w (ζ) + w (ζ),ζ = eıθ where w = ∞ w ζ−n,w =
P + P− − n=0 −n +
thetwo-termboundaryvelocity[19],whileforL > 4closedform ∞ w∗ζ−n arebothanalyticoutsidetheunitcircle. Substituting
expressionsarecumbersome, andnumerical solutionswerederived [22n]=i1nton[21]andmatchingontheunitcircPlewefind
instead. Analogouslytothetheoryfora2-Dswimmer(seethenext P
ζ(1+αζ2)
section),wheretheexplicitoptimalsolutionwasshowntoacquirean g(ζ)=w (ζ),f(ζ)=w (ζ)+ w′ (ζ).
− + ζ2 α −
infinitenumberofharmonicsintheexpansionfortheboundaryveloc- −
ity,increasingthetruncationlevelLin[19]willfurtherimprovethe In particular the swimming velocity is determined by the constant
efficiencyofswimming,thoughtheenhancementappearstobeminor. terminthisexpansionU = w0. Thecorrespondingdissipationis
−
calculatedusing[20]as
Toillustratethis,wecalculatetheoptimalsolutionuponvaryingL.
TheoptimalboundaryvelocityuponvaryingLisdepictedinFigure ∞
= Re v¯dF =2µIm w¯d(2g w)=4πµ n w 2.
3 for the elongation of c = 2.5a and compared with the one-term P − − | || n|
evxs.prce/ssaiounpo[1n0v]a.rTyihnegsLcaliendFdigisusriepa4t.ioInticnatnegbrael,rePad/iµlyause2se,nistphlaottttehde Let usInext focus on thIe case of an ellipse swnim=X−m∞ing by sur-
convergencewithrespecttoLisratherfast;thedeviationbetweenthe facetreadmilling. Theboundaryvelocityw(θ)beingtangenttothe
resultscorrespondingtoL=8and10islessthan1%forallc/aand swimmerboundaryisexpressibleasw= ddzθu(θ)=i(ζ−α/ζ)u(θ)
itvanishesatbothlimitsc=0andc/a . Thus,the‘intuitive’ for somereal-valuedfunctionu(θ). Sinceweconsider onlyswim-
one-term boundary velocity [10], that y→ield∞s δ (εlogε)−2 (see mers symmetric with respect to the x axis, we assume u(θ) to
Figure2)isnearlyoptimalforawiderangeofelo∼ngationsandlikely be an odd function allowing to write it−as u = unsin(nθ) =
s2o-Dfomrailclreolosnwgiamtiomnes.r.ThetwodimensionalStokesequationsiscon- i21nitoPUu=n(ζ−nw−0 =ζ−21n()1. +Inαt)eurm1swohfilethtihsethdeisssiwpiamtiomnPintagkveseltohceitfyortmurn
venientlyhandledbyemployingcomplexvariables[14,19,21,22]. =2πµ n (1+α2)u2 2αu u .
Thisallowsexplicitsolutionoftheoptimizationproblemfortheel- P n− n−1 n+1
lipticaltreadmiler. WhichmayalsobewXritten`asP = 12 Pijuiujforaco´rresponding
tionsDbeencootmineg2vµ=∂∂¯vvx=+i∂¯vpy,anRde∂∂v==210(∂.xTh−eim∂yo)s,ttgheenSertoalkessoleuqtuioan- tridiTaghoenoalptmimatarlixswPiimj.ming techniquPe for a given α is the one that
tothis(withpreal)isv=g+f¯ zg′, p= 4µRe(g′)whereg,f minimizesthedissipationwhilekeepingtheswimmingvelocityU =
areanypairofholomorphicfunc−tions[20]. S−olutionscorresponding 12(1+α)u1fixed.Theminimizeristhesolutionof∂∂ui(P−λu1)=0
tomultivaluedg, f arealsolegitimate(provided theresultingv, p for iwithλbeingaLagrangemultiplier,orjust
∀
aresinglevalued). Itcanbeshown(using[20]below)thatthemon-
u =λδ . [23]
odromy of g (and of f¯) around aclosed curve givesthe totalforce Pij j i,1
j
exertedbythefluidontheinteriorofthecurve. Inparticularinswim- X
Itisreadilyseenthatthecoefficientsu withevenkarenotrele-
mingproblemsthisforcemustvanishandg,f arethereforealways k
vanttotheoptimalswimmingandshouldbesettozerotominimize
singlevalued.
viscousdissipation. (Thisisalsoclearfromthefactthatu corre-
TheelementofforcedF dF +idF actingonalengthele- 2k
≡ x y spondtoflowswhichareantisymmetricwithrespecttothey axis.)
mentdz = dx+idy ofthefluidcanbeexpressedintermsofv,P Denotingb u , k 0andwritingλ=2u πµ(1−+α2)
andhenceintermsofg,f. Straightforwardcalculationshowsthatthe k ≡ 2k+1 ≥ s
withu anarbitrarynormalizationconstanthavingdimensionsofve-
relationis s
locityweobtainfrom[23]therecursionrelation
dF =ipdz+(2iµ∂¯v)dz¯=2iµd(v 2g). [20]
− (2k+1)bk ξ(kbk−1+(k+1)bk+1)=usδk,0,
−
Notethathere(dx,dy)istangentratherthenthenormaltotheseg- where ξ = 2α . Multiplying by xk and summing over k this
ment4. transforms in1to+αa2differential equation for the generating function
Weconsider a2-D swimmer shaped asan ellipseof semi-axes B(x)= 1 ∞ b xk,
us k=0 k
b,a=1 α(with0 α<1)situatedinthecomplexz =x+iy
plane. It±isthenconve≤nienttodefineanewcomplexcoordinateζby Ph(x)B′(x)+ 1h′(x)B(x)+1=0,
2
therelationz = ζ +α/ζ. Asζ rangesovertheregion ζ > 1the
| | wherewedefinedh(x) ξ(x α)(x 1). Thegeneralsolution
corresponding z rangesover theareaoutsidetheswimmer. Inpar- ≡ − − α
ticulartheswimmerboundarycorrespondstotheunitcircleζ =eiθ. tothisisB(x) = 1 x dx′ whereC isaconstantofin-
−√h(x) C √h(x′)
Notethatifweconsiderg, f asfunctionsofζratherthenzthenthe tegration. RequiringthecoeRfficientsbk todecayforlargekimplies
generalsolutionoftheStokesequationsbecomes
ζ+α/ζ
v=g+f¯ g′. [21] 4ThesignconventionhereisthatdFistheforceexertedbythel.h.softhe(oriented)segment
− 1 α/ζ¯2 dzonitsr.h.s.
−
FootlineAuthor PNAS IssueDate Volume IssueNumber 5
thatB(x)mustbeanalyticinsidetheunitdiscandhenceitspotential that[24]mustbemodifiedtoaboundedexpression. Thus(incontrast
singularityatx=αmustbeavoided. Thisdeterminestheintegration tothe3-Dcase)onecannotobtainthecorrectasymptoticefficiency
constanttobeC =α,sothatwemaywrite withoutretainingallthemodes.
Theoptimalboundaryvelocitymaybefoundexplicitlyas
1 x dx′ 2 √x−+√x+
B(x)= = log ,
− h(x)Zα h(x′) ξ√x−x+ „ √x+−x− « w(θ)= ddθz uksin(kθ)= ddθz bk (ei(2k+1)θ−2ie−i(2k+1)θ)
wherex+ =pα1 −xandpx− =α−x. Thecorrespondingswimming X =u dzXIm eiθB e2iθ .
velocityandthedissipationare,respectively, sdθ
n “ ”o
U = (1+α)u1 = us(1+α)B(0)= us(1+α)log 1+α , UsingtheexplicitexpressionwehaveforB(x)and ddθz wefindthe
2 2 2ξ 1 α absolutevalueofwisgivenby
„ − «
1 1
P = 2 Pijuiuj = 2λu1 =πµ(1+α2)u2sB(0)= w =u 1+α2 log √α−e2iθ+ 1/α−e2iθ ,
X πµu2(1+α2)2log 1+α . | | s √α ˛˛ 1/αp−α ˛˛
2α s 1 α ˛ ˛
„ − « whileitsdirectionisknownto˛betangepntial. Thus,inthelim˛itε 0
˛ ˛ →
Therefore,combiningthelasttworesultsyields onefinds(providedε θ, π θ)that
≪ | − |
2α 4πµU2 log 2sinθ
= w/U 1+ | |,
P (1+α)2log 1+α | |≃ log1/ε
1−α
“ ” andtheboundaryvelocityapproachestheconstantU thoughonlyat
Werecallthatin2-Dthedraggingproblemadmitnoregularsolution alogarithmicrate.
withintheStokesapproximation5. Thusdefiningtheswimmingeffi-
Concludingremarks.Inthispaperweexaminedthepropulsionof
ciencyasδ=(F U)/ = U2/ makesnosenseinthepresent elongatedmicroswimmerbyvirtueofthecontinuoussurfacetread-
2-Dcontextinwhi·chF,P arRenotdPefined. Thismaybeconsidered milling. Astheslendernessincreases,thehydrodynamicdisturbance
R
asamereissueofnormalization. Wethereforeusehereanalternative
created by thesurface motion diminishes, i.e. themicrobot ispro-
definitionofswimmingefficiency[19]where
pelled forward with the velocity of the surface treadmilling, while
surface,exceptthenearvicinityofthepoles,remainsstationaryinthe
4πµU2 (1+α)2 1+α
δ⋆ = = log . [24] laboratory frame. Asa result of that, the‘cigar-shaped’ treadmiler
2α 1 α
P „ − « isself-propelledthroughoutalmostquiescentfluidyieldingverylow
In the slender limit, α 1, or, a 1−α = ε 0, as the el- viscous dissipation. Thecalculation of optimal hydrodynamic effi-
→ b ≡ 1+α → ciencyofthe3-Dandthe2-Dmicroswimmersrevealsthatthepro-
lipsedegeneratesintoaneedle,theefficiencygrowslogarithmically
posed swimming technique is not only superior to various motility
unboundedas
δ⋆ 2log(1/ε). mechanismsconsideredinthepast,butalsoperformmuchbetterthan
≃ draggingundertheactionofanexternalforce.
Itmaybeofinteresttonotethattruncatingourexpansiontoinclude
any finitenumber of modes would lead to B(x) which isnot only
ThisworkwaspartiallysupportedbyIsraelScienceFoundationandtheEU
polynomial inxbutalsoalgebraicinα. Thisthenimpliesthatthe grant HPRN-CT-2002-00277 (to J.E.A.)and by the Fund of Promotion of
(truncated)efficiencyδ⋆ B(0)wouldbealgebraicinαimplying ResearchattheTechnion(toJ.E.A.andA.M.L.).
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6 www.pnas.org—— FootlineAuthor