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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 isatypicalvèlocityof´surfàcetrea´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. WhichmayalsobewXrittenàsP = 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.). ∝ 1. Purcell,E.M.,(1977)Am.J.Phys.45,3-11. 12. Kim,S.andKarrila,S.J.(1991)inMicrohydrodynamics(Butterworth–Heinemann,Boston). 2. Berg,H.C.,(2000)Phys.Today54,24-29. 13. 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