A recently introduced model for an autonomous swimmer at low Reynolds number
that is comprised of three spheres connected by two arms is considered when one
of the spheres has a large radius. The Stokes hydrodynamic flow associated with
the swimming strokes and net motion of this system can be studied analytically
using the Stokes Green's function of a point force in front of a sphere of
arbitrary radius $R$ provided by Oseen. The swimming velocity is calculated,
and shown to scale as $1/R^3$ with the radius of the sphere.Comment: 4 pages, 1 figur

Golestanian, Ramin

The European Physical Journal E

English

arXiv

A recently introduced model for an autonomous swimmer at low Reynolds number that is comprised of three spheres connected by two arms is considered when one of the spheres has a large radius. The Stokes hydrodynamic flow associated with the swimming strokes and net motion of this system can be studied analytically using the Stokes Green's function of a point force in front of a sphere of arbitrary radius R provided by Oseen. The swimming velocity is calculated, and shown to scale as 1/R3 with the radius of the sphere

Golestanian, R

Oxford University Research Archive (ORA)

Three-sphere low-Reynolds-number swimmer with a cargo container.

A recently introduced model for an autonomous swimmer at low Reynolds number that is comprised of three spheres connected by two arms is considered when one of the spheres has a large radius. The Stokes hydrodynamic flow associated with the swimming strokes and net motion of this system can be studied analytically using the Stokes Green's function of a point force in front of a sphere of arbitrary radius R provided by Oseen. The swimming velocity is calculated, and shown to scale as 1/R 3 with the radius of the sphere. © 2008 EDP Sciences, Società Italiana di Fisica and Springer-Verlag

Golestanian, R.

MPG.PuRe

Three-sphere low-Reynolds-number swimmer with a cargo container

A recently introduced model for an autonomous swimmer at low Reynolds number that is comprised of three spheres connected by two arms is considered when one of the spheres has a large radius. The Stokes hydrodynamic flow associated with the swimming strokes and net motion of this system can be studied analytically using the Stokes Green's function of a point force in front of a sphere of arbitrary radius R provided by Oseen. The swimming velocity is calculated, and shown to scale as 1/R
3 with the radius of the sphere

R. Golestanian

EDP Sciences OAI-PMH repository (1.2.0)

White Rose Research Online

This is a repository copy of Three-sphere low-Reynolds-number swimmer with a cargo container.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/121589/Version: Accepted VersionArticle:Golestanian, R. (2008) Three-sphere low-Reynolds-number swimmer with a cargo container. European Physical Journal E, 25 (1). pp. 1-4. ISSN 1292-8941 https://doi.org/10.1140/epje/i2007-10276-2eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. arXiv:0711.3772v2  [cond-mat.soft]  19 Mar 2008Three-Sphere Low Reynolds Number Swimmer with a Cargo ContainerRamin Golestanian∗Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK(Dated: February 12, 2013)A recently introduced model for an autonomous swimmer at low Reynolds number that is com-prised of three spheres connected by two arms is considered when one of the spheres has a largeradius. The Stokes hydrodynamic flow associated with the swimming strokes and net motion of thissystem can be studied analytically using the Stokes Green’s function of a point force in front of asphere of arbitrary radius R provided by Oseen. The swimming velocity is calculated, and shownto scale as 1/R3 with the radius of the sphere.PACS numbers: 87.19.ru,47.15.G-,62.25.-gSince the pioneering work of G. I. Taylor [1] we knowthat swimming at low Reynolds number is a nontriv-ial task, which is also echoed in the incredibly sophisti-cated mechanisms involved in swimming of bacteria [2, 3].While nature could overcome the difficulties of swimmingat low Reynolds number by taking advantage of a contin-uum of degrees of freedom, this will be much more chal-lenging for an artificially engineered model, as it couldoften use a finite number of degrees of freedom like mostmanmade devices. The relation between the number ofdegrees of freedom and swimming at low Reynolds num-ber was considered by Purcell in his seminal work [4],who concluded that one degree of freedom is not enoughand the minimum required is two. He further elaboratedon this by proposing a simple swimmer model made ofthree joined rods, whose motion was fully analyzed onlyrecently [5] due to the complicated nature of the hydro-dynamic problem of three rotating finite rods [6].If we take advantage of the positional degrees of free-dom rather than the orientational ones, we can constructswimmer models that can be much more easily analyzed.This is the principle used in the design of a simple swim-mer that is made of three spheres connected by two armsthat open and close [7]. The simplicity of the three-sphereswimmer model allows us to study various characteristicproperties of the swimmer itself—such as the effects ofexternal load and noise, stress distribution, power con-sumption, etc.—in great details [8, 9]. Moreover, it alsomakes it possible to study the interaction between two[10] or more of these swimmers, which could be usedtowards a systematic study of the floc behavior of a sus-pension of such swimmers [11, 12, 13, 14]. A similarprinciple has been used to propose other swimmer mod-els that can also be easily analyzed [15, 16, 17]. Thisclass of swimmer models uses a finite number of compactdegrees of freedom, and is to be contrasted from thosethat use a continuum [18, 19]. An artificial microscopicswimmer has been recently made using magnetic beadsconnected by DNA linkers, and has been shown to be ableto propel itself via beating strokes that are induced by an∗Electronic address: r.golestanian@sheffield.ac.ukoscillating magnetic field [20]. It is also possible to designswimmers with no moving parts by taking advantage ofphoretic phenomena, as has been proposed and studiedtheoretically [21, 22, 23] and also realized experimentally[24, 25, 26].The analysis of the three-sphere swimmer model has sofar been made in the limit where the radii of the spheresare the smallest length scales in the system, namely, theyare much smaller than the distances between the spheres.In practical applications, however, one can imagine thatthe swimmer could be used to carry a cargo, which wouldbe possible if one of the spheres was blown into a largehollow spherical shell, with a radius that is compara-ble to the distance between the spheres. A similar ge-ometry could also be useful for experimental testing ofthe swimmer design, as a large spherical bead as a partof the swimmer would allow optical probing of its pro-pelled motion. With these motivations, here we studythe swimming of the three-sphere model with one of thespheres having a finite radius comparable to the otherlength scales. It is possible to fully analyze the motion ofthis model swimmer along the same lines as the simplerversion, thanks to the Stokes Green’s function of a pointforce in front of a sphere of arbitrary radius calculatedby Oseen in 1927 using the method of images [27]. Wefind that the average swimming velocity scales as 1/R3with the radius of the container. Our results show thata simplistic view of “estimating” the velocity of an au-tonomous swimmer by dividing the propelling force inthe system by its Stokes friction coefficient is incorrect,and a only proper analysis of the hydrodynamic flow dueto the deformations of the swimmer should be used todetermine the propulsion velocity.Consider the geometry of Fig. 1 where one of thespheres in the swimmer is in the form of a large con-tainer of radius R and the other two have radii a1 anda2. To calculate the swimming velocity of the machine,we work in the frame of reference that is co-moving withthe large sphere. Assuming that there are two force cen-ters located at x1 and x2 exerting the forces f1 and f2,respectively, we can write the velocity of the fluid at any22 a12 a2xyzR1R2RL1L2FIG. 1: Schematics of the swimmer with a cargo container.The instantaneous distance between the two small spheresand the center of the large sphere are R1(t) and R2(t), andthere three spheres are assumed to always be positioned alongthe z-axis.point x asvi(x) = −Vi +14Rr(3 +R2r2)Vi+34Rr(1 −R2r2)xixjr2Vj+18πη[Gij(x,x1)f1j + Gij(x,x2)f2j ] , (1)where V is the velocity of the large sphere in the labora-tory frame, and Gij(x,x′) is the Oseen Green’s functionthat vanishes on the surface of the large sphere [27, 28].Assuming that the two small spheres are located on thez-axis (see Fig. 1) at x1 = (0, 0, R1) and x2 = (0, 0, R2),symmetry requires that the device swims along the z-direction, namely, V = (0, 0, V ). Using no-slip bound-ary condition on the two small spheres, and defining thequantitiesCα = 1 −32RRα+12R3R3α, (2)we can find their velocities asvα = −CαV +18πηHαβfβ , (3)where Hαβ = Gzz(xα,xβ) for α, β = 1, 2, and sum-mation over repeated indices is understood. Note thatH12 = H21 due to the symmetry of the Green’s func-tion. Assuming that the small spheres have prescribedvelocities vα = Ṙα, we can determine the forces asfα = 8πη H−1αβ (Ṙβ + CβV ), (4)where H−1αβ represent the components of the inverse ma-trix for Hαβ , with respect to the α, β indices. Note thatstrictly speaking Faxen’s theorem [29] requires that anadditional contribution proportional to the Laplacian ofthe velocity field is added to the expression for the ve-locity of the small spheres. However, these contributionswill generate terms that are higher order in the ratio be-tween the sizes of the small spheres and the distances inthe system, and can hence be neglected here.We can now impose the condition of zero external forceor the vanishing of the total Stokeslet strength as [30]Cαfα + 6πηRV = 0, (5)which yields the final equation. The above equation canbe understood as the vanishing of the sum of the forcesexerted on the fluid at the outer surface of the sphere.When we have extended objects rather than just pointforces, we should enforce the force-free condition by look-ing at the stress field in the vicinity of the swimmer andshow that its integral over any closed surface vanishes.In our case here, this stress will have a contribution fromthe sphere, which yields the familiar Stokes force term inEq. (5), plus contributions that have propagated fromthe point forces that are located at distances R1 and R2.This form of the force-free constraint can be related tothe case of three point-like spheres, where the force-freerelation reads f1 + f2 + f3 = 0. To this end, we can sim-ply take the limit of R → 0 in Eq. (5) and note that theviscous force term will produce f3 in this limit as V is thevelocity of the third sphere. Equation (5) proves that itis incorrect to assume that the velocity of the bead can beextracted from a force balance between the contributionsfrom the two force centers and the friction force at thebead, which would have (falsely) implied the vanishingof∑α fα + 6πηRV .Putting in the forces from Eq. (4), Eq. (5) can besolved to yield to propulsion velocity asV = −CαH−1αβṘβCαH−1αβCβ +34R. (6)We also feed this result back into Eq. (4) to find theforces at the two small spheres. We findfα = 8πη[H−1αβṘβ −H−1αβCβCγH−1γδ ṘδCγH−1γδ Cδ +34R], (7)where γ, δ = 1, 2.To calculate the swimming velocity we need the ex-plicit expression for Gzz(z, z′), which is presented in the1927 monograph by Oseen [27]. It readsGzz(z, z′) =2|z − z′|−2R|zz′ − R2|−R(z2 − R2)(z′2 − R2)(zz′ − R2)3, (8)3which yields the following expressions for the componentsof the H-matrixH11 =43a1−3R(R21− R2), (9)H22 =43a2−3R(R22− R2), (10)H12 =2|R1 − R2|−2R(R1R2 − R2)−R(R21 − R2)(R22 − R2)(R1R2 − R2)3, (11)where the terms involving a1 and a2 appear with a dif-ferent prefactor according to standard treatments of theOseen tensor to extract the friction coefficient. Puttingthe explicit forms of the Hαβ matrix elements as func-tions of the instantaneous separations R1(t) and R2(t) inEq. (6), we find a closed form expression for the instan-taneous swimming velocity.It is instructive to study the asymptotic limits of Eq.(6) in various limits. In the limit where (a1, a2, R) ≪(R1, R2, R1 − R2), we can use R2 = ℓ2 + u2 and R1 =ℓ1 + ℓ2 + u1 + u2 and expand the expression in powers ofuα. Keeping only terms up to the leading order, we findthe average velocity asV = −32a1a2R(a1 + a2 + R)2[1ℓ21+1ℓ22−1(ℓ1 + ℓ2)2]× (u1u̇2 − u̇1u2), (12)where the averaging over a complete swimming cycle withperiod T is defined by V = 1T∫ T0dtV (t). This is identicalto the result obtained previously [8] using the standardform of the Oseen tensor, which is a good check for ourresult. (The apparent sign difference with the previousresult [8] is due to a different convention for the direc-tion of the z-axis.) Note that in this case the swimmingvelocity scales as the inverse of the second power of thelargest length scale in the system, namely 1/ℓ2.In the limit where (a1, a2) ≪ (R1 − R, R2 − R) ≪ R,we can use R1 = R + L1 + L2 and R2 = R + L2 (see Fig.1), and expand the expression in powers of Lα/R. Wefind the instantaneous velocity asV =92(a1a2R3) L22(3L1 + 4L2)L1(L1 + 2L2)3×[(L̇1 + L̇2)L22 + L̇2(L1 + L2)2], (13)to the leading order. Expanding for small deformationsusing L1 = ℓ1 + u1 and L2 = ℓ2 + u2, yieldsV =94(a1a2R3) ℓ22(3ℓ1 + 2ℓ2)ℓ21(ℓ1 + 2ℓ2)(u̇1u2 − u̇2u1). (14)Equations (13) and (14) are proportional to 1/R3, whichis in marked contrast to the 1/ℓ2 scaling in the case of a(more) symmetric swimmer. The difference in the scalingsuggests that it might be more efficient to distribute thesize of the swimmer across the three spheres, rather thanconcentrate all of it on one sphere. Such a statement,however, needs to be checked in the limit of a symmetricswimmer with spheres of large radii compared to theirsurface-to-surface separations. Equation (14) also showsthat it is beneficial for the two small spheres to be close toeach other and relatively farther away from the surface ofthe sphere, i.e. ℓ2 ≫ ℓ1. Note, however, that this result[Eq. (14)] is only valid for (a1, a2) ≪ (ℓ1, ℓ2) ≪ R, whichmeans that the previous prescription for creating optimalswimming efficiency should be written as (a1, a2) ≪ ℓ1 ≪ℓ2 ≪ R rigorously speaking.The present calculation can be readily generalized tothe case of one large sphere with many small spheres.For a line-up of all the spheres along the z-axis, Eqs.(2), (3), (4), (5), (6), and (7) will hold true providedthe indices are run over the number of small spheres inthe system. For more complicated assortments of smallspheres, one can still use this formalism provided theright tensorial structure is maintained along the way. Auseful particular case could be a line-up of small spheresthat oscillate laterally, i.e. perpendicular to the z-axis,which would mimic the beating motion of a (finite) tail.It will also be interesting to consider the possibility ofa number of such structures attached to the bead inter-acting with each other hydrodynamically. These “activetails” could potentially synchronize with each other un-der certain conditions, to generate an efficient propul-sion. Our systematic approach might be able to provideuseful insight into the intriguing phenomenon of collec-tive hydrodynamic patterns and phases [31, 32, 33]. Onecan also study the efficiency of the swimmer in terms ofpower consumption and useful work, which for this swim-mer follows very closely similar studies that are presentedelsewhere [8].In conclusion, we have generalized the simple modelof a low Reynolds number swimmer to the case whereone of the spheres is large, and analyzed its motion usingthe Oseen tensor of a point force in front of a spherewhich imposes no-slip boundary condition on the Stokesflow. The results found here could help us in possibleexperimental probes of self-propelled devices.I would like to acknowledge fruitful discussions withA. Ajdari and A Najafi.[1] G. I. Taylor, “ Analysis of the swimming of microscopicorganisms,” Proc. Roy. Soc. London A 209, 447-461(1951).4[2] H. C. Berg, Random Walks in Biology (Princeton Uni-versity Press, Princeton, NJ, 1983).[3] D. Bray, Cell Movements 2nd edition (Garland, NewYork, 2001).[4] E. M. Purcell, “Life at low Reynolds number,” AmericanJournal of Physics 45, 3-11 (1977).[5] L. E. Becker, S. A. Koehler, and H. A. Stone, “On self-propulsion of micro-machines at low Reynolds number:Purcells three-link swimmer,” J. Fluid Mech. 490, 15(2003).[6] For movies of an experimental realization of the three-linked Purcell swimmer by A. Hosoi and B. Chan, see:http://web.mit.edu/chosetec/www/robo/3link/.[7] A. Najafi and R. Golestanian, “Simple Swimmer at LowReynolds Number: Three Linked Spheres,” Phys. Rev. E69, 062901 (2004).[8] R. Golestanian and A. Ajdari, “Three-sphere lowreynolds number swimmer revisited,” arXiv:0711.3700.[9] R. Golestanian and A. Ajdari, “Mechanical Response of aSmall Swimmer Driven by Conformational Transitions,”Phys. Rev. Lett. 100, in press (2008); arXiv:0711.4473.[10] C. M. Pooley, G. P. Alexander, and J. M. Yeomans,“Swimming with a friend at low Reynolds number,”arXiv:0705.3612.[11] T. J. Pedley and J. O. Kessler, “Hydrodynamic phenom-ena in suspensions of swimming microorganisms,” Ann.Rev. Fluid Mech. 24, 313 (1992).[12] R. A. Simha and S. Ramaswamy, “Hydrodynamic fluc-tuations and instabilities in ordered suspensions of self-propelled particles,” Phys. Rev. Lett. 89, 058101 (2002).[13] J. P. Hernandez-Ortiz, C. G. Stoltz, and M. D. Gra-ham, “Transport and collective dynamics in suspensionsof confined swimming particles,” Phys. Rev. Lett. 95,204501 (2005).[14] T. B. Liverpool and M. C. Marchetti, “Bridging the mi-croscopic and the hydrodynamic in active filament solu-tions,” Europhys. Lett. 69, 846 (2005).[15] J. E. Avron, O. Kenneth, and D. H. Oaknin, “Push-mepullyou: an efficient micro-swimmer.,” New J. Phys.7 234 (2005).[16] B. U. Felderhof, “The swimming of animalcules,” Phys.Fluids 18, 063101 (2006).[17] D. J. Earl, C. M. Pooley, J. F. Ryder, I. Bredberg andJ.M. Yeomans, “Modelling microscopic swimmers at lowReynolds number,” J. Chem. Phys. 126 064703 (2007).[18] H. A. Stone and A. D. T. Samuel, “Propulsion of mi-croorganisms by surface distortions,” Phys. Rev. Lett.77, 4102 (1996).[19] A. Najafi and R. Golestanian, “Propulsion at lowReynolds number,” J. Phys.: Condens. Matter 17, S1203(2005).[20] R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A.Stone, J. Bibette, “Microscopic artificial swimmers,” Na-ture 437, 862 (2005).[21] R. Golestanian, T. B. Liverpool, and A. Ajdari, “Propul-sion of a molecular machine driven by asymmetric dis-tribution of reaction–products,” Phys. Rev. Lett. 94,220801 (2005).[22] R. Golestanian, T. B. Liverpool, and A. Ajdari, “Designrules for phoretic micro- and nano-swimmers,” New J.Phys. 9, 126 (2007).[23] G. Rückner and R. Kapral, “Chemically powered nan-odimers,” Phys. Rev. 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Three-Sphere Low Reynolds Number Swimmer with a Cargo Container

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