Propulsion of a two-sphere swimmer
Daphne Klotsa, Kyle A. Baldwin, Richard J. A. Hill, Roger M. Bowley, Michael R. Swift
PPropulsion of a two-sphere swimmer
Daphne Klotsa,
1, 2, 3
Kyle A. Baldwin, Richard J. A. Hill, R. M. Bowley, and Michael R. Swift School of Physics and Astronomy, University of Nottingham, UK Department of Chemistry, Lensfield Rd., University of Cambridge, Cambridge CB2 1EW, UK Department of Applied Physical Sciences, University of NorthCarolina at Chapel Hill, North Carolina 27599-3290, United States (Dated: September 7, 2018)We describe experiments and simulations demonstrating the propulsion of a neutrally-buoyantswimmer that consists of a pair of spheres attached by a spring, immersed in a vibrating fluid. Thevibration of the fluid induces relative motion of the spheres which, for sufficiently large amplitudes,can lead to motion of the center of mass of the two spheres. We find that the swimming speedobtained from both experiment and simulation agree and collapse onto a single curve if plotted as afunction of the streaming Reynolds number, suggesting that the propulsion is related to streamingflows. There appears to be a critical onset value of the streaming Reynolds number for swimmingto occur. We observe a change in the streaming flows as the Reynolds number increases, from thatgenerated by two independent oscillating spheres to a collective flow pattern around the swimmeras a whole. The mechanism for swimming is traced to a strengthening of a jet of fluid in the wakeof the swimmer.
The mechanism by which self-propulsion through afluid is achieved has fascinated scientists of many dis-ciplines, and the public alike, for aesthetic, practicaland fundamental scientific reasons [1–5]. In biology andbiomechanics the mechanisms behind the way organismsswim gives insight into their biological function and pur-pose [1, 2, 6, 7]. Recently, the design of efficient “robots”able to navigate themselves through various fluids has be-come an important technological and medical challengethat brings together elements of physics, chemistry, biol-ogy, engineering and fluid mechanics [8–10]. Microscopicartificial swimmers have been proposed for use in tar-geted drug-delivery, see for example [11–13].Purcell’s scallop theorem states that at zero Reynoldsnumber an object cannot swim using a time-reversiblestroke: it will end up going back and forth with no netdisplacement [3]. Many types of small creatures, for ex-ample, insects and aquatic invertebrates, swim at inter-mediate Reynolds numbers (1-100) [14]. In these cases,time-reversal symmetry is broken by non-linearities in thefluid dynamics rather than by the nature of the stroke.For such swimmers, an interesting question arises: howdoes the motion evolve as the Reynolds number is in-creased from zero? It has been argued that symmet-rical objects with symmetrical strokes such as flappingwings have an onset for motion at a critical Reynoldsnumber [15–18], whereas asymmetrical objects or strokeshave a continuous transition as the Reynolds number isincreased [19].A central problem when designing a practical arti-ficial swimmer is how to get energy into the system.Methods based on electromagnetic or chemical actuationhave been developed [13] and currently there is inter-est in using acoustic techniques to generate propulsionthrough the oscillation of entrapped air bubbles [20, 21].Vladimirov proposed an alternative mechanism that may lead to swimming based on a deformable object which isneutrally buoyant, but composed of coupled spheres withdifferent sizes and densities [22]. Such an object can gen-erate relative motion of its parts if immersed in a vibrat-ing fluid; this motion may lead to swimming. However,his calculations in the absence of fluid and particle inertiapredicted that such an object will not swim if subjectedto unidirectional oscillation. Here we pose the question:can an experimental realisation of this object be made toswim at higher Reynolds numbers, and, if so, what is themethod of propulsion and the nature of the transition toswimming?In this Letter we describe experiments and simulationsdemonstrating the propulsion of a pair of spheres at-tached by a spring, immersed in a vertically vibratingfluid. We consider two particular realisations of this ob-ject: one with unequal-sized spheres and the other withequal-sized spheres. In both cases, the density of thespheres is different from one another and from the liq-uid in which they are immersed, however, the objectas a whole is neutrally-buoyant. We find that both de-signs swim for sufficiently high amplitudes of vibration;the unequal-sized spheres swim upwards, in the direc-tion of the larger, less dense sphere, whereas the equal-sized spheres swim downwards, in the direction of thehigher density sphere. The data for the swimming speedare found to collapse both in experiment and simulationwhen scaled appropriately with the streaming Reynoldsnumber, suggesting that the streaming flows induced byfluid non-linearities play a central role [23]. Furthermore,the apparent onset of motion appears to be governed bya critical value of the streaming Reynolds number. Themechanism for propulsion is traced to a change in thetopology of the streaming flows that transition from thoseof two noninteracting spheres when the dimer is station-ary, to a collective flow around the object, at the apparent a r X i v : . [ phy s i c s . f l u - dyn ] J u l FIG. 1. Main panel shows the experimental data collapsefor the unequal-sized swimmer which swims upwards. Thedriving conditions are the following: blue stars f = 65Hz,light blue triangle down f = 75Hz, pink hexagons f = 85Hz,cyan circles f = 95Hz, yellow triangle up f = 105Hz, greentriangles right f = 125Hz, green diamond f = 135Hz. Inall cases the viscosity was 1 . /s except for one dataset (turquoise diamond f = 135Hz) where the viscosity was2 . /s. Simulations for Γ between 2 −
20 and frequencies f = 65 , , , f = 125Hz) shown in the main panel near theapparent onset. The upper inset shows a photograph of theswimmer when stationary. onset of swimming. The flow field shows a strengtheningof a jet of fluid behind the swimmer.The dimers were constructed from two spheres joinedtogether by a small coil of wire. Examples of the asym-metric and symmetric dimers are shown in the insets toFigs. 1 and 2 respectively. Details of their constructionand the experimental set-up are given in SupplementalMaterial [24]. The dimers were designed so that theycould be made neutrally buoyant in a salt-water solution.The solution was vibrated vertically at a given frequency, f , amplitude, A . The dimensionless acceleration of thecell Γ = A (2 πf ) /g was varied between 1 −
20, where g is the gravitational acceleration. The frequency rangedfrom 30Hz to 135Hz.As the cell vibrated, each sphere had a different am-plitude and phase relative to the fluid motion due to dif-ferences in the size and/or densities of the two spheres.At low amplitudes of vibration of the cell the spheresoscillated vertically, but no net time-averaged motionof the center of mass of the spheres could be observedwithin experimental error. Beyond a certain thresh-old the dimer started to move; increasing the amplitudemade the dimer swim faster.To obtain the velocity of the dimer, the vibration wasinitiated abruptly under fixed Γ and f , and the motionof the dimer was filmed using a high-speed camera. The movies show that the separation of the spheres varied si-nusoidally (indicating that the coiled wire acted as a lin-ear spring to a good approximation). From such moviesthe steady-state velocity of the dimer, v , and the relativeamplitude of the two spheres with respect to each other, A r , was obtained. A r is the amplitude of the relativemotion of the two spheres that comprise the swimmer.Note that A r and the driving amplitude of the cell aredifferent; A r increases approximately linearly with A . Asfar as the motion of the spheres is concerned, in the restframe of the cell, A r and f are the only relevant drivingparameters. As can be seen from the movies [24], the mo-tion of the spheres was predominantly along the verticalline through their centers; there was very little sideways‘waggling’ movement.Fig. 1 shows the data obtained for the two asymmetricdimers, which swim upwards in the direction of the largersphere. The data collapse (within the scatter) when plot-ted in terms of the dimensionless combinations v/f L , andthe streaming Reynolds number Re s = A r /δ . Here L is the diameter of the larger sphere and δ = ( ν/ πf ) is the viscous length in terms of the kinematic viscosity ν . Re s is one of three dimensionless ratios that can bedefined from the length scales A r , L and δ and charac-terises the time-averaged (steady) flow [23, 29]. In ourexperiments L (cid:29) δ which results in a configuration ofthe time-averaged flow around each sphere that has in-ner and outer loops [23, 30]. The data are consistentbetween the measurements obtained from two nominallyidentical, asymmetric dimers, indicating that small dif-ferences in construction such as variations in the shapeof the loop of wire and of the shape and amount of gluehave little effect. The collapse in terms of Re s showsthat the motion is related to streaming flows generatedby the vibration of the dimer. The lower inset showsdata taken at low amplitudes of vibration and suggests asharp increase in velocity at Re s ≈ FIG. 2. Main panel shows the experimental data collapsefor the equal-sized swimmer. The driving conditions are thefollowing: blue stars f = 30Hz and green triangles right f =35Hz. Simulations for Γ between 2 −
12 and frequency f =30Hz are shown by the red plus symbols for comparison. Theinset shows a photograph of the swimmer when stationary. Fig. 3 (a) shows a photograph of the asymmetric dimertaken with an exposure time of one period of oscillation,revealing the motion of the tracer particles. In this imagethe dimer is close to the onset of motion. A downward jetoriginates from the vicinity of the lower sphere [24]. Sim-ilar behaviour was found for equal-sized spheres, exceptthat the strong jet was generated by the upper, lightersphere, causing the swimmer to swim downwards.In order to investigate the motion of the spheres andthe fluid in detail we used simulations which were basedon an embedded boundary method described previ-ously [30–34]. The fluid was assumed to obey the Navier-Stokes equations which were discretised on a staggeredmesh [35] and solved using the projection method [36]to ensure incompressibility of the fluid. The interac-tion between the fluid and the rigid spheres was achievedthrough the template model, which introduces a two-waycoupling between the particles and the fluid [33]. Thespheres were joined by a linear spring as in the experi-ments. An equal and opposite force was applied verticallyto the spheres to mimic the effects of static buoyancy,rather than imposing the effect of gravity directly on thefluid. The influence of vibration was introduced by ap-plying a sinusoidal acceleration to the fluid and particles,so that the simulations were carried out in the frame ofreference of the vibrated cell.The computational parameters of the swimmer (size,density and gap) and fluid (viscosity and density) werechosen to match the experiments. However, any interac-tion of the wire with the fluid was ignored and the dimerswere assumed to be made of perfect spheres. Details ofthe parameters used are given in the Supplemental Mate-rial. One difference between experiment and simulation
FIG. 3. Illustrations of the fluid flows generated by the vi-bration of the spheres from experiment and simulation. Panel(a) shows an image taken from experiment showing the flowaround the spheres. The arrow illustrates the direction of ajet of fluid evident from the movies (supplemental informa-tion [24]). Panels (b) and (c) show the direction of the time-averaged velocity field (i.e. the normalised velocity vectors)in the plane of the spheres. In (b) the swimmer is stationary( Re s = 15) while in panel (c) it is swimming ( Re s = 60).These figures illustrate the change in topology of the flows asthe amplitude of vibration increases. Note that the magni-tude of the flows is much greater around the smaller spherethan around the larger sphere, as seen in panel (a). is that the simulated cells are smaller due to computa-tional limitations. Examples of the simulated data areshown in Figs. 1 and 2 by the large red plus symbols.There is clearly good agreement between the simulationsand experiment despite the numerical limitations arising FIG. 4. Data collapse from simulations confirming the scalingbehaviour for different viscosities (red 1 . × − m /s, blue2 × − m /s, green 3 × − m /s). Each data set includessimulations for Γ between 2 −
20 and f = 65 , , , f = 65Hz (red line) and one thatswims for Γ = 12, f = 75Hz (blue line). The upper left insetis a snapshot from simulations showing the swimmer and thesimulated cell. from the simulated cell size and possible fluid lattice ef-fects.The simulations allow us to determine in more detailthe fluid flows generated by the motion of the spheres in-duced by the vibration of the cell. This flow is best illus-trated by plotting the direction of the velocity field in thevertical plane through the center of the two spheres. Ex-amples of these flows for the two unequal-sized spheres,time-averaged over a cycle, are shown in Fig. 3 (b) and(c). At low amplitudes, Fig. 3 (b), there are two outervortex rings around each sphere, marked by crosses. Thisis the flow pattern expected if the flows of the two spheresdo not interact strongly [29]. Under these conditions thetime-averaged center of mass of the two spheres remainsstationary: the dimer does not swim.As the amplitude increases, the flows grow in strength,but more importantly, the flows around each sphere startto interact strongly. The lower loop of the upper sphereis forced towards the surface of the sphere and reduces insize. Eventually, for sufficiently high amplitudes, thereare only three vortex loops, as shown in Fig. 3 (c). A jetof fluid directed downwards from the smaller sphere canbe observed from the plot of the normalised velocity field,Fig. 3 (c), and from experiment Fig. 3 (a). Under theseconditions, the swimmer moves upwards, in the oppositedirection to the jet.Simulations also allow us to vary parameters whichare not easily accessible experimentally, such as a widerrange of fluid viscosities, as shown in Fig. 4. When thedimer is moving there are four independent length scales: v/f , A r , L and the viscous length δ . We obtain the best data collapse if v/f is made non-dimensional by dividingby L rather than either of the other two length scales(see Supplemental Material [24]). Fig. 4 shows the simu-lation data plotted in this way indicating data collapse,the same way as the experimental data collapse shown inFig. 1. The lower right inset to Fig. 4. shows typicaltrajectories after vibration has been applied. There area few seconds of transient motion before the steady-statevelocity is reached.Figs. 1, 2 and 4 all show that v/f L scales approx-imately linearly with the streaming Reynolds number Re s for sufficiently large amplitudes A r . This behaviouris different from that observed for magnetic granularsnakes [37] and rigid dimers on surfaces [31]. A sim-ple argument can be constructed to explain the scalingbehaviour. Taking the unequal-sized swimmer as an ex-ample, the smaller sphere has a much larger amplitudeof motion than the larger sphere, (see movie in the Sup-plemental Material [24]). The smaller sphere acts as apump, imparting downward momentum to the fluid. Thereaction force on the small sphere is equal and oppositeto the rate of momentum transfer to the fluid. Its mag-nitude is proportional to the square of its speed ( f A r ) ,the fluid density, ρ , and a geometric factor proportionalto L . In this simple model, the force is balanced by theStokes’ drag on the larger sphere which scales as 6 πLηv with v the velocity of the swimmer and η is the dynamicviscosity of the fluid ( ρν ). By equating the two forces weobtain v/f L proportional to Re s = A r /δ as observed inthe data for large amplitudes.Note, however, that this particular scaling behaviouris not expected to hold generally as there are four in-dependent length scales in this problem, and thereforethree independent dimensionless ratios of lengths. Theargument presented above is only expected to hold in thelimit L (cid:29) δ .The analysis given above assumes a strong asymme-try of the flows around both spheres, an assumption thatbreaks down at lower Reynolds numbers, as shown fromthe flow patterns in Fig. 3. In both experiment andsimulation there appears to be a critical onset value of Re s ≈
20 for swimming to occur, obtained by extrap-olation of the data to v = 0. It has been argued thatasymmetric objects have a continuous transition to swim-ming [19]. This is not necessarily inconsistent with ourobservations. For Re s below the apparent onset it is dif-ficult to determine whether v is strictly zero or is justsmall: experimentally it is hard to ensure that any smallcentre-of-mass motion is not due to residual buoyancy;in simulation, lattice effects may influence the motionwhen the amplitudes of the spheres become comparableto the lattice spacing. The existence of an apparent on-set to motion has also been observed in an asymmetricflapping wing [17] and the ‘acoustic scallop’ [20]. Thegood agreement between the experiment and simulationfor our system allows us to conjecture that the apparentonset of motion arises from the change in topology of thestreaming flows.The examples presented here show a rich variety ofbehaviour but only represent a small part of the param-eter space. A systematic investigation into the influenceof the overall size of the dimer, the ratio of the spherediameters, the sphere density ratios and the gap widthwould be informative. It would be of interest to make afully self-propelled swimmer based on the relative vibra-tion of two spheres, driven by an internal linear motor,since such swimmers would not be constrained to movealong one axis. Collections of such swimmers might beexpected to exhibit interesting cooperative behaviour in-duced by interacting streaming flows [30, 32, 33, 38, 39].D.K. would like to thank Sharon Glotzer for sup-port and guidance. D.K. acknowledges FP7 Marie CurieActions of the European Commission (PIOF-GA-2011-302490 Actsa). R.J.A.H. acknowledges support from anEPSRC Fellowship; Grant No. EP/I004599/1. [1] S. Vogel, Life in Moving Fluids: The Physical Biology ofFlow (Princeton University Press, 1996)[2] H. Berg, Physics Today , 24 (2000)[3] E. M. Purcell, Am. J. Phys. , 3 (1977)[4] E. Lauga and R. E. Goldstein, Physics Today , 30(2012)[5] E. Lauga and T. R. Powers, Rep. Prog. Phys. , 096601(2009)[6] R. E. Goldstein, M. Polin, and I. Tuval, Phys. Rev. Lett. , 148103 (2011)[7] K. C. Leptos, K. Y. Wan, M. Polin, I. Tuval, A. I. Pesci,and R. E. Goldstein, Phys. Rev. Lett. , 158101 (2013)[8] R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A.Stone, and J. Bibette, Nature , 862 (2005)[9] B. J. Williams, S. V. Anand, J. Rajagopalan, andM. T. A. Saif, Nature Comm. , 3081 (2013)[10] Y. Bar-Cohen, Bioinspir. Biomim. , P1 (2006)[11] L. Zhang, J. J. Abbott, L. Dong, B. E. Kratochvil,D. Bell, and B. J. Nelson, App. Phys. Lett. , 064107(2009)[12] I. S. M. Khalil, H. C. Dijkslag, L. Abelmann, andS. Misra, App. Phys. Lett. , 223701 (2014) [13] J. Feng and S. K. Cho, Micromachines , 97 (2014)[14] S. Childress and R. Dudley, J. Fluid Mech. , 257(2004)[15] N. Vandenberghe, J. Zhang, and S. Childress, J. FluidMech. , 147 (2004)[16] S. Alben and M. Shelley, PNAS , 11163 (2005)[17] N. Vandenberghe, S. Childress, and J. Zhang, Physics ofFluids , 014102 (2006)[18] L. Xi-Yun and L. Qin, Phys. Fluids , 098104 (2006)[19] E. Lauga, Phys. Fluids , 061703 (2007)[20] R. J. Dijkinc, J. P. van der Dennen, C. D. Ohl, andA. Prosperetti, J. Micromech. Microeng. , 1653 (2006)[21] D. Ahmed, M. Lu, A. Nourhani, P. E. Lammert, Z. Strat-ton, H. S. Muddana,V. H. Crespi, and T. J. Huang, Sci-entific Reports , 9744 (2015)[22] V. A. Vladimirov, J. Fluid Mech. (2013)[23] N. Riley, Ann. Rev. Fluid Mech. , 43 (2001)[24] See Supplemental Materials for additional methods,movies, figures and references [25-28][25] V. S. Sorokin, I. I. Blekhman and V. B. Vasilkov, Non-linear Dynamics , 147 (2012).[26] F. H. Harlow and J. E. Welch, Phys. Fluids , 2182(1965).[27] K. H¨ofler and S. Schwarzer, Phys. Rev. E , 7146(2000).[28] K. D. Klotsa, PhD Thesis, University of Nottingham(2009).[29] F. Otto, E. K. Riegler, and G. A. Voth, Phys. Fluids ,093304 (2008)[30] D. Klotsa, M. R. Swift, R. M. Bowley, and P. J. King,Phys. Rev. E , 056314 (2007)[31] H. S. Wright, M. R. Swift, and P. J. King, Phys. Rev. E , 036311 (2008)[32] D. Klotsa, M. R. Swift, R. M. Bowley, and P. J. King,Phys. Rev. E , 021302 (2009)[33] K. D. Klotsa, PhD Thesis(2009)[34] H. A. Pacheco-Martinez, L. Liao, R. J. A. Hill, M. R.Swift, and R. M. Bowley, Phys. Rev. Lett. , 154501(2013)[35] F. H. Harlow and J. E. Welch, Phys. Fluids , 2182 (1965)[36] K. H¨ofler and S. Schwarzer, Phys. Rev. E , 7146 (2000)[37] M. Belkin, A. Snezhko, I. S. Aranson, and W. K. Kwok,Phys. Rev. Lett. , 158301 (2007)[38] R. Wunenburger, V. Carrier, and Y. Garrabos, Physicsof Fluids , 2350 (2002)[39] G. A. Voth, B. Bigger, M. R. Buckley, W. Losert, M. P.Brenner, H. A. Stone, and J. P. Gollub, Phys. Rev. Lett.88