Propulsion of a two-sphere swimmer

We describe experiments and simulations demonstrating the propulsion of a neutrally-buoyant swimmer that consists of a pair of spheres attached by a spring, immersed in a vibrating fluid. The vibration 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 speed obtained from both experiment and simulation agree and collapse onto a single curve if plotted as a function of the streaming Reynolds number, suggesting that the propulsion is related to streaming flows. There appears to be a critical onset value of the streaming Reynolds number for swimming to occur. We observe a change in the streaming flows as the Reynolds number increases, from that generated by two independent oscillating spheres to a collective flow pattern around the swimmer as a whole. The mechanism for swimming is traced to a strengthening of a jet of fluid in the wake of the swimmer.

The way objects propel themselves through a fluid has fascinated scientists of many disciplines, and the public alike, for aesthetic, practical and fundamental scientific reasons [1][2][3][4][5]. In biology and biomechanics the mechanisms behind the way organisms swim gives insight into their biological function and purpose [1,2,6,7]. Recently, the design of efficient robots able to navigate themselves through various fluids has become an important technological and medical challenge that brings together elements of physics, chemistry, biology, engineering and fluid mechanics [8][9][10]. Robot design covers scales from the microscopic, for targeted drug-delivery within the body [11,12] to the macroscopic, used for exploration, for example, in the ocean [13,14].
Purcell's scallop theorem states that at zero Reynolds number an object cannot swim using a time-reversible stroke: it will end up going back and forth with no net displacement [3]. Swimming micro-organisms such as bacteria and sperm therefore use flagella and cilia with non time-reversible strokes. A lot of experimental and theoretical research has gone into designing low Reynolds number 'microswimmers' that can break timereversal symmetry in various ways such as non-reciprocal strokes [8,9,15], collective hydrodynamic interactions for groups of swimmers [16,17], or asymmetric deformation of the swimmer shape [18].
Many types of small creatures, for example, insects and aquatic invertebrates, swim at intermediate Reynolds numbers . In fact some even change as they grow, from having a non-reciprocal stroke when they are small and the Reynolds number is low, to having a reciprocal stroke at higher Reynolds numbers [19]. In the latter case, time-reversal symmetry is broken by non-linearities in the fluid rather than by the nature of the stroke. For such swimmers, an interesting question arises: how does the motion evolve as the Reynolds number is increased from zero? It has been suggested that symmetrical objects with symmetrical strokes such as flapping wings have an onset for motion at a critical Reynolds number [19][20][21][22][23], whereas asymmetrical objects or strokes have a continuous transition as the Reynolds number is increased [24].
A central problem when designing an artificial swimmer for use in experiment is how to get energy into the system. Recently Vladimirov proposed a simple mechanism that may lead to swimming [25]. He suggested that a deformable object which is neutrally buoyant, but composed of coupled spheres with different densities can generate relative motion of its parts if immersed in a vibrating fluid. However, calculations in the creeping flow limit predicted that such an object will not swim if subjected to unidirectional oscillation. Here we ask the question can such a object be made to swim at higher Reynolds numbers, and, if so, what is the nature of the transition to swimming?
In this Letter we describe experiments and simulations demonstrating the propulsion of a neutrally-buoyant pair of spheres attached by a spring, immersed in a vertically vibrating fluid. Here we consider two particular realisations of such an object: one with unequal-sized spheres and the other with equal-sized spheres. In both cases, the density of the spheres is different from one another and from the liquid in which they are immersed. We find that both designs swim for sufficiently high relative amplitudes of vibration; the unequal-sized spheres swim upwards, in the direction of the larger, less dense sphere, whereas the equal-sized spheres swim downwards. The data for the swimming speed are found to collapse both in experiment and simulation when scaled appropriately with the streaming Reynolds number, suggesting that the streaming flows induced by fluid non-linearities play a central role [26]. Indeed, the onset of motion appears to be governed by a critical value of the streaming Reynolds number. The mechanism for propulsion is traced to a jet of fluid generated by the broken symmetry of these streaming flows.
The dimers were constructed from two spheres joined together by a small coil of wire. Examples of the asym-  [27]. The dimers were designed so that they could be made neutrally buoyant in a salt-water solution. The solution was vibrated vertically at a given frequency and the amplitude of vibration was slowly increased. The dimensionless acceleration of the cell Γ = Aω 2 /g, where A is the driving amplitude, ω the oscillation frequency and g the gravitational acceleration, varied between 1− 20. The frequency f = ω/2π ranged from 30Hz to 135Hz. At low amplitudes the spheres oscillated vertically, relative to each other, but no net time-averaged motion of the center of mass of the spheres could be observed. Beyond a certain threshold the dimer started to move; increasing the amplitude made the dimer swim faster.
To obtain the velocity of the dimer as a function of vibration amplitude, the vibration was initiated abruptly at a particular amplitude and the motion of the dimer was filmed using a high-speed camera. From such movies the steady-state velocity of the dimer, v, and the relative amplitude of the two spheres with respect to each other, A r , was obtained. As can be seen from the movies [27], the motion of the spheres was predominantly along the vertical line through their centers; there was very little sideways 'waggling' movement. Fig. 1 shows the data obtained for the two asymmetric dimers, which swim upwards in the direction of the larger sphere. The data is seen to collapse (within the scatter) when plotted in terms of the dimensionless combinations v/f L, and the streaming Reynolds number Re s = A 2 r ω/ν, where L is the diameter of the larger sphere and ν is the kinematic viscosity of the liquid. The data is consistent between the measurements obtained from two nominally identical, asymmetric dimers indicating that small differences in construction such as variations in the shape of the loop of wire and of the shape and amount of glue have little effect. The lower inset shows data taken at low amplitudes of vibration and suggests a sharp increase in velocity at Re s ≈ 20. Fig. 2 shows the behaviour of the symmetric dimer. The main reason for considering this system is that it is arguably one of the simplest objects that can be made to swim. In this case it moves in the opposite direction, downwards, with the heavier sphere at the front. The data illustrates that the speed and direction of motion depends on the densities and sizes of the spheres, as well as the gap between them; if the two spheres are sufficiently far apart, the dimer will not swim. Note that it was difficult to design dimers made of equal-sized spheres of different densities that can be made neutrally buoyant in salt solutions, and have sufficient mass difference between the spheres to generate enough relative motion to induce swimming.
We now ask the question: what is the cause of the motion? To address this we first imaged the flow using tracer particles illuminated by a planar laser sheet in the plane containing the centres of the two spheres of the dimer. Fig. 3 (a) shows a snapshot of the experiment on the asymmetric dimer close to the onset of motion. Clearly a downward jet can be seen originating from the vicinity of the lower sphere. Similar behaviour is found for equal-sized spheres, except that the strong jet is generated by the upper, lighter sphere, causing the swimmer to swim downwards.
In order to investigate in more detail the motion of the spheres and the fluid we used simulations, which were based on an embedded boundary method described previously [28][29][30][31][32]. The fluid was assumed to obey the Navier-Stokes equations which were discretised on a staggered mesh [33] and solved using the projection method [34] to ensure incompressibility of the fluid. The interaction between the fluid and the rigid spheres was achieved through the template model, which introduces a two-way coupling between the particles and the fluid [31]. The spheres were joined by a linear spring as in the experiments. An equal and opposite force was applied vertically to the spheres to mimic the effects of static buoyancy, rather than imposing the effect of gravity directly on the fluid. The influence of vibration was introduced by applying a sinusoidal acceleration to the fluid and particles, so that the simulations were carried out in the frame of reference of the vibrated cell.
The computational parameters of the swimmer (size, density and gap) and fluid (viscosity and density) were chosen to simulate an idealised version of the experiments as closely as possible, in which any interaction of the wire and the fluid was ignored and the dimers were assumed to be made of perfect spheres. Details of the parameters used are given in the Supplemental Material. One difference between experiment and simulation is that the simulated cells are smaller due to computational limitations. Examples of the simulated data are shown in Figs. 1 and 2 by the large red plus symbols. There is clearly good agreement between the simulations and experiment despite the numerical limitations arising from the simulated cell size and possible lattice effects.
The simulations allow us to determine the fluid flows generated by the motion of the spheres induced by the vibration of the cell. This flow is best illustrated by taking the curl of the velocity field in the vertical plane through the center of the two spheres. Examples of these flows for the two unequal-sized spheres, time-averaged over a cycle, are shown in Fig. 3 (b)-(d). At low amplitudes, Fig. 3 (b), there are two outer vortex rings around each sphere. The flows appear to be symmetric above and below each sphere. This is the flow pattern to be expected if the flows of the two spheres do not interact [35]. Under these conditions the time-averaged center of mass of the two spheres remains stationary; the dimer does not swim.
As the amplitude increases the flows grow in strength, Fig. 3 (c); more importantly, the flows around each sphere start to interact and break the symmetry above and below each sphere. This broken symmetry becomes very clear at the highest amplitude shown in Fig. 3 (d): a jet of fluid directed downwards from the small sphere can be inferred from the plot of the curl of the velocity. Under these conditions, the swimmer moves upwards, in the opposite direction to the jet.
Simulations also allow us to vary parameters which are not easily accessible experimentally, such as a wider range of fluid viscosities. When the dimer is moving there are four independent length scales: v/f , A r , L and the viscous length δ = (ν/ω) 1 2 . We obtain the best data collapse if v/f is made non-dimensional by dividing by L rather than either of the other two length scales (see Supplemental Material [27]). Fig. 4 shows the simulation data plotted in this way indicating data collapse, in the same way as the experimental data collapse shown in Fig. 1. The lower right inset to Fig. 4. shows typical trajectories after vibration has been applied. There are a few seconds of transient motion before the steady-state velocity is reached.
Figs. 1, 2 and 4 all show that v/f L scales linearly with the streaming Reynolds number Re s for sufficiently large amplitudes A r , above a critical onset. This behaviour is different from that observed for magnetic granular snakes [36] and rigid dimers on surfaces [29]. A simple argument can be constructed to explain the scaling behaviour. Taking the unequal-sized swimmer as an example, the smaller sphere has a much larger amplitude of motion than the larger sphere, (see movie in the Supplemental Material [27]). The smaller sphere acts as a pump, imparting downward momentum to the fluid. The reaction force on the small sphere is equal and opposite to the rate of momentum transfer to the fluid. Its magnitude is proportional to the square of its speed (f A r ) 2 , the fluid density, ρ, and a geometric factor proportional to L 2 . In this simple model, the force is balanced by the Stokes' drag on the larger sphere which scales as 6πLηv with v the velocity of the swimmer and η is the dynamic viscosity of the fluid (ρν). By equating the two forces we obtain v/f L proportional to A 2 r ω/ν as observed in the data for large amplitudes.
Note, however, that this particular scaling behaviour is not expected to hold generally as there are four independent length scales in this problem, and therefore three independent dimensionless ratios of lengths. The argument presented above is only expected to hold in the limit L δ. The analysis given above assumes a strong asymme-try of the flows around both spheres, an assumption that breaks down at lower Reynolds numbers, as shown from the flow patterns in Fig. 3. In both experiment and simulation there appears to be a critical onset value of Re s ≈ 20 for swimming to occur. Experimentally it is difficult to ensure that small centre-of -mass motion is not due to residual buoyancy, and in simulation lattice effects may influence motion for small velocities. Nevertheless, the good agreement between the apparent onset observed in experiment and simulation strongly supports our conjecture of a critical onset for swimming in this way.
The examples presented here show a rich variety of behaviour but only represent a small part of the parameter space. A systematic investigation into the influence of the overall size of the dimer, the ratio of the sphere diameters, the sphere density ratios and the gap width would be informative. It would be of interest to make a fully self-propelled swimmer based on the relative vibration of two spheres, driven by an internal linear motor since such swimmers would not be constrained to move along one axis. Collections of such swimmers might be expected to exhibit interesting cooperative behaviour induced by interacting streaming flows [28,30,31,37,38]. It would also be interesting to investigate whether streaming flows are present in systems of biological swimmers at lower Reynolds number. A possible example is Synechococcus, a cyanobacterium whose swimming mechanism remains unknown (it has no flagella or cilia) [39][40][41].
In summary, we have demonstrated the ability of a neutrally-buoyant pair of spheres, energised by vibration of the fluid, to swim. Such a system is arguably one of the simplest forms of swimmer. Simulations showed that the mechanism for propulsion arises from asymmetries in the streaming flow patterns induced by the spheres. Experiments and simulations suggest the existence of a critical value of the streaming Reynolds number for the onset of motion.