A pump up close, a weak field far away
Pooley, Alexander and Yeomans study the three-sphere swimmer with separate stroke amplitudes for its front and back arms. Averaged over a cycle, its far flow is a sum of point-force singularities. When the two amplitudes are equal, the stroke looks the same if you play it backward in time and flip the swimmer front to back, a property the paper calls TP invariance. Then the dipolar part of the mean flow vanishes and the far field falls as the inverse cube of distance, in line with Golestanian and Ajdari’s symmetric case.
Close to the swimmer the averaged flow looks different. Near the middle sphere the fluid moves backward: the swimmer acts as a pump that carries fluid from in front of it to behind it, and that is how it moves forward. The paper’s analytic far-field expression matches its numerical flow to within ten percent beyond about three swimmer lengths.
Dead and live terms
Now place a second swimmer nearby; the authors call the two Alice and Bob. Bob’s effect on Alice has two parts. The dead term is what Alice would feel if Bob’s cycle-averaged flow simply carried and turned her. It depends only on Bob’s flow and takes the expected dipolar form. Its size is proportional to the product of Bob’s front and back amplitudes times their difference, so it vanishes when they are equal.
The live term comes from both swimmers moving at once. Bob’s flow pushes on Alice’s spheres at particular moments of her own stroke, and the products of those instantaneous pushes leave a residue after averaging. Its size depends on the relative phase of the two strokes through a phase function that the paper plots in Figure 3. Equation 8 compares the two terms and finds the live term dominant.
Worked example
How many strokes before they meet
Figure 4(i) of the paper starts Alice at the origin and Bob five arm lengths above her, both aligned and in phase. All lengths are in units of the arm length: the sphere radius is one tenth and the stroke amplitude three tenths. The paper gives the step per cycle as seven twelfths of radius times amplitude squared over arm length squared, about 0.005 arm lengths.
This is the small-loop area law of Chapter 5. With both arms equal to the arm length, the coefficient is seven twelfths of radius over arm length squared, and the loop is a square whose side is the amplitude. The step comes out at 0.00525 arm lengths. The swimmers travel about 25 arm lengths before they collide, so the run lasts about 4,800 cycles, consistent with the paper’s figure of about five thousand. The trajectories stop when the swimmers come so close that the point-force approximation fails.
Now put in sizes. With ten-micrometre arms, one-micrometre spheres, three-micrometre strokes and arms moving at thirty micrometres per second, a cycle takes 0.4 seconds, so 4,800 cycles take about 32 minutes. In that time a single one-micrometre sphere in water wanders about 28 micrometres along any one direction by thermal motion. The three-sphere body wanders somewhat less, still several arm lengths. This estimate is ours, made with the diffusion coefficient from Chapter 2, and it shows how slowly the pair effects build compared with Brownian motion at this size.
The phase decides the sign
For in-phase swimmers with equal amplitudes, Figure 4(a) divides Bob’s starting positions into four regions: attraction, repulsion, oscillation and parallel paths. Besides the attracting pair above, a repelling pair turns each swimmer through about seven degrees, mostly within 3,000 cycles. In the oscillating region the swimmers oscillate relative to each other, one oscillation taking about 10,000 cycles, with a slow drift on top. The authors trace the oscillation to the rotational part of the live term and the drift to its translational part.
Shift the swimmers half a cycle out of phase and the live term changes sign. In Figure 4(b) the attracting region becomes repelling, and the oscillating region turns into parallel paths, tilted slightly from the swimming direction, along which the two swimmers wobble exactly half a period apart. With a quarter-cycle lag, Figure 4(c), the lowest-order live term vanishes. The behaviour still differs for a partner in front and a partner behind, which the authors read as a sign that live terms still dominate.
Unequal amplitudes break the perfect periodicity of the in-phase oscillations. A smaller front amplitude makes the oscillations grow until the swimmers diverge. A larger front amplitude makes them shrink and the swimmers close in, which suggests that in-phase suspensions might form chains. The authors add that attraction between two swimmers is insufficient evidence for clustering in larger groups.
What the pair Lab reproduces
Our calculation: the pair Lab uses the sizes above and couples all six spheres of the two swimmers as point forces. Side by side, five arm lengths apart and in step, the two swimmers turn toward each other and meet after about 2,700 strokes. For a stroke of three tenths of an arm, the full point-force step is 0.0824 micrometres per cycle, about 1.6 times the small-loop estimate. At that step, 25 arm lengths take about 3,000 cycles, close to the Lab’s count. The paper quotes the small-loop step for its runs, and we have not traced how its figure of five thousand was computed.
Half a cycle out of step, the same pair turns apart, as the sign flip of the live term predicts. Far apart, Alice’s turn per stroke varies with Bob’s bearing, measured from her swimming direction, with the angular shape of the leading-order result of Pooley and colleagues. The Lab plots the two side by side.
Fixed clocks and real synchrony
Every stroke in this paper runs on its own fixed clock. The calculations show what a given phase does to the trajectories, and the phases themselves stay put. Pooley and colleagues note that externally driven artificial swimmers could be held strictly in phase, which makes their predictions testable, while autonomous organisms would let their phases drift, perhaps toward phase locking.
Lauga and Powers review when phases do lock. Taylor showed that two swimming sheets with the same prescribed waveform dissipate least when they beat in phase. Later simulations found that such sheets drift until they are exactly in phase or exactly out of phase, with both states stable. For real flagella the waveform itself responds to the forces from a neighbour, and that feedback is expected to allow perfect phase locking. Rotating helices tell a similar story: perfectly rigid ones fail to synchronise, and some flexibility is needed. A model of synchrony therefore lets each stroke respond to the force it meets.