Low Reynolds number · Swimmer interactions

Two Swimmers: Stroke Timing Decides Attraction

Why do two identical microswimmers drift together in step but apart out of step? Each swimmer stirs the water, and that stirring turns its neighbour slightly with every stroke. Whether the turns steer the pair together or apart depends on how their strokes line up. In this model two swimmers side by side and in step turn toward each other and meet after about 2,700 strokes; with strokes half a cycle apart they turn away and separate.

Set two three-sphere swimmers side by side and choose how their strokes line up. In step they draw together; half a cycle apart they push each other away.

Where Bob starts, and his timing

Pair the swimmers

Bob starts ahead by

\Delta x_0

0\,\mu\mathrm{m}

Sideways gap at the start

\Delta z_0

50\,\mu\mathrm{m}

Bob’s stroke runs ahead by

\varphi

0^{\circ}

Drag Bob in the scene to choose where he starts relative to Alice.

Time-lapse

Time-lapse speed

N

400\,\mathrm{cycles/s}

Strokes per second of animation. The arms are drawn slowed down to one stroke per second, so you can see whether the two strokes move together or half a cycle apart; a real stroke here takes 0.4 s.

How the swimmers are computed

Each swimmer has spheres of radius 1 µm on arms of 10 µm. Its arms shorten by 3 µm one at a time and then lengthen one at a time, at 30 µm/s, so each of the four moves takes the same time. Bob’s stroke can run ahead of Alice’s by a multiple of 45°.

One stroke of both swimmers is integrated exactly in time (fourth-order Runge–Kutta, 16 steps, each move starting on a step). The net change of both positions and headings is then integrated over thousands of strokes in steps of at most 25 strokes.

Model assumptions

Two swimmers move in one plane of unbounded water. All six spheres act as point forces coupled by the Oseen tensor, and each swimmer exerts no net force or torque on the water. The model fails when spheres of different swimmers come within a few radii, so a run stops there. Positions are shown once per stroke and accumulated over thousands of strokes. Thermal noise, which would scramble the timing of real swimmers, is left out; here the strokes stay exactly in step.

Sideways gap

47.4\,\mu\mathrm{m}

Alice’s heading

2.03^{\circ}

Bob’s heading

-2.03^{\circ}

Strokes so far

900

Pink is Alice, blue is Bob. Right: Alice’s turn with Bob at each bearing, 10 arm lengths away, against the leading-order shape of Pooley et al., matched at 90°.

More numbers and the raw record

Swimming time at 0.4 s per stroke $360\,\mathrm{s}$

Distance swum by the pair $74.1\,\mu\mathrm{m}$

Closest sphere-to-sphere distance $46.7\,\mu\mathrm{m}$

The JSON record lists the model version, units, geometry, every start or parameter change, both sampled tracks and the angular check values.

Swimmers, stroke and starting arrangements follow Pooley, Alexander and Yeomans (2007). Read the 2007 paper

Physics tutorial

Swimmers interact through the timing of their strokes

BackgroundA swimmer at low Reynolds number leaves the water around it stirred but not flowing on average, at least at long range. Yet two swimmers close together still steer each other, slowly, over thousands of strokes.

Why it mattersPooley, Alexander and Yeomans followed two identical three-sphere swimmers and found that whether they attract, repel or weave depends on the relative phase of their strokes as well as on where they sit. The largest effect comes from both swimmers moving at the same moment.

Start with the essentials

Focus question: Why do two identical microswimmers drift together when they stroke in step, but apart when their strokes are half a cycle apart?
One-sentence intuition: Each stroke turns the neighbour by a tiny angle, and the angle depends on where the neighbour sits and on how the two strokes line up. Side by side and in step the turns point the swimmers at each other; shift one stroke by half a cycle and the turns point them apart.

Core mathematical model

Every sphere feels the flow of all the others

\mathbf{v}_i=\frac{\mathbf{f}_i}{6\pi\eta a}+\sum_{j\ne i}\frac{1}{8\pi\eta\,r_{ij}}\left(\mathbf{I}+\hat{\mathbf{r}}_{ij}\hat{\mathbf{r}}_{ij}\right)\cdot\mathbf{f}_j

Six spheres, each pushed by its own force and carried by the Oseen flow of the other five. Each swimmer’s three forces add to zero, and so do their torques, so the swimmers can only rotate and translate in response to each other’s flow.

Turn per stroke for parallel swimmers

\Delta\theta_{A}\propto\Phi(\varphi)\,\frac{aD\xi^{2}}{r^{4}}\,\sin\alpha\left(3-30\cos^{2}\alpha+35\cos^{4}\alpha\right)

Pooley and co-workers give this leading term for two parallel swimmers with equal arm amplitudes ξ, arm length D, separation r and bearing α. The phase factor Φ changes sign between strokes in step and half a cycle apart. The angular factor vanishes near 31° and 70°; the lower chart compares it with the full model.

Common difficulties

Not just the average flow

Typical misconceptionEach swimmer only feels the flow that the other leaves behind on average.

Better mental modelAveraged over a stroke, these swimmers leave only a weak flow that fades as the cube of distance. The turn comes from both swimmers moving at the same moment, which is why shifting one stroke in time can reverse it.

What this model leaves out

Typical misconceptionReal swimmers keep their strokes locked together like this.

Better mental modelTwo swimmers move in one plane of unbounded water. All six spheres act as point forces coupled by the Oseen tensor, and each swimmer exerts no net force or torque on the water. The model fails when spheres of different swimmers come within a few radii, so a run stops there. Positions are shown once per stroke and accumulated over thousands of strokes. Thermal noise, which would scramble the timing of real swimmers, is left out; here the strokes stay exactly in step.

Run the experiment

01
Side by side, in step
Start with In step side by side and watch the two noses.
What to observe: The noses tilt toward each other, by about 19° at the end, and the gap closes over about 2,700 strokes, until the spheres come too close for the model.
02
Shift one stroke by half a cycle
Choose Opposite phase side by side, or set Bob’s stroke to run 180° ahead.
What to observe: The same two swimmers in the same places now turn about 4° away from each other and drift apart. The arms in the scene move half a cycle apart.
03
Put Bob ahead
Choose In step, one ahead, or drag Bob in front of Alice.
What to observe: Bob weaves across Alice’s line and back with a period of roughly 9,000 strokes, and the gap chart shows a repeating wave.

Two Swimmers: Stroke Timing Decides Attraction

Swimmers interact through the timing of their strokes

Core mathematical model

Every sphere feels the flow of all the others

Turn per stroke for parallel swimmers

Common difficulties

Not just the average flow

What this model leaves out

Run the experiment

Side by side, in step

Shift one stroke by half a cycle

Put Bob ahead

Related experiments and articles

Swimming without inertia, from Purcell’s scallop to pairs of swimmers