Low Reynolds number · Strokes as loops

Three-Sphere Swimmer: Why Back-and-Forth Strokes Fail

Why can a microswimmer not move by repeating one stroke forward and back? At this size water stops a body almost instantly, so the arm lengths alone fix how far it has moved. Running a stroke backward undoes every step, at any speed. A swimmer gains ground only when its two arms change out of step and trace a loop; a small loop moves it in proportion to the area enclosed.

Drive the two arms of a three-sphere swimmer in water. Compare a back-and-forth stroke with loops of different size and direction, and measure how far each cycle carries the body.

Two arms, one loop

Choose a stroke

Sphere radius

a

1\,\mu\mathrm{m}

Stroke size

s

1\times

Arm speed

W

30\,\mu\mathrm{m/s}

Drag an end sphere along the axis to stroke by hand, or drag a corner of the loop below the stage.

Fluid

Fluid viscosity

\eta

1\,\mathrm{mPa\,s}

Viscosity scales every force and the energy per cycle. The path, the distance per cycle and the period stay the same.

How the swimmer is computed

Three equal spheres on one line exert forces that sum to zero. Each sphere moves with its own drag plus the point-force flow of the other two:

v_i=\frac{f_i}{6\pi\eta a}+\sum_{j\ne i}\frac{f_j}{4\pi\eta\,|x_i-x_j|}

The body position is the centroid of the three spheres. Distances per cycle use eight-point Gauss–Legendre quadrature along each straight piece of the loop.

Model assumptions

Three equal spheres on one line in unbounded water, coupled by point-force (Oseen) hydrodynamics; the swimmer exerts no net force. The model needs the radius to be much smaller than the arms, so arms stay longer than three radii. Brownian motion is left out: in one cycle a 1 µm sphere diffuses farther than it swims, so a real swimmer of this size only shows its drift over many cycles. Fluid tracers use the same point-force field and are only qualitative within about one radius of a sphere.

Distance per cycle

0.172\,\mu\mathrm{m}

Loop area, with direction

16\,\mu\mathrm{m}^2

Small-loop area law

0.146\,\mu\mathrm{m}

Energy dissipated per cycle

6.93\,\mathrm{aJ}

Pink marks the body: its current shape on the loop and its position over time. A brighter background means more distance per unit of enclosed area; the dashed line is the average drift.

More numbers and the raw record

Cycle period $0.533\,\mathrm{s}$

Average swimming speed $0.323\,\mu\mathrm{m/s}$

Reynolds number $3\times10^{-4}$

Finite-size correction (Rotne–Prager minus point force) $-0.00707\,\mu\mathrm{m}$

The JSON record lists the model version, units, loop vertices, tracer seed, every parameter change and the sampled centroid positions.

Swimmer and stroke follow Najafi and Golestanian (2004); the area law is Eq. 13 of Golestanian and Ajdari (2008). Read the 2004 paper · Read the 2008 paper

Physics tutorial

Distance comes from the loop, not the speed

BackgroundA bacterium coasts about a tenth of a nanometre after it stops pushing. Water forces balance the swimmer’s own forces at every instant, so its velocity follows the current change of shape.

Why it mattersNajafi and Golestanian built the simplest body that can still swim: two arms whose lengths change. Plotting the two lengths as a point turns every stroke into a path, and the question of swimming becomes a question about that path.

Start with the essentials

Focus question: Why can a microswimmer not move by repeating one stroke forward and back?
One-sentence intuition: A path that retraces itself returns the body to its start. A loop leaves a net shift set by what it encloses; for a small loop that shift is the enclosed area times a coefficient fixed by the sphere size and arm lengths.

Core mathematical model

Force balance with no inertia

v_i=\sum_j M_{ij}f_j,\qquad M_{ii}=\frac{1}{6\pi\eta a},\quad M_{ij}=\frac{1}{4\pi\eta\,|x_i-x_j|},\qquad \sum_i f_i=0

Each sphere moves with the Stokes drag of its own force plus the flow created by the other two. The swimmer pushes on the water with zero total force. Given the arm rates, these three equations fix all three velocities.

Displacement is a line integral in shape space

\Delta X=\oint\left(A_1\,dL_1+A_2\,dL_2\right)=\iint\left(\frac{\partial A_2}{\partial L_1}-\frac{\partial A_1}{\partial L_2}\right)dL_1\,dL_2

Time has dropped out: only the path matters, so changing the arm speed changes the period but not the distance per cycle. A path enclosing no area gives zero. The background colour of the shape panel shows the integrand.

Small loops: the area law

\Delta X\approx K\,\mathcal{A},\qquad K=\frac{a}{3}\left[\frac{1}{\ell_1^{2}}+\frac{1}{\ell_2^{2}}-\frac{1}{(\ell_1+\ell_2)^{2}}\right]

For equal spheres and a loop small compared with the arms, Golestanian and Ajdari obtained this coefficient at leading order in radius over arm length. The readout compares it with the full model; the gap grows with loop size and with sphere radius.

Common difficulties

Faster is not further

Typical misconceptionA swimmer that pushes harder on the forward stroke than on the return should drift forward.

Better mental modelWithout inertia, force and velocity are proportional at each instant. A faster stroke covers the same shape path in less time, so it moves the body by exactly the same amount. Only the sequence of shapes counts.

What this model leaves out

Typical misconceptionThe drawn distance is what a real micron-sized swimmer would show in each cycle.

Better mental modelThree equal spheres on one line in unbounded water, coupled by point-force (Oseen) hydrodynamics; the swimmer exerts no net force. The model needs the radius to be much smaller than the arms, so arms stay longer than three radii. Brownian motion is left out: in one cycle a 1 µm sphere diffuses farther than it swims, so a real swimmer of this size only shows its drift over many cycles. Fluid tracers use the same point-force field and are only qualitative within about one radius of a sphere.

Run the experiment

01
Run the stroke backward
Choose Back-and-forth goes nowhere and watch a tracer near a sphere for several cycles.
What to observe: Every tracer and the body retrace their paths. The position trace oscillates around a flat line, and the distance per cycle reads zero.
02
Reverse a loop
Compare Four strokes swim forward with Reverse the loop, reverse the swim. Then drag one corner of the loop in the shape panel.
What to observe: The same loop run the other way gives the same distance with the opposite sign. Enlarging the loop toward short arms, where the panel is brighter, increases the distance most.
03
Make it slower or faster
Change Arm speed, then Stroke size. Finally drag the right-hand sphere back and forth in the scene to draw your own stroke.
What to observe: Arm speed changes the period and the energy per cycle but not the distance per cycle. Stroke size changes the distance roughly as the square of the size for small strokes. Moving one arm alone draws a line and gains nothing.

Three-Sphere Swimmer: Why Back-and-Forth Strokes Fail

Distance comes from the loop, not the speed

Core mathematical model

Force balance with no inertia

Displacement is a line integral in shape space

Small loops: the area law

Common difficulties

Faster is not further

What this model leaves out

Run the experiment

Run the stroke backward

Reverse a loop

Make it slower or faster

Related experiments and articles

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