---
title: "Three-Sphere Swimmer: Why Back-and-Forth Strokes Fail Interactive Lab | Sandbox Physics"
description: "Why can a microswimmer not move by repeating one stroke forward and back?"
canonical_url: "https://sandboxphysics.com/labs/three-sphere-swimmer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/three-sphere-swimmer/"
last_modified: "2026-10-29"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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](https://arxiv.org/abs/cond-mat/0402070) · [Read the 2008 paper](https://arxiv.org/abs/0711.3700)

Physics tutorial

## Distance comes from the loop, not the speed

**Background** A 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 matters** Najafi 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 misconception** A swimmer that pushes harder on the forward stroke than on the return should drift forward.

**Better mental model** Without 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 misconception** The drawn distance is what a real micron-sized swimmer would show in each cycle.

**Better mental model** 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.

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

To swim without inertia, a body must change shape along a loop. The loop’s direction sets the direction of travel, its enclosed area sets the distance, and its speed only sets how long the trip takes.

## Related experiments and articles

Microswimming · Reading path

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

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **A microbe stops the moment it stops pushing**](https://sandboxphysics.com/stories/microswimming/life-without-coasting/)
2. [Lab **Flapping Wing** Compare an inertial flapping wing](https://sandboxphysics.com/labs/flapping-wing/)
3. [Article **Stokes flow answers every push at once**](https://sandboxphysics.com/stories/microswimming/stokes-flow-has-no-memory/)
4. [Lab **Rotational Rheometer & Couette–Poiseuille Flow** Measure viscosity in a sheared gap](https://sandboxphysics.com/labs/couette-poiseuille/)
5. [Article **Opening and closing a shell gets you nowhere**](https://sandboxphysics.com/stories/microswimming/the-scallop-theorem/)
6. [Article **Three beads, two arms and a mobility matrix**](https://sandboxphysics.com/stories/microswimming/three-spheres-and-a-mobility-matrix/)
7. [Article **Distance per stroke is an area in shape space**](https://sandboxphysics.com/stories/microswimming/swimming-is-an-area/)
8. [Article **Flexible tails and turning corkscrews escape the scallop theorem**](https://sandboxphysics.com/stories/microswimming/flexible-oars-and-corkscrews/)
9. [Article **Reading Najafi and Golestanian’s three-sphere paper**](https://sandboxphysics.com/stories/microswimming/reading-najafi-golestanian/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)