---
title: "Swimming without inertia, from Purcell’s scallop to pairs of swimmers"
description: "Nine chapters on swimming where viscosity wins, from Stokes flow and the scallop theorem to walls, flexible tails and swimmer pairs."
canonical_url: "https://sandboxphysics.com/stories/microswimming/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/microswimming/"
last_modified: "2026-10-29"
---

[Microswimming · A course with experiments](https://sandboxphysics.com/stories/microswimming/)

SWIMMING WHERE VISCOSITY WINS

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

A bacterium lives where viscosity wins and inertia is negligible, so it stops the moment it stops pushing. Start with why a back-and-forth stroke goes nowhere, then build a three-sphere swimmer and take it to a wall and to a partner.

Sandbox Physics · 2026-10-29

One swimmer in open water · Chapters 1–5

## Why a stroke has to trace a loop

See why a microbe stops the moment it stops pushing, prove the scallop theorem, then build the three-sphere swimmer and measure its step as an area.

[Start with a microbe that cannot coast →](https://sandboxphysics.com/stories/microswimming/life-without-coasting/)

Walls, tails and pairs · Chapters 6–9

## What changes beside a wall or a neighbour

Follow a swimmer beside a wall, see how flexible tails and corkscrews escape the theorem, time two swimmers against each other and read the original paper.

[Start beside a wall →](https://sandboxphysics.com/stories/microswimming/walls-change-the-flow/)

## Nine chapters, with examples and experiments

1. [01 L1 A microbe stops the moment it stops pushing What changes when a swimmer is so small that it cannot coast? Read and try →](https://sandboxphysics.com/stories/microswimming/life-without-coasting/)
2. [02 L2 Stokes flow answers every push at once Which equations govern the water around a microswimmer, and what do they predict for a single sphere? Read and try →](https://sandboxphysics.com/stories/microswimming/stokes-flow-has-no-memory/)
3. [03 L2 Opening and closing a shell gets you nowhere Why does any back-and-forth stroke bring a microswimmer exactly back to where it started? Read and try →](https://sandboxphysics.com/stories/microswimming/the-scallop-theorem/)
4. [04 L2 Three beads, two arms and a mobility matrix How do you find how fast three spheres on a line move when you prescribe only how their arms change? Read and try →](https://sandboxphysics.com/stories/microswimming/three-spheres-and-a-mobility-matrix/)
5. [05 L3 Distance per stroke is an area in shape space Why does a small stroke loop carry the swimmer a distance proportional to the area it encloses in the plane of arm lengths? Read and try →](https://sandboxphysics.com/stories/microswimming/swimming-is-an-area/)
6. [06 L3 A wall slows each sphere and turns the swimmer How does a nearby wall change the mobility of each sphere, and what paths does a three-sphere swimmer follow beside it? Read and try →](https://sandboxphysics.com/stories/microswimming/walls-change-the-flow/)
7. [07 L3 Flexible tails and turning corkscrews escape the scallop theorem How do real microorganisms make strokes that never retrace themselves, and how fast do those strokes push them? Read and try →](https://sandboxphysics.com/stories/microswimming/flexible-oars-and-corkscrews/)
8. [08 L4 Two swimmers feel each other through the timing of their strokes When two three-sphere swimmers stroke near each other on fixed clocks, how does the phase between their strokes decide whether they attract, repel or oscillate? Read and try →](https://sandboxphysics.com/stories/microswimming/two-swimmers-keep-time/)
9. [09 L4 Reading Najafi and Golestanian’s three-sphere paper What does the 2004 paper show, which of its numbers can you reproduce, and where do later papers correct or extend it? Read and try →](https://sandboxphysics.com/stories/microswimming/reading-najafi-golestanian/)

## Keep the experiments beside the chapters

Three Labs drive three-sphere swimmers: alone, beside a wall and in pairs. The flapping wing and the rheometer show the inertial and viscous sides of the comparison.

[Stroke a three-sphere swimmer ↗](https://sandboxphysics.com/labs/three-sphere-swimmer/) [Release a swimmer beside a wall ↗](https://sandboxphysics.com/labs/swimmer-near-wall/) [Set two swimmers in or out of step ↗](https://sandboxphysics.com/labs/swimmer-pair-phase/) [Compare an inertial flapping wing ↗](https://sandboxphysics.com/labs/flapping-wing/) [Measure viscosity in a sheared gap ↗](https://sandboxphysics.com/labs/couette-poiseuille/)