A microbe stops the moment it stops pushing

For a swimmer one micrometre across, the water’s viscosity outweighs its inertia tens of thousands of times over. When the swimmer stops pushing, it stops within less than the width of an atom. All of its progress has to come from what its body is doing right now, and that forces a microbe to swim in ways a fish never needs.

Sandbox Physics · 2026-10-29

One number compares gliding with sticking

Push off from the wall of a swimming pool and you glide for about a body length. The water lets you travel because your momentum takes time to drain away. Shrink yourself to the size of a bacterium and the same water grips you hard enough to stop you almost at once.

Physicists compare these two effects with the Reynolds number. Multiply a swimmer’s size by its speed and by the density of the water, then divide by the viscosity, the property that makes honey harder to stir than water. A large result means inertia wins and bodies glide. A small result means viscosity wins and motion ends when the push ends.

Edward Purcell put rough values on this scale in a 1977 lecture. A person swimming sits near ten thousand, a goldfish near one hundred and a swimming microbe near one ten-thousandth. Eight powers of ten separate the person from the microbe.

\mathrm{Re}=\frac{\rho\,v\,a}{\eta}=\frac{v\,a}{\nu}

Less than an atom of gliding

Purcell worked one example in detail. Take an animal one micrometre across, swimming through water at thirty micrometres per second. Its Reynolds number is about 0.00003. If it suddenly stops swimming, it glides about a tenth of an ångström before it comes to rest, a small fraction of the width of one atom.

You can check the size of this yourself. A ball with the density of water and a radius of one micrometre loses its speed in about a fifth of a microsecond. At thirty micrometres per second it moves less than a tenth of an ångström in that time, the same tiny scale that Purcell found.

Purcell also turned the picture around. To feel what a microbe feels, imagine swimming in a pool of molasses where no part of your body may move faster than a centimetre a minute. If you could still cover a few metres in a couple of weeks, you would be swimming the way a microbe does.

Only the present push counts

Purcell summed up the consequence in one sentence: “what you are doing at the moment is entirely determined by the forces exerted on you at that moment, and by nothing in the past.” A microbe has no stored momentum to spend. Its velocity at each instant follows from the forces acting at that instant.

This has a surprising effect on swimming strokes. A fish can flick its tail quickly one way and bring it back slowly, and the fast flick wins. For a microbe the pace of a stroke drops out. Moving an arm slowly or quickly carries the body the same distance, so a stroke that simply runs backward brings the body back to where it started. Purcell called this the scallop theorem, and the third chapter proves it.

Microbes swim anyway. Bacteria turn corkscrew-shaped tails, sperm cells send bending waves along their tails, and the simple model in this course uses three beads that change their spacing around a loop. Each of these motions looks different going forward from coming back.

A wing leaves a wake behind

At high Reynolds number a moving body hands momentum to the fluid, and the fluid keeps it. A flapping wing sheds swirling vortices that drift on long after the wing has passed. Those vortices carry the momentum the wing gave away.

In a microbe’s world nothing lingers like that. When the stroke stops, the flow around the body stops with it. That is why the two simulations linked below look so different. The wing’s wake keeps evolving after every beat, while the three spheres of the swimmer move only while an arm is changing length.

An experiment to think through

Predict, change, explain

Before you open the three-sphere Lab, predict where the swimmer ends up if both arms shorten together and then lengthen together.
Run “Back-and-forth goes nowhere”, then “Four strokes swim forward”. Then open the flapping-wing Lab and watch the vortices that keep moving after each beat.
Explain why the wing leaves a moving trail behind, while the three spheres move only while an arm changes length.

Stroke a three-sphere swimmer Compare an inertial flapping wing

The three-sphere Lab couples three equal spheres through point-force Stokes flow in unbounded water. It leaves out walls, neighbours and thermal noise. The flapping-wing Lab uses a two-dimensional inviscid vortex-panel model, the inertial side of the comparison.

Try explaining it yourself

1. A goldfish and a bacterium stop swimming at the same moment. Which one glides farther, measured in its own body lengths?

Read an explained answer

The goldfish. Its Reynolds number is about a million times the bacterium’s, and the gliding distance in body lengths grows with the Reynolds number. The goldfish visibly glides on, while the bacterium halts within a fraction of an atom.

2. If Purcell’s microbe swam twice as fast, how would its Reynolds number change? Would it begin to glide?

Read an explained answer

The Reynolds number doubles to about 0.00006. The gliding distance also roughly doubles, to about two tenths of an ångström, still far below the size of an atom. Even a thousandfold increase in speed would leave viscosity far ahead.

3. Why does Purcell’s molasses swimmer have to move so slowly?

Read an explained answer

Bringing a person’s Reynolds number down toward a microbe’s takes a large viscosity and a small speed. Molasses raises the viscosity, and the limit of a centimetre a minute lowers the speed, so together they push a human-sized swimmer toward the range where a bacterium lives.

Sources to read alongside

Purcell, Life at low Reynolds number, American Journal of Physics 45, 3 (1977)
Pages 3 to 6 were read: the Reynolds-number estimates of Figure 3, the coasting example of Figure 4, the scallop theorem of Figure 6, the two-hinge swimmer of Figure 7 and the flexible oar and corkscrew of Figure 9. Pages 7 to 11 were not read for this course.
Lauga & Powers, The hydrodynamics of swimming microorganisms, arXiv:0812.2887v1 (2008); Reports on Progress in Physics 72, 096601 (2009)
Preprint version read. Sections 3 and 4 supply the Stokes equations, the point-force flow and the proofs of the scallop theorem; Sections 5 to 7 cover resistive-force theory, flexible filaments, walls and synchronisation.