



What is a Reynold’s number

In fluid dynamics Reynold’s number is the ratio between inertial and viscous forces for an object moving through a fluid. It depends on the object’s dimensions – usually just one significant dimension – its velocity, and the fluid’s density and viscosity.


Reynold's number

In the formula, ‘ρ’ is the fluid’s density, ‘v’ is the object’s velocity, ‘L’ its significant dimension, and ‘μ’ the fluid’s kinematic viscosity.

For a person swimming in water ‘Re’ would be in the order of 104, for a small fish it can go as far down as 102. Which might be why they’re so difficult to catch with your bare hands. Water just feels more solid to them than to you.

But at the microscopic scale, for something like a paramecium or a bacteria, the ‘Re’ of water will go as far down as 10-4. At that scale, inertia simply stops existing.

At Reynold’s numbers that small, the only thing that matters are forces currently exhibited on a body. Past activity doesn’t matter and even time itself becomes irrelevant.

What does it mean to swim


Think about what swimming is like for a person in water. You push yourself off of the water and then, while you’re still moving ahead, move your arms and legs back into position so you can push off again. The reciprocal motion slows you down, but you make up for it on the next stroke.


Scallop theorem

Scallop theorem


Take for instance a scallop: Scallops swim by closing their shell quickly and opening it slowly, which produces a jet of water they can ride. At a low Reynold’s number they’d be moving back and forth in place.

To solve this problem, animals have to use cyclical motion that isn’t reciprocal.

The flagellum



One common solution for locomotion at a low Reynold’s number is the flagellum – a lash-like growth on the outside of the cell that can move like a whip.


  • the flexible oar
  • the corkscrew

The flexible oar works because it can bend in such a way it doesn’t push in the wrong direction when recoiling and a corkscrew works much like a ship’s propeller. Propellers are complicated and Purcell says it’s merely coincidental that our intuitive understanding matches what actually goes on.

Perhaps even more surprisingly, E. Coli uses its flagellum as a corkscrew. It’s odd for a living organism to have a rotary motor and joint, but experiments performed by Silverman and Simon showed this is the case.


Velocity components of glued E. Coli

Velocity components of glued E. Coli


How a corkscrew works

To be honest I don’t fully understand this part of the paper, but the basic tenet seems to be that because we’re talking about low Reynold’s number conditions, everything is linear. This means we can use matrices to describe the behaviour of a corkscrew.

We’re interested in force exhibited on the surrounding liquid and the screw’s torque. The linear equations for these can be expressed in a four element matrix.

Corkscrew as a matrix

Corkscrew as a matrix


A slinky works much better than a round wire though. Supposedly because it’s flat, which creates a better difference between the perpendicular and horizontal thing.


1% efficiency of a corkscrew

1% efficiency of a corkscrew

Why swim at all?


Nutrient differentials.

Microorganisms swim in their nutrients, which means they technically don’t have to do anything to feed themselves. In fact, they can’t do anything because viscosity is so high compared to other forces that their environment sort of sticks to them.

What this means is that no matter how much an organism stirs the fluid it’s surrounded by, nothing will change. The environment will be exactly the same unless they can stir at 20 times the speed they’re able to move at. Not practical.



Movement patterns of E. Coli

Movement patterns of E. Coli

As it turns out, observed bacteria choose a direction at random, then keep moving until the situation is better than what it was where they left off. If you hit a rough patch, it doesn’t make sense to stop moving, and if conditions are improving, they might improve a bit more if you keep moving.


Outrunning diffusion

Outrunning diffusion

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