Haw Far Can Bacteria Coast?

In last week’s blog post, I told you about the recent death of Howard Berg, author of Random Walks in Biology. This week, I present a new homework problem based on a topic from Berg’s book. When discussing the Reynolds number, a dimensionless number from fluid dynamics that is small when viscosity dominates inertia, Berg writes

The Reynolds number of the fish is very large, that of the bacterium is very small. The fish propels itself by accelerating water, the bacterium by using viscous shear. The fish knows a great deal about inertia, the bacterium knows nothing. In short, the two live in very different hydrodynamic worlds.

To make this point clear, it is instructive to compute the distance that the bacterium can coast when it stops swimming.

Here is the new homework problem, which asks the student to compute the distance the bacterium can coast.

Section 1.20

Problem 54. When a bacterium stops swimming, it will coast to a stop. Let us calculate how long this coasting takes, and how far it will go.

(a) Write a differential equation governing the speed, v, of the bacterium. Use Newton’s second law with the force given by Stokes law. Be careful about minus signs.

(b) Solve this differential equation to determine the speed as a function of time.

(c) Write the time constant, τ, governing the decay of the speed in terms of the bacterium’s mass, m, its radius, a, and the fluid viscosity, η.

(d) Calculate the mass of the bacterium assuming it has the density of water and it is a sphere with a radius of one micron.

(e) Calculate the time constant of the decay of the speed, for swimming in water having a viscosity of 0.001 Pa s.

(f) Integrate the speed over time to determine how far the bacterium will coast, assuming its initial speed is 20 microns per second.

I won’t solve all the intermediate steps for you; after all, it’s your homework problem. However, below is what Berg has to say about the final result.

A cell moving at an initial velocity of 2 × 10^-3 cm/sec coasts 4 × 10^-10 cm = 0.04 Å, a distance small compared with the diameter of a hydrogen atom! Note that the bacterium is still subject to Brownian movement, so it does not actually stop. The drift goes to zero, not the diffusion.

Berg didn’t calculate the deceleration of the bacterium. If the speed drops from 20 microns per second to zero in one time constant, I calculate the acceleration to be be about 91 m/s², or nearly 10g. This is similar to the maximum allowed acceleration of a plane flying in the Red Bull Air Race. That poor bacterium.