Intermediate Physics for Medicine and Biology: Life at Low Reynolds Number

“Life at Low Reynolds Number,”
by Edward Purcell.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite Edward Purcell’s wonderful article “Life at Low Reynolds Number” (American Journal of Physics, Volume 45, Pages 3–11, 1977). This paper is a transcript of a talk Purcell gave to honor physicist Victor Weisskopf. The transcript captures the casual tone of the talk, and the hand-drawn figures are charming. Below I quote excerpts from the article, including my own versions of a couple of the drawings. Notice how his words emphasize insight. As Purcell says, “some essential hand-waving could not be reproduced.” Enjoy!

I’m going to talk about a world which, as physicists, we almost never think about. The physicist hears about viscosity in high school when he’s repeating Millikan’s oil drop experiment and never hears about it again, as least not in what I teach. And Reynolds’s number, of course, is something for the engineers. And the low Reynolds number regime most engineers aren’t even interested in… But I want to take you into the world of very low Reynolds number—a world which is inhabited by the overwhelming majority of the organisms in this room. This world is quite different from the one that we have developed our intuitions in…

Based on Figure 1 from
“Life at Low Reynolds Number.”

In Fig. 1, you see an object which is moving through a fluid with velocity v. It has dimension a,… η and ρ are the viscosity and density of the fluid. The ratio of the inertial forces to the viscous forces, as Osborne Reynolds pointed out slightly less than a hundred years ago, is given by avρ/η or av/ν, where ν is called the kinematic viscosity. It’s easier to remember its dimensions; for water, ν = 10⁻² cm²/sec. The ratio is called the Reynolds number and when that number is small the viscous forces dominate… Now consider things that move through a liquid... The Reynolds number for a man swimming in water might be 10⁴, if we put in reasonable dimensions. For a goldfish or a tiny guppy it might get down to 10². For the animals that we’re going to be talking about, as we’ll see in a moment, it’s about 10⁻⁴ or 10⁻⁵. For these animals inertia is totally irrelevant. We know F = ma, but they could scarcely care less. I’ll show you a picture of the real animals in a bit but we are going to be talking about objects which are the order of a micron in size… In water where the kinematic viscosity is 10⁻² cm²/sec these things move around with a typical speed of 10 μm/sec. If I have to push that animal to move it, and suddenly I stop pushing, how far will it coast before it slows down? The answer is, about 0.1 Å. And it takes it about 0.6 μsec to slow down. I think this makes it clear what low Reynolds number means. Inertia plays no role whatsoever. If you are at very low Reynolds number, what you are doing at the moment is entirely determined by the forces that are exerted on you at that moment, and by nothing in the past…

Based on Figure 18 from
“Life at Low Reynolds Number.”

Diffusion is important because of [a] very peculiar feature of the world at low Reynolds number, and that is, stirring isn’t any good… At low Reynolds number you can’t shake off your environment. If you move, you take it along; it only gradually falls behind. We can use elementary physics to look at this in a very simple way. The time for transporting anything a distance ℓ by stirring is about ℓ divided by the stirring speed v. Whereas, for transport by diffusion, it’s ℓ² divided by D, the diffusion constant. The ratio of those two times is a measure of the effectiveness of stirring versus that of diffusion for any given distance and diffusion constant. I’m sure this ratio has someone’s name but I don’t know the literature and I don’t know whose number that’s called. Call it S for stirring number. It’s just ℓv/D. You’ll notice by the way that the Reynolds number was ℓv/ν. ν is the kinematic viscosity in cm²/sec, and D is the diffusion constant in cm²/sec, for whatever it is that we are interested in following—let’s say a nutrient molecule in water. Now, in water the diffusion constant is pretty much the same for every reasonably sized molecule, something like 10⁻⁵ cm²/sec. In the size domain that we’re interested in, of micron distances, we find that the stirring number S is 10⁻², for the velocities that we are talking about (Fig. 18). In other words, this bug can’t do anything by stirring its local surroundings. It might as well wait for things to diffuse, either in or out. The transport of wastes away from the animal or food to the animal is entirely controlled locally by diffusion. You can thrash around a lot, but the fellow who just sits there quietly waiting for stuff to diffuse will collect just as much.