Life at Low Reynolds Number - Part 1
Introduction.
Life at Low Reynolds Number is the title of a really fun paper, one which I think should be mandatory reading for chemical engineering undergraduates. The personal tone and sporadic humor is all part of the original, since the ‘paper’ is in fact a transcript of a lecture. The foreword ends with the statement: “Some essential hand waving could not be reproduced.”
I want to take you into the world of very low Reynolds number. This world is quite different form the one that we have developed our intuitions in.
The counter-intuitive nature of low-Re systems is something chemical engineers need to come to grips with sooner rather than later. To that end, this paper is insightful stuff. I like it because it illustrates the following really well:
- Momentum transfer and its counter-intuitive nature
- Combined mass and momentum transfer
- The usefulness of dimensional analysis
I started writing a post about it, and it quickly became a three part series. Let’s dive right in.
Who Cares?
We will answer that indirectly – by looking at forces. Remember – the Reynolds number is a ratio of inertial (push/pull) forces vs viscous forces. As humans (i.e. with our characteristic length and time scales), inertial forces dominate all the time. When is the last time you went swimming in a purely laminar regime? Never. That’s right, you inertial bad-boy.
Back to forces. What’s the force required to produce a (low) Reynolds number of 1?
\[Re = \frac{\rho v R}{\eta}\]Put \(Re = 1\) and we get
\[vR = \frac{\eta}{\rho}\]Now, the viscous force on a sphere is given by Stokes Law:
\[F = 6 \pi \eta v R\]Substitution for \(vR\) gives
\[F = \frac{6 \pi \eta^2}{\rho}\]That’s the force which pulls a sphere along with a Reynolds number of one. Drop the scaling factor for the sphere to get some force that pulls some object in a regime that has significant viscous forces.
\[\textrm{some force} = \frac{\eta^2}{\rho}\]Notice that viscosity and density are both fluid properties, hence some force is independent of the object being pulled. For water
\[\textrm{some force} = ~ 1 \times 10^{-9} \textrm{Newtons}\]Is a force of this magnitude significant for your system? If you are a submarine moving in the Indian ocean, clearly it’s not. If you are an E. Coli bacterium, then it probably is. Let’s talk more about E. Coli in Part 2!