# Life at Low Reynolds Number

This is part of my “journal club for credit” series. You can see the other computational neuroscience papers in this post.

## Unit: Diffusion

Organized by Ben Regner

1. Standard Diffusion
2. Anomalous Diffusion
3. Life at Low Reynold’s Number

## Papers

Life at Low Reynold’s Number. By Purcell in 1977.

## Introduction

This is one of my favorite papers. The presentation style is extremely fun and readable without sacrificing any scientific integrity. I think it serves as a great introduction to fluid mechanics at low Reynold’s number. I don’t have too many comments since I think the paper explains it the best, but I will provide a few supplementary details for a more in depth exploration of the ideas from the paper.

And just to get you excited about fluid dynamics, I present an example of laminar flow:

## Basics of Fluid Mechanics

The fundamental equation of fluid mechanics is Navier-Stokes. The relevant version for this paper is the incompressible flow equations with pressure but no other external fields:

$\frac{\partial \vec{u}}{\partial t}+ \vec{u}\cdot\nabla\vec{u} +\frac{1}{\rho}\nabla p -\nu\nabla^2\vec{u}=0$

where $\vec{u}$ is the velocity vector, $\vec{x}$ is position, $\rho$ is density, $p$ is pressure, and $\nu$ is the kinematic viscosity. This equation can be made non-dimensional by the introduction of a characteristic velocity $U$, length $L$, and introducing the dynamic viscosity $\eta=\nu/\rho$. This gives the following dimensionless variables:

$u^* = \frac{u}{U}$

$x^* = \frac{x}{L}$

$p^* = \frac{pL}{\eta U}$

$t^* = \frac{L}{U}$

Substituting in these characteristic length scales and doing some algebra, one arrives at the simplified equations:

$R\frac{\partial \vec{u^*}}{\partial t^*}+ R\vec{u^*}\cdot\nabla^*\vec{u^*} +\nabla^* p^*-(\nabla^*)^2\vec{u^*}=0$

with only one dimensionless constant, the Reynold’s number, defined as:

$R = \frac{UL\rho}{\eta} = \frac{UL}{\nu}$

As explained in the paper, Reynold’s number is one of the essential constants describing a flow. High Reynold’s number leads to turbulent (chaotic) flow, while low Reynold’s number leads to laminar (smooth) flow. For extemely small Reynold’s number, Navier-Stokes simplifies to:

$\nabla^* p^* = (\nabla^*)^2\vec{u^*}$

which is also just called Stoke’s equation.

At the end of the paper, Purcell describes another dimensionless number which he calls $S$ and in a footnote identifies as the Sherwood number. However, Ben Regner pointed out, that Purcell’s $S$ would actually be called the Peclet number today.

## Basics of Ecoli Chemotaxis

Chemotaxis and cellular sensing really deserves its own series of papers. But in the meantime, I recommend the following resources

## Video Proof of Purcell’s Scallop Theorem

Reversible kicking does fine in water (high Reynold’s number)…

… but the same motion has issues in corn syrup (low Reynold’s number).

Here is a solution similar to what Ecoli and other bacteria employ.

## Fundamental Questions

• Purcell does an amazing job, so I have nothing to add.