Filtering with inertia

The guy with the great sideburns is George Stokes, a 19th physicist who made many contributions to physics, and after whom the Stokes number is named. In this blog post, I’ll show how his work helps us to understand how to filter out corona-virus laden droplets.

The Stokes number* is one of many dimensionless ratios in fluid mechanics. It tells us about the competition between two timescales, and it applies to particles, eg a droplet of mucus containing corona virus, moving in a flowing fluid, eg our breath.

The first timescale is basically the cornering timescale for a droplet, i.e., the time a droplet in flowing air, needs in order to go round a corner. In this case the corner will be provided by the fibres inside a mask. Masks are basically meshes of tiny fibres, and when we breathe through a mask the air has to go around these fibres inside the mask.

The second timescale is the cornering timescale for the air itself, i.e., the time the air needs to go round an obstacle such as fibre. As the Stokes number is just the first timescale divided by the second, when it is small, this means that droplets change direction faster than the air. Then the droplets follow the air. So when the air of our breath flows through a mask, droplets with small Stokes numbers tend to follow the air flow, and so go through the mask. This is bad of course, as we want the mask to filter out the droplets.

However, droplets with large Stokes numbers change direction more slowly than the air does, and so when the air changes direction to go around a fibre inside a mask, the droplet tends to carry on going in a straight line, and so crash into the material of the fibre. It is then collected by the fibre, and so filtered out. So the mask does its job here.

The Stokes number increases as the square of the droplet diameter –because large droplets have much more inertia than small ones, and this makes them slow to turn. This means that masks can efficiently filter out large droplets, by exploiting the fact that these larger droplets can’t follow the tortuous path of the air through the filter.

Here large droplets means a few micrometres and above. Unfortunately, the corona virus is only 100 nanometres across — ten times smaller. So we cannot use their inertia to filter out all the dangerous droplets, but their own inertia will remove the larger droplets. It is rather hard to determine what size of droplets we need to filter out, to reduce the transmission of viruses, but removing the larger ones is almost certainly useful.

* The Stokes number Wikipedia article was not so clear about what it was, and did not have a derivation, so I rewrote the introduction and first section this morning. Let me know if anything is unclear in those bits, and I will see if I can improve them.