I’ve been reading
The Big One: How We Must Prepare For Future Deadly Pandemics, by
Michael Osterholm and
Mark Olshaker. As is my wont, while I read I watched for physics applied to biology and medicine. And sure enough I found it in Chapter 2, where Osterholm and Olshaker discuss the difference between disease transmission by
droplets compared to
aerosols.
Droplets are tiny globs of liquid that come out of your nose or mouth when, say, you sneeze or cough… As small and generally unnoticable as these particles may be, they’re heavy enough to fall to the ground by force of gravity. Droplets travel short distances or sink to the nearest surface. This is where the guidance for maintaining six feet of social distancing came from during the Covid pandemic...
Aerosol particles come out of your nose and mouth as droplets do… If I’m in a room speaking, within minutes, small particles expelled from my mouth and nose will be floating in the air, even though no one may see or feel them. If you’re in that room, you’re going to inhale my particles and exhale particles of your own…
Droplets come largely from coughing or sneezing, and the droplet hits you in your nose, eyes, or mouth, like an incoming projectile. Compare these droplets to the free-floating aerosol particles circulating from that same cough, sneeze, or even just breathing. The aerosols are present in that same six-foot ‘social distance’ zone as the droplets are, but aerosols are also potentially present even yards away. You can see how the transmission of a respiratory pathogen via an aerosol versus a droplet is a game changer in terms of the ease with which a virus can be spread.
Because this blog is about physics in medicine and biology, let’s examine some of the physics that distinguishes droplets from aerosols. Much of the physics we need is in Intermediate Physics for Medicine and Biology.
Consider the motion of a particle in still air. For the moment, we’ll neglect gravity. Newton’s second law gives us an equation for the particle’s speed, v,
On the left is mass, m, times acceleration, dv/dt. One the right is Stokes’ law for the force of air friction. This doesn’t look exactly like Stokes’ law as written in IPMB because here we use the droplet’s diameter, d, instead of its radius, a, so the leading factor of six becomes three. The frictional force depends on the viscosity of the air, η. Anyone who has studied Chapter 2 of IPMB will recognize this differential equation as governing exponential decay. The velocity will decay with a time constant τ equal to m/3πηd. We prefer to write the mass in terms of the droplet’s density, ρ, where
so
This is the time needed for the particle’s speed to decay to zero relative to the air. Think of it as a relaxation time. If we had included the gravitation force, it would be the time constant for approaching a terminal speed. We know that the particle density will be close to the density of water, 1000 kg m–3, and the viscosity of air is about 1.8 × 10–5 N s m–2. So, we can make a table giving the relaxation time for different particle diameters.
| d (μm) |
τ (s) |
| 1 |
0.000003 |
| 10 |
0.0003 |
| 100 |
0.03 |
| 1000 |
3 |
After several of these time constants, the droplet will essentially flow with the air along a streamline. Because fluid flows parallel to a surface (a wall or ceiling), the particle will rarely hit a surface and adhere to it. Instead, it becomes part of the air we breathe.
What should we compare this relaxation time to? If you are in a room of size L, in which air is moving at a speed u, you can compare it to the time required to move across the room, L/u. If the room is 5 m long and has an air speed of 0.1 m s–1 (typical for indoor air circulation), 50 s would be needed to cross that room. We could call this the circulation time. All the relaxation times in the above table are shorter than 50 s, so all these particles flow along streamlines (ignoring gravity). The ratio of the relaxation time to the circulation time is called the Stokes’ Number. It is a dimensionless number—like the Reynolds Number discussed in Chapter 1 of IPMB—that governs the particle motion. If you had a really big particle, say a centimeter in diameter, the relaxation time would be 300 s, and it would move more like a ballistic billiard ball or a bullet; it would not have time to approach the speed of the moving air before it slammed into the wall of the room. The air motion would then be more or less irrelevant.
Now let’s put gravity in. Newton’s second law for the particle speed becomes
where g is the acceleration of gravity (for our purposes, take it as 10 m s–2). The particle will approach a terminal speed equal to gτ. If it approaches its terminal speed quickly (as the table above indicates it will), we can calculate the time T required for the particle to fall a distance H to the ground: T = H/gτ. Below I reproduce the table shown earlier, but with a column added for the fall time T. I’ll assume the fall distance is H = 2 m.
| d (μm) |
τ (s) |
T (s) |
| 1 |
0.000003 |
67,000 |
| 10 |
0.0003 |
670 |
| 100 |
0.03 |
6.7 |
| 1000 |
3 |
0.067 |
For the 1 and 10 micron aerosols, the fall time is much longer than the time for the particle to travel across the room. It take minutes or even hours to fall. For a 100 micron particle, the fall time is somewhat less than the time to cross the room. It will travel a ways, but not too far. For the giant 1 mm droplet, it falls in less than a second. The time listed above is probably too small, because the particle would not have time to reach its terminal speed. But the time would still be much smaller than the time to cross the room. It would get the floor dirty, but a person a couple meters away would not breath it.
The key question is, how big are the particles we spew out when we have Covid? If they are 1 or 10 microns in size, they are truly aerosols and would spread throughout the room as the air circulated. If they are tiny, say 0.1 micron in size, they become similar to the size of the Covid virus itself, so our model begins to break down. If they are large 1 mm droplets, they fall to the ground quickly, and are not carried by the air. If they are about 100 microns in diameter, they are in a transition zone, and would behave a little like droplets and a little like aerosols (but, I think, mostly like droplets).
This all seems rather simple. Indeed, it’s a toy model of particle spread, useful for getting insights into the important parameters, but not terribly accurate. Stokes’ Law is not universally valid. The model does not include such features as diffusion, turbulence, buoyancy, and evaporation. But the model does confirm Osterholm and Olshaker’s main conclusion: If the particles are big (droplets), they fall quickly and may causes surfaces to become infectious but social distancing will probably prevent infection through the air. If the particles are small (aerosols), they move with the air circulation and can carry the virus to anyone in the same room or building. Apparently Covid produces aerosols, and that’s why it’s so transmissible. Osterholm and Olshaker’s fear is that the “big one”—a viral disease that causes some future horrible pandemic—will similarly be spread by aerosols and therefore be easily transmissible, but will be more deadly than Covid. Yikes!
