I work on how water droplets start form in the air but got distracted today by the other end of the process: Why don’t they grow to huge sizes? Why don’t you see rain droplets the size of your head? The maximum size of a rain droplet as it falls through the Earth’s atmosphere is a few millimetres.

The answer is that a falling rain droplet that is more than a few millimetres in diameter is pulled apart by the force of air resistance as it falls through the Earth’s atmosphere.

Showing that droplets bigger than a few millimetres across are unstable is via a simple comparison of competing forces – this is something that is very common in physics. There are two forces at work: the force of air resistance, often called drag, and the force that tries to keep droplets together and spherical – the force of surface tension.

First, surface tension is a force per unit length (units of N/m) that tends to minimise surface areas of liquids. Here is an insect (water strider):

You can see that the water surface is deforming under the weight of the insect but it is supporting the weight: this force that is supporting the weight is the surface tension force. The surface tension of water,  γ, is around γ = 0.03 N/m. To get the force of surface tension for a droplet of a radius R we just multiply γ and R to get a force γR N (N = Newton the unit of force).

The other force is the drag. Lord Raleigh showed (some time ago) that for a falling object in air (that is not too small) the drag is roughly the mass density of air, ρ, times the area of the object, A,  times its velocity, u, squared. This velocity is here the terminal or maximum velocity a droplet reaches falling under gravity, which is around 100√R m/s.

The forces of drag and surface tension then balance when

γR = ρR²u²

where I took the area A to be roughly R². This is just a rough estimate. As u = 100√R we have

γR = ρR²10,000 R

or rearranging

R²  =  γ / ( 10,000 ρ )

here ρ is the mass density of air (not water) which is around 1 kg/m³. So we have that

R = √(0.03/10,000) = 2 mm    (approximately)

There ends the maths/physics lesson. The force of surface tension balances the drag force for droplets around approximately 2 mm across.

If you look at the second equation above you’ll see that the surface tension force increases as R while that of drag increases as R³. So if we double the droplet radius the drag force that tries to pull the droplet apart increases by a factor of 8 (=2³) while the surface tension force that holds the droplet together only increases by a factor of 2. Thus for bigger droplets drag wins and the droplets come apart into smaller droplets, while for smaller droplets surface tension wins and the droplets remain intact.

Incidentally, on another planet with lower gravity (but with air at the same density) the terminal velocity would be smaller and so droplets would be larger. Or on a planet with methane rain it would be more of a drizzle than rain as methane’s surface tension is lower than that of water so the droplets would be smaller.