Terminal Velocity

Raindrops Keep Falling on My Head

Small drops (drizzle size stuff) fall according to Stokes Law, i.e. terminal velocity (maximum velocity reached) is a function of the radius of the drop squared (r^2)! So, where a 0.01 mm drops falls at 0.0001 m/s [I don't think that is even falling]. Actually at this rate it would take some 80 days to reach the ground. A 0.05 mm drop falls at 0.1 m/s. If it were as simple as this, then a 1 mm modest sized drop would fall at a hold-on-to-your-hard-hat terminal velocity of something > 10,000 m/s. It is to our good fortune that our world was crafted such that Mr. Stokes and his law don't apply after about .08 mm drop size. Bigger drops fall by Mr. Euler's rule where terminal velocity is proportional to the square root of the radius of the drop (r^0.5). So even a giant 10 mm drop only falls at about as fast as a sprinter sprints: 10 m/s. Even a nylon umbrella can with stand the pounding at this speed.

Mr. Stokes rain drops are tiny, slow falling and spherical. These drops leave no wake of turbulence to slow their terminal velocity. Mr. Euler’s falling drops do leave a wake in their fall. The turbulence is a drag on the fall and slows its progress. The drag also tends to flatten out the rain drops in Mr. Euler’ world. The rain drops are no longer spherical. Rather, they resemble a falling hamburger. Or, in the Shenandoah Valley with its German heritage, the whoopie pie would be the proper cauterization of shape Euler’s falling raindrop. It is out with the tear drop shape, in with the sphere and the burger!

Aerosols in the atmosphere that serve as nuclei of condensation are generally less than 1 micron in size (one millionth of a meter). Aerosols without condensed water around them give rise to the blue of the Blue Ridge Mountains. If there is water around the aerosols then the whitish of the Great Smokey Mountains to our self. These nuclei of condensation have the property of hydroscopisity. That means they are wetable or hydrophylic. Aerosols that shun getting wet are said to be hydrophobic.

Creeping of cloud drops is followed by drizzle and in turn is followed showers. A sudden shower is made up of drops falling at about the same fall speed. In my opinion, IMO, in modern tweet-speak, it is the sudden shower that is most refreshing.

In Mr. Stokes’ region droplet terminal velocity increases with the square of the radius while Mr. Euler’s region terminal velocity increases with the square root of the radius. As a result, Mr. Euler’s drops speed up much more slowly than Mr. Stokes’ drops. So, at birth, raindrops speed up faster and faster while big drops it is “not so much”. At 500 microns, an Euler drop is falling at 2.2 mph while Mr. Stokes drop is falling at 222 mph. That is painfully fast. It would be best not to go outside. Soundproofing of buildings would be advised. It would be a different world if big drops were subjected to Mr. Stokes region. Rain-wise it is the best of all worlds.


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