Question 13 of 13: Shape of a Falling Raindrop

Question 13 — Shape of a Falling Raindrop (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

A raindrop's shape is set by the competition between surface tension, which always acts to minimise surface area for a given volume (favouring a sphere), and the aerodynamic pressure the falling drop experiences, which scales with dynamic pressure ½ρairV² and grows with drop size because larger drops fall faster at terminal velocity. The relevant dimensionless grouping is essentially a Weber number, We=ρairV²D/σ, comparing aerodynamic force to surface-tension force.

For a very small raindrop (roughly <1 mm), terminal velocity is low and surface tension σ=0.0728 N/m easily dominates the small aerodynamic pressure acting over its tiny surface area — the drop remains essentially spherical, the shape of minimum surface energy.

For a very large raindrop (several millimetres), terminal velocity is much higher and the dynamic pressure of the airflow acting on the underside of the falling drop becomes comparable to, and eventually exceeds, the surface-tension pressure holding it together. The drag pressure is greatest at the stagnation point on the drop's underside and pushes that face inward/flat, while surface tension can no longer pull the whole shape back to a sphere fast enough; the result is a flattened, hamburger-bun shape — oblate, with a flat or even slightly concave base and a rounded top, not the pointed "teardrop" often drawn in cartoons. Very large drops flatten enough that the aerodynamic pressure eventually overcomes surface tension entirely and the drop becomes unstable and breaks apart into smaller droplets, which is why raindrops rarely exceed about 5–6 mm in practice.

The classic "pointed prolate/teardrop" shape shown in the sketch is not physically realised by a drop falling freely under gravity and aerodynamic drag — the pointed tail requires a trailing attachment or a very different force balance (such as a drop still forming and detaching from a surface), not free fall.