Question 11 of 13: Shape of a Falling Raindrop

Question 11: 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 very large raindrop falling at its terminal velocity assumes an oblate (flattened) shape, not a sphere and certainly not the pointed "tear-drop" prolate shape so often drawn in cartoons. The competing influences on a falling drop's shape are surface tension, which always acts to minimise surface area and pull the drop toward a sphere, and aerodynamic pressure, which builds up on the drop's underside as it falls through the air at increasing speed. For a large drop the aerodynamic (dynamic) pressure acting over its relatively large cross-sectional area becomes comparable to, and eventually exceeds, the stabilising surface-tension pressure (which scales as 2σ/r and therefore weakens as the drop grows). The high pressure on the flat underside pushes the base of the drop inward and outward radially, flattening it into a shape resembling a hamburger bun; at the largest stable sizes the underside can even develop a concave dimple before the drop becomes aerodynamically unstable and breaks apart into smaller droplets.

A very small drop, by contrast, has a much larger surface-tension pressure (2σ/r grows as r shrinks) relative to the aerodynamic pressure it experiences at its much lower terminal velocity, so surface tension dominates completely and the drop remains essentially spherical. The familiar pointed "tear-drop" silhouette is not physically observed in free fall at all — it is an artefact of drops clinging to and stretching from a surface (a tap or a leaf) under gravity before they detach, not of drops falling freely through air.