Question 13 of 13: Shape of a Falling Raindrop — Large vs. Small

Question 13: Shape of a Falling Raindrop — Large vs. Small (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 assumes an OBLATE (flattened) shape, not the popular pointed "teardrop"/prolate shape often drawn in cartoons — that pointed shape does not actually occur on a freely falling drop at all. The correct comparison is between a small drop (spherical) and a large drop (oblate, flattened, often with a slightly concave or dimpled underside at the very largest sizes, resembling a hamburger bun rather than a teardrop).

The shape a falling drop assumes is set by a competition between surface tension, which always acts to minimize surface area and pull the drop toward a sphere, and the aerodynamic pressure distribution generated by its motion through the air, which acts to deform it. For a very small drop, the terminal velocity is low (drag scales with the projected area and the square of velocity, but a small drop also has a very high surface-tension-to-size ratio), so the aerodynamic force is far too weak to overcome surface tension — surface tension wins outright and the drop remains essentially spherical, exactly as shown in the middle sketch.

A very large drop, however, falls much faster (terminal velocity increases with size until drag limits it), so the dynamic (aerodynamic) pressure acting on the drop's underside grows substantially, while the restoring surface-tension force per unit area actually falls as the drop gets bigger (surface tension force scales with the drop's circumference/radius, while its weight and cross-sectional area scale with radius squared and up, so the ratio shifts in favour of aerodynamic deformation as size increases). The relatively high pressure built up on the flat-on underside of the falling drop pushes the base outward and flat, while the top, moving through relatively undisturbed air, retains a rounded profile — producing the characteristic oblate, flattened-bottom shape. At even larger sizes this aerodynamic pressure can push the base concave (dimpled), and beyond a critical size the drop becomes aerodynamically unstable and breaks apart into smaller droplets, which is part of why raindrops rarely grow much beyond a few millimetres in diameter.

Comparison. Small drop: spherical, surface-tension-dominated, negligible aerodynamic deformation. Large drop: oblate/flattened, aerodynamic-pressure-dominated on the underside, surface tension too weak (relative to drop size) to hold a sphere. The transition between the two regimes is governed by the balance of these two effects (captured by the drop's Weber/Bond number), not by the fanciful pointed teardrop shape, which would require a force pulling the TOP of the drop into a point — a shape no physical mechanism on a falling drop actually produces.