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04-BS-7 · May 2013

Question 13 of 13: Shape of a Falling Raindrop

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examination, 2013-May. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Analytical, 4 questions, do 3); every question is answered below regardless of the exam's "do N of M" instruction, so the set is a complete study resource.

Reference texts: Crowe, C.T., Elger, D.F. & Roberson, J.A., Engineering Fluid Mechanics (the exam's own Moody chart and drag-coefficient chart are reproduced from this text); Douglas, J.F., Gasiorek, J.M., Swaffield, J.A. & Jack, L.B., Fluid Mechanics; White, F.M., Fluid Mechanics.

Check — assumptions used across this paper:
  • Where a question does not restate an ambient temperature, air is taken at the ISA sea-level standard of 15°C, giving ρair = 1.21 kg/m³ (the Constants table's 15°C value) — used in Q5, Q6, Q7 and Q9.
  • Q8's Moody chart and Q9's sphere drag-coefficient chart are supplied as attachments. Both are solved via the equations the charts themselves plot: the Colebrook–White equation for Q8 (the Moody chart is a graphical solution of Colebrook–White) and the Morrison (2013) curve-fit for sphere drag versus Reynolds number for Q9 (which reproduces the published "Sphere" curve to within a few percent over this Re range).
  • Q1, Q3 and Q11 depend on their figures; the relevant crop of the exam page is reproduced beside each, and every reading used is taken from it.

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.

Popular illustration shows raindrops as a pointed "teardrop," but that shape is a physics myth: no natural raindrop is pointed-prolate. The two forces that actually compete to shape a falling drop are surface tension, which always acts to minimise surface area and so pulls the drop toward a sphere, and aerodynamic pressure from the drop's own motion through the air, which pushes in on the leading (bottom) face and tends to flatten it.

Which force wins depends on drop size. Surface tension's restoring "stiffness" scales with the drop's radius (a smaller drop has proportionally more surface curvature per unit volume, so surface tension dominates), while the aerodynamic distorting force scales with the square of the fall (terminal) velocity and with the drop's cross-sectional area — and larger drops fall measurably faster and present more frontal area. The competition between the two is captured by the Weber number, $We=\rho_{air}V^2D/\sigma$, the ratio of aerodynamic (disruptive) force to surface-tension (restoring) force.

Evaluating $We$ at two representative sizes, using each size's own approximate terminal velocity, shows the trend clearly: for a small drop ($D\approx0.5\ \text{mm}$, $V_t\approx2.0\ \text{m/s}$), $We\approx0.03$ — surface tension utterly dominates and the drop is essentially spherical. For a large drop ($D\approx5\ \text{mm}$, $V_t\approx9.1\ \text{m/s}$), $We\approx6.9$ — aerodynamic pressure is now comparable to or exceeding surface tension, and the bottom of the drop flattens under the stagnation pressure while surface tension can only round off the edges, producing the classic "hamburger bun" oblate shape (flat or even concave underneath, rounded on top). Beyond about 4–5 mm, real raindrops become aerodynamically unstable in this oblate form and break apart into smaller drops, which is why raindrops rarely grow much larger than this.

Conclusion: a very large raindrop assumes an oblate (flattened) shape, not prolate/pointed and not perfectly spherical, because aerodynamic pressure on its underside overwhelms surface tension at that size; a very small raindrop, where surface tension dominates, remains essentially spherical.

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