Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Legendre — where it appears

The Legendre polynomials, the family of functions that describe how a quantity varies with latitude on a sphere. A drop's small oscillations, a sphere's surface pressure and the far field of an axisymmetric body are each expanded in them, one mode per polynomial.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A falling drop is a bun, not a tear. The shape of a drop pressed on by the air it is falling through, computed by matching the Legendre component of a sphere's own potential-flow pressure distribution against the change in curvature it produces. The result is oblate — flattened along the direction of travel — because the pressure is high at the poles and low round the equator. The teardrop of every illustration has the wrong symmetry entirely: a drop in free fall has no up, and one at terminal speed is being pressed on from in front.

The drop that is not a tear

A falling raindrop is flattened along the direction it is going, by the pressure of the air passing it rather than by its own weight, and the group that decides is the Weber number. The teardrop of every illustration has the wrong symmetry entirely — there is no up in the problem it is drawn for.

regimes · Drop shape The first three ways a drop can ring. A drop's first three modes of oscillation, each drawn at the two ends of its swing: the shaded outline and the red one are half a period apart. The fundamental, l = 2, alternates between a lemon and a lentil. The next two ring at 1.94 and 3.00 times its pitch, with three and four lobes. The swings are drawn at a fifth of the radius so they can be seen; the frequencies belong to swings much smaller than that.

A drop rings like a bell

Disturb a drop and it swings between a lemon and a lentil at a pitch set by its surface tension and its size, and it keeps swinging for dozens of cycles because water is nearly frictionless at that scale. The usual story has a falling raindrop's wake ringing it. The wake strikes several times too fast for the note the drop plays.

regimes · Drop shape

Named alongside it

The objects these essays reach for when they reach for this one.

DropSurface tensionDampingDimensionlessKinetic energyModel limitOscillationPerturbationPotential flowPressure coefficientResonanceStrouhal number

All concepts