Concept

Terminal velocity — where it appears

The speed at which a body falling or rising through a fluid has its weight exactly balanced by drag, so that it stops accelerating. It depends on which drag law applies, and the same body has different terminal velocities in different regimes.

Named by 6 essays across 3 fields — 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 $Two-thirds to one, and nothing outside it. The drag on a spherical drop as a fraction of Stokes' drag on a rigid sphere of the same size, against the ratio of the viscosities. It runs from exactly two-thirds for a clean gas bubble to exactly one for a rigid particle and takes no value outside that range, however light or heavy the drop is — the formula has no density in it anywhere. A drop of water in air and a drop of mercury in water sit at opposite ends of it.$

The surface that moves with the flow

A clean gas bubble feels two-thirds of the drag a rigid sphere of the same size would, and the formula has no density in it anywhere. What buys the third is that the bubble's surface is free to move — and real bubbles in ordinary water do not get it, for a reason that is a millionth of a per cent of the water by mass.

viscous · Mobile interface

Every ball's best angle is on one curve. The launch angle that carries a ball furthest over level ground, against β, the square of its launch speed measured in terminal speeds. At small β the air does nothing and the answer is 45°. It falls slowly as β grows: 41° at β = 1, 32.6° at 10 and 20.3° at 1000. A shot put sits at β = 0.009 and 44.9°, a golf ball struck without spin at β = 2.39 and 38.1°, a shuttlecock hit at 30 m/s at β = 19.5 and 30.2°.

The terminal speed sets the best launch angle

In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.

applied · Sport ball

Two spheres settle faster together than apart, faster in line than side by side, and still faster at thirty radii than either alone. The settling speed of a pair of equal spheres over a single sphere's, against the distance between their centres in radii: along the line of centres, exactly (the bispherical solution) and by Batchelor's far-field series; across it, by the series; dashed, each sphere's Stokeslet alone, 1 + 3/(2ρ) and 1 + 3/(4ρ). Touching in line the pair falls 1.55 times as fast; at 5 radii, 1.286 in line and 1.154 side by side; at 30, 1.05 and 1.025.

Two spheres settling at a slant keep their slant for ever, and three do not

Two equal spheres settling through a viscous liquid fall faster together than either alone — half as fast again if they touch in line, and still five per cent faster thirty radii apart. Slanted, the pair drifts sideways as it falls, by up to three and a half degrees, and keeps the slant, the separation and the drift for as long as it falls: the two spheres always move with exactly the same velocity. Release a third and the shape is lost at once, the middle sphere drawing ahead and the outer two closing in behind it. What a pair keeps and a trio does not is a statement about symmetry, and reciprocity is half of it.

viscous · Reciprocity

A sphere two per cent larger overtakes its neighbour round a path that is its own mirror image, and leaves at the offset it came in at. The centre of a sphere two per cent larger than its neighbour, relative to the smaller sphere, as it overtakes it in a viscous liquid, from four horizontal offsets; the circle is the distance at which they would touch. Each path bulges outward as it passes and is symmetric about the smaller sphere's level: from 0.85, 1.5, 2.5, 4 radii the larger sphere leaves at 0.8499, 1.5, 2.5, 4, after closest approaches of 2.27, 2.83, 3.74, 5.15 radii.

A larger sphere overtakes its neighbour and leaves on the line it came in on

Two equal spheres settling together never change their shape. Make one of them two per cent larger and the pair becomes an encounter: the larger sphere sinks past the smaller round a path that bulges out and comes back, slowed sixteenfold at contact but never stopped, and it leaves at exactly the offset it arrived with. The smaller sphere swings seven radii sideways while the larger is above it and gives every radius back while the larger is below. In Stokes flow, meeting a neighbour displaces nothing — unless the spheres touch, and a hundredth of a radius of roughness is enough to make every touching path leave on the same line.

viscous · Reciprocity $A flicker of a few per cent empties the suspension band and drains Stommel's traps more slowly. The share of a cloud released over one cell that is still within four cells of its start after the run, against the flicker's standard deviation as a fraction of the cells' mean strength, with a memory of three turnovers: inertialess particles falling at 0.3 of the cells' speed in still air, held in Stommel's traps, and particles of Stokes number 3 falling at 0.1, held in the suspension band. Stommel's traps hold 0.52, 0.41, 0.33, 0.16, 0.03, 0; the suspension band 0.22, 0.055, 0.015, 0.015, 0.005, 0, at flickers of 0, 0.05, 0.1, 0.2, 0.35, 0.5.$

Flickering vortices cannot hold a settling particle up, and still make it fall faster

A steady lattice of vortices can do two things to a heavy particle settling through it: hold it up, on closed paths or on orbits of its own, and speed its fall, by flinging it into the down-draughts. Let the vortices strengthen and fade at random, as real eddies do, and the two come apart. A flicker of five per cent of the cells' strength empties three quarters of the band of particles held up by their inertia; a flicker of a third drains Stommel's traps almost entirely. The speed-up hardly notices a flicker of a tenth, and is halved only when the cells swing by half their strength and remember it for a few turnovers. Trapping needs a steady cell; the speed-up does not.

regimes · Particle

Named alongside it

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

Creeping flow Drag Drop Model limit Reciprocity Reversibility Stokes flow Symmetry Boundary condition Bubble Dimensional analysis Dimensionless

All concepts