Illustrated Physics
physics, drawn
Concept

Vorticity — where it appears

A measure of the local spinning of a fluid, defined as the curl of its velocity and equal to twice the angular velocity of a small fluid element. It is created at walls and by density differences, carried and stretched by the flow, and conserved in suitable forms on a rotating planet.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

The whirlpool that comes in one size

Spin a bucket of ordinary liquid and it ends up turning with the bucket. Spin a bucket of superfluid helium slowly and it does not turn at all. Spin it faster and it does not turn either — until a threshold, at which a single line of circulation appears, carrying not some amount but exactly h/m. There is nothing in between, because the velocity is the gradient of a phase and a phase has to come back to itself.

fluids · Superfluidity Why the meander has to move west. A line of fluid columns at 45° pushed into a meander 3,000 km long and 300 km in amplitude, seen from above with north up and east to the right. Columns pushed north find the planet spinning faster beneath them and turn clockwise; columns pushed south turn anticlockwise. Each turning column drives the fluid north on one side of it and south on the other, and the arrows are that flow, from integrating the columns' spin along the line: it peaks at 2.32 m/s, a quarter of a wavelength west of each crest, so the columns just west of a crest are carried north and the crest itself is rebuilt further west. The dashed meander is the same pattern 2 days later, moved 638 km west at β/k² = 3.69 m/s. Nothing in the argument depends on which way the meander was first pushed.

The wave that can only travel west

A column of air or water on a spinning planet keeps one quantity as it moves: its total spin divided by its depth. Carry it north, where the planet's own spin about the local vertical is larger, and it must turn the other way to compensate. That single bookkeeping rule makes a wave that exists nowhere else — one whose crests can only ever move west — and the same rule holds the jet stream's meanders still over the mountains, makes the ocean take decades to notice a change in the wind, and pins the Gulf Stream to the western side of the Atlantic.

mechanics · Circular motion A rod in a shear flow turns in fits and starts. The angle between a rigid rod's axis and the flow direction, unwrapped, against the strain of a steady shear flow (shear rate times time), for spheroids of aspect ratio 1, 5, 20 lying in the plane of the shear, each starting along the flow. A sphere (r = 1) turns steadily at half the shear rate. A rod spends most of its time nearly aligned with the flow, turning slowly, and then flips through half a turn quickly; at r = 5 it is within 10° of the flow direction 46 per cent of the time, at r = 20, 82 per cent. Each flip is half of Jeffery's period, π(r + 1/r) units of strain.

The rod that tumbles for ever in a stream

Drop a short rod into honey being sheared between two plates and it does not settle into line with the flow. It lies nearly along the stream for a long time, then flips end over end in a moment, then lies along it again, and it keeps doing that for as long as the shear lasts. George Jeffery worked out the motion in 1922: every orientation belongs to a closed orbit, every orbit takes the same time, and the flow has no way to choose between them — so a suspension of rods never reaches a steady state, and its thickness rises and falls in time with the tumbling.

fluids · Viscosity

Named alongside it

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

ViscosityAngular momentumAspect ratioCirculationCondensateConservation lawCoriolis effectDispersion relationIrrotational flowPhasePotential vorticityQuantisation

All concepts