Circulation against how fast the bucket turns

At its defaults it draws 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.

vortex-lattice is one function in lib/figures/fluids.js — matter that will not hold a shape, and the forces in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

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 head of liquid that a temperature difference of 1 millikelvin can support across a superleak — a plug fine enough that the normal fluid cannot pass and the superfluid can — against the temperature it is done at. Only the normal component carries entropy, so warming one side makes the superfluid flow toward the warm side until the pressure difference balances, and equilibrium is at ΔP = ρSΔT. The height that supports is SΔT/g, in which the density cancels exactly; computed both ways here the two agree to machine precision. The numbers are the striking part: 2.0 mm at 1.2 K, 4.6 mm at 1.4 K, 9.2 mm at 1.6 K, 19.4 mm at 1.8 K, 46.9 mm at 2 K, from a temperature step a thousand times smaller than anything a hand could feel. Aim a light at the warm side and the liquid does not merely rise but jets out of the tube, which is the fountain effect Allen and Jones found in 1938 and the most direct demonstration that helium II is two fluids rather than one. The entropies used are measured values; everything else on this chart is computed from them.

The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do.

Helium II does not conduct heat; it carries it. All the entropy is in the normal component, so a heat flux q is transported by that component moving at v = q/ρST, with the superfluid flowing back underneath it carrying none — two fluids passing through each other in opposite directions in the same tube. The curves are that speed against temperature for fluxes of 0.1 W/cm², 0.5 W/cm², 1 W/cm², 2 W/cm². At 1.8 K and 1 W/cm² the normal fluid runs at 0.202 m/s and the superfluid returns at 0.108 m/s. The consequence is the one that made helium II famous before anybody had a theory of it: the same flux through a metre of copper would need a temperature difference of 25 K, and helium II moves it on a difference of millikelvin — a thermal conductance four orders of magnitude better than the best metal. That is why helium II does not boil — no bubbles form, because no part of it is hotter than any other for long enough — and it is why superconducting magnets are cooled with it. The speed rises steeply at low temperature because the entropy falls: the colder the liquid, the less each kilogram of normal fluid can carry, and the faster it has to go.

A heat pulse released at one end of a 20 mm channel, and what a thermometer at the far end reads. In helium II the disturbance is a wave: it arrives at 1.00 milliseconds, which is the path divided by the second-sound speed of 20 m/s near 1.6 K, and it arrives as a pulse with a front. In an ordinary liquid the same disturbance diffuses, and the curve for helium I's thermal diffusivity peaks at 0 milliseconds and has no front at all — a diffusive signal is nonzero at the far end immediately and merely very small, so there is no arrival to time. That is the operational difference between the two, and it is how second sound was found: Peshkov released heat pulses in 1944 and timed them. Measuring the speed then measures the ratio of superfluid to normal density, so the model's one parameter is read off one of its own predictions.

Second-sound speed against temperature, computed from the two-fluid equations with the normal component treated as a phonon gas — which it is below about six-tenths of a kelvin. The upper line is ordinary sound at 238 m/s, which moves the two components together. The lower curve is the other mode, and at low temperature it sits at 137.4 m/s, which is 238/√3 to three figures: a result with no adjustable constant in it, and the reason to believe the two-fluid model rather than merely to use it. What oscillates in this wave is not the density — the two components move in opposite directions and their sum stays put — but the fraction that is normal, which is a temperature. So a temperature disturbance in helium II *propagates*, with a speed, a reflection and a resonance, where in every ordinary liquid it diffuses and has none of those. Above a kelvin the rotons take over from the phonons and the measured curve falls to about 20 m/s; the model here is the low-temperature one and it is drawn only where it holds.

physicscheck asserts something about vortex-lattice that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.