Illustrated Physics
physics, drawn
Concept

Two-fluid model — where it appears

The description of a superfluid as two interpenetrating components with two densities and two velocity fields. One carries all the entropy and viscosity and the other carries neither, which is bookkeeping rather than two kinds of atom — and it predicts the fountain effect, the counterflow and second sound from one measured ratio.

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

The height a 1 mK difference lifts helium. 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 fountain a lamp can drive

Below two degrees above absolute zero, liquid helium behaves as though it were two fluids occupying the same space — one carrying all the entropy and all the viscosity, the other carrying neither. It is not a metaphor and not a mixture. Shine a light on one side of a fine plug and the liquid jets out of the tube, because a temperature difference of a thousandth of a degree is a pressure of a hundred pascals.

fluids · Superfluidity The speed of a wave that carries no pressure. 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.

The heat that arrives as a wave

Two fluids with two velocities give two wave equations, not one. In the first the components move together and the density oscillates, which is ordinary sound. In the second they move oppositely, the density stays put, and what oscillates is the temperature — so a heat pulse in liquid helium has a speed, a front and a reflection.

fluids · Superfluidity A beaker that empties at a steady rate. The height of liquid helium inside a beaker 1 cm across, held 1 cm above the level of the bath it stands in, against time, as the film carries liquid up the inner wall, over the rim and down into the bath. The film carries a fixed volume per centimetre of rim — 7.5·10⁻⁵ cm³/s at 1.4 K — whatever the difference of levels, so the level falls in a straight line, 180 μm a minute, and the beaker is level with the bath in 56 minutes. At 1.9 K, with less superfluid, it takes 95 minutes; at 2.1 K, 279. Dashed, a flow driven by the difference of levels, like a siphon, starting at the same speed: it slows exponentially as the difference shrinks and never quite finishes. The film's straight line is the signature of a flow set by a critical speed, not by the push.

The film whose thinnest point sets the flow

Leave a beaker of superfluid helium standing in a bath and it empties itself, the liquid creeping up the inside wall as a film eighty atoms thick, over the rim and down the outside. The film itself is nothing special: ordinary liquid helium coats the wall just as thickly. What is special is that it flows without friction, and so its flow is set not by how hard it is pushed but by how fast it can go. The level falls in a straight line rather than slowing as it nears the bath's, and the whole flow is decided at one place, the rim, where the film is thinnest and has to run fastest.

quantum · Superfluidity

Named alongside it

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

CounterflowEntropyLambda pointSuperfluidityCondensate fractionCritical velocityDiffusionHeat transportNormal fluidPhase transitionPhononSecond sound

All concepts