Illustrated Physics
physics, drawn
Concept

Heat equation — where it appears

The equation saying that temperature changes at a point at a rate proportional to the curvature of the temperature profile there. Its solutions spread and smooth, and a disturbance reaches a distance growing as the square root of time, scaled by the thermal diffusivity.

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

Heat that arrives everywhere at once, and heat that waits. The temperature spread from a point of heat released at time zero, in one dimension, at times 0.5, 2, 8 relaxation times, from the diffusion equation (dashed) and from Cattaneo's telegraph equation (solid) with the same diffusivity, distances in units of the signal speed times the relaxation time. At 0.5, the diffusion equation has put 62 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 78 per cent is still in the two fronts; at 2, the diffusion equation has put 32 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 37 per cent is still in the two fronts; at 8, the diffusion equation has put 5 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 1.8 per cent is still in the two fronts. The fronts, drawn as the vertical marks, carry heat outward at the signal speed and decay; behind them the telegraph solution fills in and, after many relaxation times, becomes the diffusion equation's Gaussian.

The heat that arrives before it could

Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.

relativity · Relativistic thermodynamics Where skin and the thing it touches meet. Temperature across the contact between skin at 33 °C (left) and a block at 20 °C (right), against distance from the contact in millimetres, at 0.1, 1 and 10 s after touching, for steel (blue) and pine wood (red). Each pair of curves is two error functions meeting at the contact, and the meeting temperature does not change with time: 21.0 °C against steel and 30.3 °C against wood. The nerve endings a fraction of a millimetre under the skin report something close to that interface temperature, so the steel feels like an object at 21 °C and the wood like one at 30 — although both are at 20. The profiles spread as the square root of time, much further into the steel than into the wood.

The metal that feels colder than the wood

A steel railing and a wooden bench in the same shade are at the same temperature, and the railing feels much colder. The usual explanation, that metal conducts heat better, is half the answer, and it gets water wrong. When two bodies touch, the surface between them jumps at once to a temperature that then stays fixed, a weighted average of the two in which each body's weight is the square root of its conductivity times its heat capacity per volume. Skin at 33 °C touching steel at 20 °C meets it at 21; touching wood, at 30. The nerves report the meeting temperature, not the object's.

thermodynamics · Diffusion

Named alongside it

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

DiffusionCausalityError functionHeat capacityHeat conductionMean free pathRelativistic hydrodynamicsRelaxation timeSecond soundTelegraph equationThermal diffusivityThermal effusivity

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