Illustrated Physics
physics, drawn
Concept

Transport — where it appears

The carrying of a quantity — momentum, energy, or particles — across a gradient by whatever moves through it, which is what viscosity and conduction measure. All three coefficients in a gas are the same product of a density, a mean speed and a free path, with a different cargo each time.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The viscosity of a gas, over 8 decades of pressure. The viscosity of 3 gases at 300 K against pressure, on a logarithmic pressure axis and a linear viscosity one. The lines are flat, and that is the whole figure. Viscosity is the rate at which momentum is carried across a shear, which is the density of carriers times the distance each one carries it: ⅓ρv̄λ. Doubling the pressure doubles the density and halves the free path, and the two cancel exactly, so the same gas at a hundredth of an atmosphere is exactly as viscous as at one — which is not what anybody expects of a thinner gas and is what is measured. nitrogen comes out at 17.9 μPa·s against a measured 17.9, helium comes out at 19.3 μPa·s against a measured 19.9, argon comes out at 21.7 μPa·s against a measured 22.7. The flatness ends when the free path reaches the apparatus rather than the next layer of gas: at a vessel 10 mm across that is around 0.68 Pa for nitrogen, 1.94 Pa for helium, 0.69 Pa for argon, below which there is no gas-to-gas hand-off left to make.

The viscosity that does not care how much gas there is

Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.

thermodynamics · Kinetic theory Where a drying drop loses its liquid. The rate at which liquid leaves the surface of a drying drop, against distance from the centre in units of the drop's radius, for 5 contact angles. The flux is not uniform, and it is not a property of the liquid: it is set by how vapour diffuses away from a lens-shaped object, which is the same boundary-value problem as the field around a charged lens and has the same answer — a power law in the distance from the rim, with an exponent that depends only on the contact angle. at 10° the exponent is 0.471, and the loss has doubled by 87.8 per cent of the way out, at 40° the exponent is 0.357, and the loss has doubled by 92.5 per cent of the way out, at 70° the exponent is 0.182, and the loss has doubled by 98.9 per cent of the way out, at 90° the exponent is 0.000 and the drop dries evenly everywhere, at 120° the exponent is -0.500 and the flux falls toward the rim. Below a right angle the flux diverges at the contact line; at exactly a right angle it is uniform; above it the edge is the slowest-drying part of the drop. Since a pinned edge must be resupplied from the interior, that sign decides which way the liquid inside the drop flows — and therefore whether everything suspended in it ends up in a ring at the rim or in a spot at the centre.

The ring the drop leaves behind

A drop of coffee dries into a ring rather than a disc, and nothing about coffee is responsible. The pattern is produced by a boundary condition — an edge that cannot move — and it survives replacing the coffee with anything else that will stay suspended.

fluids · Surface tension An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

The distance that takes the treble out

Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles.

waves · Attenuation Effusion rate against molecular mass. The rate at which a gas escapes through a small hole, against its molar mass, normalised to hydrogen. The rate is a quarter of the number density times the mean speed times the area, and the mean speed goes as the inverse square root of the mass — so the rate does too, which is Graham's law of 1848. Hydrogen escapes four times faster than oxygen and 13.3 times faster than uranium hexafluoride. The practical consequence is isotope separation, and its difficulty is on this chart. The two uranium hexafluorides differ by 3 out of 352 in mass, so a single stage enriches by a factor of only 1.00429 — four parts in a thousand. Reaching 90 per cent from natural uranium's 0.72 per cent therefore takes about 1665 ideal stages, and a real cascade needs more because each stage is imperfect. That number is why gaseous-diffusion plants were among the largest industrial structures ever built, and why centrifuges — which separate by mass directly rather than by the square root of it — replaced them.

The gradient that drives the other thing

A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.

thermodynamics · Diffusion The fraction of the pressure a membrane can hold. The osmotic pressure actually developed across a membrane against a 300 mol/m³ solution at 298 K, as the reflection coefficient runs from a membrane the solute passes freely to one it cannot pass at all. The line is straight with the van 't Hoff pressure 743.69 kPa as its slope — checked against the drawn line — because the coefficient enters as a simple factor. At σ = 1 the pressure is 743.69 kPa; At σ = 0.6 the pressure is 446.21 kPa; At σ = 0.2 the pressure is 148.74 kPa. Van 't Hoff's law is the ceiling rather than the answer, and a membrane's coefficient against a given solute is as much a property of the pair as the concentration is of the solution.

The membrane that almost holds

Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper.

fluids · Osmosis A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation An absorption that goes up when the photon gets harder. Mass attenuation coefficients for 4 materials against photon energy, both logarithmic, from tabulated measurements interpolated between. Away from a threshold every curve falls steeply — a harder photon is less easily absorbed, which is what makes X-rays penetrating. At a K edge the curve jumps upward: the photon has become able to eject an innermost electron, a whole new population becomes available, and the absorption multiplies by 4.4 for iodine at 33.17 keV and 4.0 for lead at 88 keV. That is a discontinuity in a material property as a function of frequency, and nothing in the classical description of absorption produces one. It exists because absorption at these energies is a transition of a bound electron rather than a loss in a medium.

The steps in an absorption curve

Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

waves · Attenuation 14 decades of fluid, and a floor none of them reaches. The ratio of viscosity to entropy density for 8 substances, in units of the proposed lower bound, on a logarithmic axis. The quantity is a viscosity divided by how much entropy a cubic metre of the substance holds, and it has the dimensions of Planck's constant over Boltzmann's — so a bound on it is a statement with no material properties in it at all. The span here is a factor of 1.6e+14, from pitch at 20 °C at the top to the quark–gluon plasma at the bottom, which sits a factor of 1.6 above the floor. The values marked as computed are worked out from a viscosity and a tabulated entropy; the rest are quoted from the compilations, because a minimum along an isobar is an inference from many measurements rather than a single one.

Whether a fluid can be made arbitrarily thin

Viscosity has no units anybody would call fundamental, and nothing obviously stops it being as small as you like. Divide it by the entropy in a cubic metre and the units become Planck's constant over Boltzmann's — and a conjecture from 2005 says that ratio has a floor. Across fourteen decades of ordinary fluid nothing has been measured below it, the closest thing to it is the hottest matter ever made, and a kinetic-theory argument reaches the same number from the opposite direction.

fluids · Viscosity

Named alongside it

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

DiffusionViscosityAbsorptionAttenuationDissipationEntropyEquilibriumIntensityKinetic theoryMean free pathSpectrumThermal energy

All concepts