The largest bulk viscosity is the first to expire

The viscosity nobody uses made the case for a second coefficient. A fluid resists a change of shape through its shear viscosity and a change of volume through its bulk viscosity, and Stokes set the second to zero in 1845 with, by his own account, no argument for doing so. The essay then quoted what the second coefficient actually is: zero for argon and helium, a little under the shear viscosity for nitrogen and oxygen, twenty-eight times it for hydrogen, fifteen hundred times it for carbon dioxide. It drew a careful line under those numbers. The excess absorption of sound in humid air, caused by oxygen and nitrogen vibrating, is “not a larger bulk viscosity”, because “a transport coefficient is a property of a fluid and this is a property of a fluid and a frequency together”.

The line is right about the humid air and wrong about where it is drawn. Carbon dioxide’s fifteen hundred is a vibrational relaxation too, of exactly the kind the essay declined to call a viscosity, and nitrogen’s value of 0.8 is a rotational relaxation, the same mechanism with a shorter clock. There is no bulk viscosity that is not a relaxation. The only question is how short the clock is compared with whatever the flow is doing, and the size of the coefficient is a measurement of the clock.

Which sound speed a gas has is itself a question about a derivative. What a signal travels at is the square root of the pressure’s derivative with respect to density, and the classic lesson is that the derivative must be taken at constant entropy rather than constant temperature — Laplace’s correction to Newton. A relaxing mode adds a second version of the same lesson. Taken fast, the derivative sees only the modes that keep up; taken slowly, it sees all of them. A gas with a lagging mode has two sound speeds, and the bulk viscosity is what the gap between them looks like from the slow side.

One lagging mode #

A molecule’s energy sits in several places: in its motion from place to place, in its rotation, in its vibrations. Compress a gas and the work goes first into translation, because that is what collisions exchange directly; rotation and vibration take their share only after enough collisions to pass it on. For a moment the gas behaves as if it had fewer places to put energy, which makes it stiffer — its ratio of specific heats is higher — and then it relaxes towards equilibrium as the internal mode catches up.

The simplest model has one internal mode with molar heat capacity $c_i$ and a relaxation time τ. At frequency ω the gas’s heat capacity at constant volume is

$$c_v(\omega) = c_f + \frac{c_i}{1 + i\omega\tau},$$

where $c_f$ is the part that keeps up — translation, plus rotation when the lagging mode is a vibration. Everything else follows without further assumption: the ratio of specific heats is $1 + R/c_v(\omega)$, the sound speed is its square root times $\sqrt{RT/M}$, and a wave’s wavenumber is the frequency over that complex speed. Its real part gives the phase speed; its imaginary part, the absorption.

At low frequency the absorption grows as the square of the frequency, which is precisely the signature of a viscosity in the Navier–Stokes equations. Matching the two defines the bulk viscosity this mode produces. For a weak mode it comes out in closed form, the pressure times the relaxation time times the difference between the frozen and equilibrium values of $R/c_v$, and the computed value agrees with that to about a tenth of a per cent. The coefficient is a relaxation time written in the units of a viscosity.

The quoted numbers are times #

That can be run backwards. Take each gas’s quoted ζ/μ, supply its heat capacities — rotation contributes R for a linear molecule; carbon dioxide’s vibrations contribute 0.93 R at room temperature, almost all of it from the twice-degenerate bending mode — and solve for the relaxation time that produces it. Dividing by the mean time between collisions, roughly μ/p, turns each into a count.

The counts are the ones molecular physics expects, which is the check that the inversion means something. Nitrogen’s rotation relaxes in about five collisions; the energy steps between rotational levels are small against the energy of a collision, and a few encounters are enough. Hydrogen’s rotation takes about 175, because hydrogen is so light that its rotational levels are unusually far apart and most collisions cannot bridge them. Carbon dioxide’s bending vibration takes about nineteen thousand, because a vibrational quantum is several times the typical collision energy and only the rare hard collision can deliver one. The implied relaxation time is 2.75 microseconds at one atmosphere, within a factor of about two of the value measured directly by acoustic and shock-tube methods.

The five-order spread of bulk viscosities in the table of the earlier essay is therefore not five orders of magnitude of a material property. It is five orders of magnitude of clock, with the heat capacity of the lagging mode as a modest multiplier. And a clock has a frequency.

Below its frequency, a viscosity; above it, a stiffer gas #

The figure runs the same calculation across frequency, scaled by each mode’s own time. Far below ωτ = 1 the mode keeps up, the gas has its equilibrium heat capacity and its equilibrium sound speed, and the small lag shows up as an absorption per wavelength rising in proportion to frequency — a viscosity. Far above, the mode cannot follow at all. The gas behaves as if the mode did not exist: sound travels at the frozen speed, 9.1% faster for nitrogen with its rotation frozen and 4.1% faster for carbon dioxide with its vibration frozen, and the mode absorbs nothing, because it no longer takes part. In between, where the period matches the relaxation time, the absorption per wavelength peaks — 0.27 nepers for a diatomic rotation, 0.13 for carbon dioxide’s weaker vibration.

This is the same structure as the heat of compression in a squeeze film, where heat leaving across the gap gives the gas a complex polytropic index that is isothermal at one end and adiabatic at the other and lossy between. It is the same as a viscoelastic liquid, which is a solid if it is not given time. In each case a single coefficient describes the low-frequency side of a relaxation, and the coefficient’s value is proportional to the relaxation time.

That proportionality is the whole argument in one line. The larger a bulk viscosity is, the longer the time it stands for, and so the lower the frequency at which it stops describing anything.

Carbon dioxide, measured against itself #

The figure computes the absorption of sound in carbon dioxide at one atmosphere two ways from the same bulk viscosity. The dashed line is the Navier–Stokes answer with ζ held constant at fifteen hundred times μ; the solid line is the relaxing mode that ζ was measured from. They agree through the audible band, which is why the constant value is the one quoted. At 23.6 kHz the constant coefficient is already ten per cent high. Near 76 kHz, where the absorption per wavelength peaks, it is double. At a megahertz it is 181 times too high, because the real absorption has levelled off at about seventy nepers a metre while the constant coefficient’s $\omega^2$ keeps climbing.

Ultrasound in carbon dioxide is not exotic. Gas-flow meters, ultrasonic level sensors and the acoustic cells used to measure gas composition run at tens to hundreds of kilohertz, and carbon dioxide is one of the gases they are most often asked about. A designer who took the largest bulk viscosity in the table at its word would predict a signal lost within millimetres at a megahertz, where the real gas lets it through. The same designer, estimating shock thickness, would be making the mistake that a gas that has not finished being shocked documents: behind a strong shock the vibrational energy takes thousands of shock thicknesses to arrive, and a coefficient that folds it into the shock’s own diffusion puts it in the wrong place.

The expiry dates #

Doing the same for every gas gives an expiry date for each coefficient: the frequency at which the constant bulk viscosity overstates the absorption by ten per cent. Air and nitrogen hold to about a hundred megahertz, which is to say everywhere a fluid dynamicist will ever need them; at those frequencies the wavelength is approaching the mean free path and the continuum description is failing for other reasons first. Hydrogen’s holds to about five megahertz. Carbon dioxide’s expires at 24 kHz — the edge of human hearing.

The one liquid in the earlier essay’s table makes the same point from the other side. Water’s bulk viscosity, about three times its shear viscosity, is measured by Brillouin scattering at gigahertz frequencies, which would be an odd place to measure a coefficient meant for kilohertz acoustics if the coefficient could expire in between. It does not, because the relaxation behind it is the rearrangement of water’s hydrogen-bonded structure, which takes picoseconds. A short clock is what makes a measurement at one frequency transferable to every lower one — and carbon dioxide’s long clock is exactly why its value, measured in the audible band, cannot be carried upwards.

So the conclusion of the earlier essay, that air’s bulk viscosity is a legitimate transport coefficient worth a quarter of the absorption of sound in dry air, stands, and it stands for a reason the essay did not give: air’s rotational relaxation is so fast that no acoustic frequency comes near it. The vibrational relaxations of oxygen and nitrogen that the essay set apart — the ones that set a sonic boom’s rise time — are the same thing with clocks from microseconds to milliseconds, which is why they cannot be folded into a coefficient anywhere in the audible band. The distinction between “a viscosity” and “a relaxation” is not a distinction of kind. It is a statement about which side of the clock’s frequency a flow sits.

Mars #

Every relaxation time here is a number of collisions, and the time between collisions goes inversely as the pressure. Take carbon dioxide from one atmosphere down to 610 pascals — the mean surface pressure of Mars — and every relaxation time is 166 times longer, while the bulk viscosity itself, pressure times time, is unchanged. The expiry date falls by the same factor, from the edge of hearing into the middle of it.

The figure computes the speed of sound in Martian air, taken as pure carbon dioxide at 610 pascals and 240 kelvin, with the relaxation time scaled from one atmosphere by pressure alone. Low notes travel at 245 metres a second, with the bending vibration keeping up; high notes at 252, with it frozen. The step between them is centred near 342 hertz, and the constant-coefficient description is ten per cent wrong by about a hundred hertz.

The Perseverance rover carried microphones to the surface of Mars, and the analysis of their recordings, published in 2022, reported exactly this: two speeds of sound, about ten metres a second apart, near 240 and 250 metres a second, with the transition at a few hundred hertz. The scaling here ignores that a colder gas relaxes more slowly, which would move the step lower, and ignores the few per cent of nitrogen and argon in the atmosphere; that it lands within a factor of two of the measured transition from a number quoted for a laboratory at room temperature is the point.

The same pressure scaling operates on Earth, only less dramatically. A sonic boom is born near the cruise altitude of the aircraft that makes it, at a fifth of sea-level pressure or less, and a boom is aged in the thin air it starts in because every relaxation time up there is five or more times longer and the molecular clocks that round off its shocks run correspondingly slower. Mars simply takes the factor to 166, with a gas whose clock was slow to begin with.

On Mars, then, the bulk viscosity is not a small correction to the absorption of sound. It is the reason a listener hears the high notes of a distant event before the low ones. The coefficient Stokes set to zero is, on another planet, a property anyone with a microphone can hear.

What a flow sees #

The practical rule that falls out is short. A bulk viscosity may be used as a constant wherever the flow’s rate of compression is slow against the mode’s relaxation, and the coefficient’s own size says how slow that has to be: ζ divided by the pressure and the lagging share of heat capacity is a time, and the flow’s compression rate times that time must be small.

For air at sea level that time is under a nanosecond, and no flow outside a shock’s own interior compresses that fast. For carbon dioxide at one atmosphere it is microseconds, and ultrasonics, strong shocks and the fine structure of high-speed jets all reach it. For carbon dioxide on Mars it is a fraction of a millisecond, and ordinary sound reaches it. A computation that carries a bulk viscosity to thicken a shock or damp its acoustics should check that number before using the coefficient, and when that number is not small, carry the mode as a separate equation instead — a relaxation equation with its own time, which is what the coefficient was always an approximation to.

What was checked #

The relaxing gas must reach its two limiting sound speeds, the equilibrium one at low frequency and the frozen one at high, and it does to a part in a hundred billion. The bulk viscosity it produces must be linear in the relaxation time and independent of the pressure at fixed pressure-times-time, and it is to rounding error. The extracted coefficient must match the weak-mode closed form, and does to 0.13 per cent. The tests also offer a negative relaxation time, a gas the model does not know, and a tolerance of zero, and all three are refused.

What the model leaves out #

More than one mode. Carbon dioxide has three vibrations and rotation; air has two vibrations, of different gases, coupled through water vapour. Each lagging mode adds its own term with its own time, and the absorption becomes a sum of relaxation peaks. Only the slowest dominant mode was carried here, which is right for carbon dioxide’s bending vibration and is the reason air’s vibrational relaxations had to be left to the boom essays.

Temperature dependence of the clock. Vibrational relaxation times fall steeply as the temperature rises, roughly as the exponential of minus the cube root of the temperature, so both the laboratory-to-Mars extrapolation and any hot flow need the time at their own temperature. Pressure scaling alone is exact only at fixed temperature.

The shear viscosity and conduction. The figures isolate the bulk part of the absorption. The classical shear and thermal parts add a further $\omega^2$ term that does not expire at these frequencies, which is why the constant-coefficient picture survives for air.

Who worked it out #

Einstein, in 1920, was the first to show that a slowly relaxing internal energy would disperse and absorb sound, and Kneser measured the effect in carbon dioxide in the early 1930s. That the bulk viscosity of kinetic theory is the low-frequency limit of such a relaxation was set out in the textbook of Landau and Lifshitz, who warn in so many words that a large second viscosity means a long relaxation time and a limited range of validity. Tisza, in 1942, made the connection to the measured absorption of polyatomic gases. The Perseverance measurements are Maurice and colleagues’, in Nature, 2022.

Still open: the relaxing shock #

A weak shock in carbon dioxide is thinner than its relaxation length as soon as its strength passes a few per cent, so it cannot be described by a constant bulk viscosity at all: it is a thin viscous jump to the frozen state followed by a long relaxation tail, and only the weakest shocks are smooth enough to be a single diffusive profile. The next calculation carries the mode’s relaxation equation through a steady shock and asks at what strength the shock splits into a frozen front and a tail — the strength at which the flow speed jump exceeds the gap between the frozen and equilibrium sound speeds — and what that says about computations that thicken shocks with the bulk viscosity the table gives.