A bulk viscosity holds a shock together until it splits

The largest bulk viscosity is the first to expire turned the table of bulk viscosities into a table of clocks. A gas resists compression beyond its shear viscosity because some of its molecular energy — rotation, or vibration — takes time to follow a change, and the coefficient is that time multiplied by the pressure and by how much energy the lagging mode holds. Carbon dioxide’s value, fifteen hundred times its shear viscosity, is the bending vibration relaxing in about nineteen thousand collisions, 2.75 microseconds at one atmosphere, and the constant coefficient it stands for is ten per cent wrong in the absorption of sound above 24 kHz.

That essay ended on a shock, because a shock is where a gradient is steepest and where a coefficient standing in for a lag is asked to do the most. Its question was at what strength a shock in carbon dioxide stops being describable by the bulk viscosity at all. The answer turns out to be a single Mach number, set by the ratio of two sound speeds, and the calculation that finds it also shows that below that number the coefficient is not an approximation but an exact limit.

A shock with nothing but relaxation in it #

The calculation strips a shock down to the one process in question. The gas has a frozen part — translation and rotation, which keep up with any change on the scale of a shock — and one lagging mode, the bending vibration, whose energy per unit mass $e$ relaxes towards its equilibrium value at the local temperature:

$$u\,\frac{de}{dx} = \frac{c_i R_s T - e}{\tau}, \qquad \tau = \tau_1\,\frac{p_1}{p},$$

the relaxation time going inversely with pressure because it is a fixed number of collisions. There is no shear viscosity and no heat conduction in it. Mass, momentum and energy fluxes are constant through a steady shock, and for each value of $e$ they fix the gas’s state as one of two roots of a quadratic: one slower than the frozen sound speed, one faster. The vibration’s energy is the only thing that changes along the shock, and the relaxation equation says how fast.

Two sound speeds matter. What a signal travels at is the square root of a derivative of pressure, and which derivative depends on what the molecules have time to do. The equilibrium speed is the one sound travels at when the vibration keeps up, 267 m/s in carbon dioxide at room temperature; the frozen speed, 4.1 per cent faster, is what it travels at when the vibration has no time to respond, which is the speed the earlier essay found ultrasound reaching in the same gas. A shock must outrun the equilibrium speed or it would not steepen. Whether it outruns the frozen one decides everything else.

Smooth, then a jump and a tail #

A shock slower than the frozen sound speed stays on the slower root the whole way. Its entire rise is the vibration catching up, and the profile is smooth: the gas ahead is warned by the frozen sound, which runs faster than the shock, and compresses gradually as energy leaks into vibration. The figure draws the velocity change through such shocks at Mach numbers of 1.01, 1.02 and 1.035, referred to the equilibrium sound speed, against distance in relaxation lengths — the equilibrium sound speed times the relaxation time, 0.737 mm at one atmosphere. They are wide and gentle, widest for the weakest.

A shock faster than the frozen sound speed cannot start that way, because no signal outruns it. The gas ahead of it arrives unwarned and must jump, at fixed vibrational energy, from the faster root to the slower one: a shock in the frozen gas, obeying the jump conditions with the frozen ratio of specific heats. Only after the jump does the vibration begin to take its share, and the gas relaxes over a tail to the same equilibrium state behind. At M = 1.1 and 1.3 the profiles in the figure have a vertical step followed by a long approach. The jump itself has a thickness — a few mean free paths, set by shear viscosity and conduction, as the discontinuity that has a thickness computed — but on the scale of the relaxation it is a discontinuity.

The dividing strength is where the shock speed equals the frozen sound speed. Referred to the equilibrium speed it is the ratio of the two, M = 1.0411 for carbon dioxide at room temperature, and through the jump conditions that is a pressure rise of 9.5 per cent. A shock in carbon dioxide gentler than that is a relaxation wave; any shock stronger is a relaxation wave with an ordinary shock at its front.

The coefficient is the weak limit, exactly #

Taylor showed in 1910 that a weak shock in a viscous gas has a hyperbolic-tangent profile whose thickness, measured by its steepest slope, is $4\delta/(\beta\,\Delta u)$ — δ the gas’s diffusivity of sound, β the nonlinearity $(\gamma+1)/2$, $\Delta u$ the velocity jump. With only a bulk viscosity in the gas, δ is ζ/ρ. That is the prediction a code makes when the tabulated bulk viscosity is added to the Navier–Stokes equations, and the figure draws it as the dashed line.

The relaxing gas’s smooth shock lies on it for the weakest shocks. At M = 1.002 the relaxing shock is 45.01 mm thick and Taylor’s, with the bulk viscosity the same vibrational mode gives the absorption of sound, is 45.11 — agreement to two parts in a thousand, from a calculation in which no viscosity appears. This is the precise sense in which a bulk viscosity is a relaxation seen from below its frequency: a shock thick enough that the vibration keeps nearly up everywhere in it is indistinguishable from a viscous shock with that coefficient.

As the shock strengthens the two part company. At M = 1.02 the relaxing shock is 5.74 relaxation lengths thick against Taylor’s 6.18, eight per cent thinner; at 1.04 it is 1.45 mm against 2.30. The shock is now thin enough that the vibration lags a finite amount, and a lag is not a viscosity. Then the split arrives and the smooth thickness vanishes altogether: the relaxing shock has a jump, which has no thickness on this scale, followed by a tail. At M = 1.3 the constant coefficient draws a smooth shock 0.34 mm thick, while the real one jumps and then relaxes over 0.72 mm — thinner at its front than the coefficient allows by a factor of about a thousand, and longer behind.

A computation that thickens shocks in carbon dioxide with the tabulated bulk viscosity is therefore right for shocks weaker than about two per cent in Mach number, qualitatively wrong beyond the split, and in between it overstates the thickness by an amount that grows with the strength. The coefficient is not a property of the gas that a shock inherits; it is the first term of an expansion in how far the vibration lags, and the split is where the expansion stops meaning anything.

The jump takes over #

Once the split has happened, how much of the shock is jump and how much is tail? The figure plots the jump’s share of the velocity change. It is zero for every dispersed shock, and rises steeply once the shock outruns the frozen sound: 57 per cent at M = 1.1, 89 per cent at 1.5, 94.5 at 3. The remainder is the compression the vibration adds as it takes its share of energy, which makes the gas behind denser than a frozen shock would leave it.

So a strong shock in carbon dioxide is mostly an ordinary shock, and the relaxation is a modest after-effect in velocity and pressure. That is why the jump conditions with the equilibrium ratio of specific heats, which describe only the far downstream state, are good enough for most purposes, and why what a shock costs — the entropy it generates — is the same whichever path the gas takes, since it depends on the two end states alone. What the relaxation changes is where the gas spends its time on the way, and for temperature that turns out to matter.

A split shock can be hotter inside than behind #

The frozen jump puts all of its heating into translation and rotation. The vibration then draws energy from them as it relaxes, and the temperature — which measures translation — falls towards the equilibrium value behind. At high Mach number this gives a temperature peak just behind the jump well above the final temperature: the air behind a Mach 6 shock is 2,382 K before its vibration starts and 2,059 K after, which is the central number of a gas that has not finished being shocked.

A weak split shock does not do this, and the figure shows why. At M = 1.1 the jump takes the gas to 304 K and the relaxation carries it on up, to 307.6 K behind: the tail’s further compression heats the gas more than the vibration’s appetite cools it. At M = 1.3 the jump reaches 339.6 K and the gas relaxes down to 335.5; at M = 2 the peak is 476 K against 444 behind. The crossover is at M = 1.19. Below it a split shock rises monotonically through its jump and tail; above it the tail cools. The overshoot is 9.8 per cent of the whole temperature rise at M = 1.3 and 22 per cent at M = 2.

The constant bulk viscosity cannot reproduce either behaviour, because a Taylor profile is monotonic in every quantity. That matters wherever a rate depends steeply on temperature — ignition, dissociation, the chemistry of a combustion shock tube — because a reaction sees the peak, and the peak is where the gas spends its first few relaxation lengths.

Every relaxing gas splits #

The split criterion is general: any gas with a lagging mode has a frozen sound speed faster than its equilibrium one, and any shock that outruns the frozen speed has a jump at its front. For nitrogen and air, whose rotation lags, the frozen speed is 9.1 per cent faster and the split comes at M = 1.091. For hydrogen, whose rotation also lags, the same. Whether the split can be seen is a different question, and the figure answers it in collisions.

Nitrogen’s and air’s rotation relax in four or five collisions. A frozen shock is itself several mean free paths thick, so the tail is inside the jump and the split is invisible: a shock in air is, to every measurement, one viscous structure, and its rotational lag is folded into its bulk viscosity without loss — which is the reason the viscosity nobody uses is a good coefficient for air. Hydrogen’s rotation takes about 175 collisions, a tail of about twenty micrometres at one atmosphere, long enough to resolve in a molecular simulation. Carbon dioxide’s vibration takes about nineteen thousand, and its tail is three-quarters of a millimetre: a structure a schlieren photograph can see. The size of a gas’s bulk viscosity is a good guide to whether its shocks will look split, because both are the length of the same clock.

On Mars the clock is long #

The relaxation time is a number of collisions, so it goes inversely as the pressure, and the Martian atmosphere — nearly pure carbon dioxide at about 610 pascals — stretches every clock by a factor of 166. The earlier essay found that this puts the expiry of the bulk viscosity inside the audible band there. It does something just as large to shocks.

The figure computes the rise time of a weak shock, its thickness divided by its speed, against overpressure. In carbon dioxide at one atmosphere a shock of one per cent overpressure rises in 0.076 milliseconds. On Mars the same shock rises in 12.4 milliseconds and is about three metres thick; a tenth of a per cent takes 125 milliseconds. A sonic boom arriving at the Martian surface from a descending vehicle would therefore reach the ground not as a crack but as a swell lasting a large fraction of its own duration, and a boom on Mars is closer to a pressure wave than to a shock. The calculation that did this for Earth’s air, where oxygen makes the crack and nitrogen the rise, found rise times of a millisecond or so set by two slow vibrations and humidity; on Mars one mode does it alone, and more slowly.

The same stretching moves the split. At 240 K less of carbon dioxide’s vibration is excited, so the frozen and equilibrium speeds are closer, and the shock splits at a lower Mach number, 1.030. The relaxation length there is fourteen centimetres, and any shock strong enough to split on Mars trails a tail about that long.

Two structures, only one of them a continuum #

A split shock in carbon dioxide is two structures at scales four orders of magnitude apart, and they fail differently. The frozen jump is a few mean free paths thick — a few hundred nanometres at one atmosphere for a moderate shock — which is exactly where the discontinuity that has a thickness found the Navier–Stokes equations producing a structure thinner than the distance between collisions, and where a fluid stops being one. The tail is nineteen thousand collisions long, deep in the continuum, and described exactly by the relaxation equation. The jump conditions, which the equations allow whatever happens inside, connect the two and are the only part of the calculation that needs no model of either.

A code that adds the tabulated bulk viscosity to the Navier–Stokes equations blurs these into one smooth structure a third of a millimetre thick at M = 1.3. It is wrong about the jump, which it makes about a thousand times too thick, and wrong about the tail, which it makes about half as long and removes the overshoot from. What it gets right is the pair of end states, because those are fixed by conservation. For a quantity integrated across the shock — the entropy, the momentum — that is enough; for anything that happens inside it, it is not.

The same split happens in a liquid with a memory. A viscoelastic liquid struck faster than its relaxation time responds first as an elastic solid, with a wave front that travels at the solid’s shear-wave speed, and only afterwards flows. A solid if it is not given time is that behaviour, and a Maxwell liquid’s viscosity is its relaxation time times its elastic modulus exactly as a gas’s bulk viscosity is its relaxation time times a difference of compressibilities. Hit either one gently and a viscosity describes it; hit it faster than its own clock and it splits into a frozen response and a relaxation, and the viscosity describes neither.

What was checked #

The integration’s end states must be the equilibrium jump conditions, with the equilibrium ratio of specific heats, and its frozen jumps the same conditions with the frozen ratio; both agree to rounding error from M = 1.02 to 2. The weak-shock thickness agrees with Taylor’s to 0.2 per cent at M = 1.002 using the bulk viscosity extracted from the absorption of sound by the same mode — two independent calculations, one acoustic and one of a shock, meeting at the coefficient. Shocks just below and just above M = 1.0411 are confirmed to be smooth and split respectively. The tests also refuse a shock slower than the equilibrium sound speed, a gas outside the table, and a tolerance of zero.

What the picture cannot show #

The relaxation time is held fixed in temperature. Vibrational relaxation times fall steeply as temperature rises — the Landau–Teller dependence makes them exponential in the inverse cube root of temperature — so the tail of a strong shock, where the gas is hot, is shorter than drawn. The split criterion does not depend on the time at all, and the weak-shock limit does not either, since the temperature barely changes; the tail lengths and the temperature profiles above M = 1.3 are upper bounds.

One mode with a constant heat capacity. Carbon dioxide has three vibrational modes, relaxing at different rates, and its vibrational heat capacity grows with temperature. One lagging mode with the room-temperature capacity is the simplest version, and it is the one the tabulated bulk viscosity describes.

No viscosity in the dispersed shock. For carbon dioxide the relaxation dominates a weak shock’s thickness by the ratio of the bulk to the shear viscosity, fifteen hundred to one. For nitrogen the two are comparable, and a relaxation-only calculation understates its weak shocks’ thickness.

Who found it, and when #

The structure of a shock in a relaxing gas — fully dispersed below the frozen sound speed, partly dispersed above it — is Lighthill’s, from 1956, in his account of viscosity effects in sound waves of finite amplitude, and Zel’dovich and Raizer set it out for gases with lagging internal energy in their 1960s monograph. Taylor’s weak-shock profile is from 1910. The bending-mode relaxation of carbon dioxide has been measured by acoustic absorption and in shock tubes since the 1950s, and the recordings of sound on Mars by the Perseverance rover’s microphones, published in 2022, measured its effect on the speed of sound directly.

Still open: the combustion shock tube #

The temperature overshoot is where the relaxation changes an answer someone relies on. A reflected shock in a shock tube is the standard way to heat a reactive mixture instantly, and ignition delay times measured that way are the data kinetic mechanisms are fitted to. In a mixture rich in carbon dioxide — oxy-fuel combustion, or exhaust-gas recirculation — the gas behind the shock spends its first relaxation lengths above or below its equilibrium temperature depending on the strength, and an ignition delay of tens of microseconds overlaps a relaxation time of a few. The next calculation gives the relaxing gas a one-step Arrhenius reaction with an activation energy typical of hydrocarbon ignition, and asks how much the ignition delay measured behind a split shock differs from the one computed at the equilibrium temperature — and whether the error changes sign at the Mach number where the overshoot appears.