The viscosity nobody uses

The dissipation function this collection has been using since the essay that derived it is $\Phi = 2\mu\,e_{ij}e_{ij}$, and it is the incompressible form. The general one is

$$\Phi = 2\mu\left(e_{ij}e_{ij} - \tfrac13(\nabla\!\cdot\!\mathbf{u})^2\right) + \zeta\,(\nabla\!\cdot\!\mathbf{u})^2.$$

Two terms, because there are two independent ways to deform a parcel of fluid. It can change shape at constant volume, which is what a shear does. And it can change volume, which is what a compression does. They are separate deformations and there is no reason for them to cost the same per unit rate, so they get separate coefficients.

The second one, $\zeta$, is the bulk viscosity. This site has set it to zero in every figure of every essay without saying so — including every one about a bearing film and a boundary layer, where the omission is exact.

Stokes’ hypothesis, and what he actually said #

Stokes set $\zeta = 0$ in 1845. What is worth knowing is that he wrote down the reason and said it was not one: he assumed the mean of the normal stresses equals the thermodynamic pressure, remarked that this was an assumption he could not justify, and got on with it.

For a monatomic gas he was right, and kinetic theory says so exactly: a molecule with no internal degrees of freedom has nothing that can lag behind a compression, so a monatomic gas has $\zeta = 0$ identically. Argon and helium are exempt.

For anything else it is not zero. Compress a diatomic gas and the translational temperature rises immediately while the rotational and vibrational modes take a finite time to catch up. During that time the pressure is not its equilibrium value, and the discrepancy — proportional to the rate of compression — is exactly a bulk viscosity.

So the bulk viscosity is a relaxation time in disguise, and it depends on frequency, and quoting one number for it is an approximation that holds while the compression is slow compared with the relaxation.

Why no essay in this collection noticed #

Because $\zeta$ multiplies $(\nabla\!\cdot\!\mathbf{u})^2$, and in an incompressible flow that is identically zero.

Every figure in this collection about a boundary layer, a bearing, a vortex, a wing at low speed or a suspension is a flow with no divergence, so its dissipation contains no $\zeta$ at all. The omission was not an approximation; it was an exact statement about those flows.

It becomes an approximation the moment the density changes, and the two places a reader is likely to meet it are the absorption of sound and the interior of a shock.

Sound: a quarter of it #

A plane sound wave loses amplitude as it travels, and the classical calculation of how fast gives

$$\alpha = \frac{\omega^2\delta_s}{2c^3}, \qquad \delta_s = \frac{1}{\rho}\left[\tfrac43\mu + \zeta + k\left(\frac{1}{c_v} - \frac{1}{c_p}\right)\right].$$

The combination $\delta_s$ is the diffusivity of sound and it is the only place any of the three transport coefficients appears. For dry air at room temperature the three contributions are 53, 24 and 22 per cent.

That is the case for taking $\zeta$ seriously, and it comes with an immediate and larger disappointment.

Where the classical calculation runs out #

Compare the classical prediction with what the atmosphere actually does and the gap is enormous.

The excess is not a larger bulk viscosity in the sense used above. It is the vibrational relaxation of oxygen and nitrogen, whose relaxation times are comparable with an acoustic period, and which are catalysed by water vapour — so atmospheric absorption depends strongly on humidity, and a constant-coefficient continuum calculation cannot contain that at all.

It could be called a frequency-dependent bulk viscosity, and sometimes is. That fits the numbers and misdescribes the physics: a transport coefficient is a property of a fluid and this is a property of a fluid and a frequency together. The honest position is that continuum fluid mechanics computes a lower bound on the absorption of sound in air, and the bound is out by a decade over most of the audible range.

The shock: a length but not an amount #

The other place is more interesting, because there the coefficient decides one thing and conspicuously fails to decide another.

A weak shock in a gas is not a discontinuity. It is a smooth transition a few hundred nanometres thick, and its thickness is set by exactly the same $\delta_s$: the steepening of the nonlinear term against the diffusion of the transport terms, which is a Burgers balance and gives a hyperbolic tangent.

Double the transport coefficients and the shock doubles in thickness. Add the bulk viscosity to a calculation that omitted it and every shock gets 32 per cent thicker.

And the heat the shock makes does not move at all.

That is worth stating carefully, because it is the sharpest instance in this collection of a distinction the whole phase is about. The amount of energy a shock destroys is fixed by its two end states, which are fixed by conservation of mass, momentum and energy across it. Viscosity does not appear in those, and cannot. What viscosity decides is where and over what distance the destruction happens.

A shock in a fluid with twice the viscosity is twice as thick and makes exactly the same amount of heat. A shock in a fluid with a tenth of it is a tenth as thick, and makes exactly the same amount of heat. What a shock costs is a thermodynamic question with a thermodynamic answer, and the transport coefficients are irrelevant to it.

The two coefficients, and one number that separates them #

There is a compact way to say what the two viscosities do, and it is worth having because it explains why one of them is famous.

The rate-of-strain tensor splits into a trace and a traceless part, and those two pieces transform independently under rotation — no change of coordinates mixes them. So the most general linear relation between stress and rate of strain in an isotropic fluid has exactly two coefficients, one for each piece, and there is no third possibility. Two viscosities is not an empirical finding; it is a consequence of the fluid being isotropic and the relation being linear.

Which of the two is famous is decided by a single number: the Mach number squared. The divergence of a flow is of order $M^2$ times the shear, so the bulk term’s contribution to the dissipation is of order $M^4$ times the shear term’s. At Mach 0.1 that is a factor of ten thousand. At Mach 1 it is one.

So the bulk viscosity is not a small coefficient that happens to be neglected. It is a coefficient whose effect is suppressed by a large power of a small number in almost everything, and which arrives with full force in the two places where the divergence is not small — a sound wave, where the compression is the whole of the motion, and a shock, where the gradients are steep enough that the Mach number is not the right measure.

The place the model eats itself #

There is a limit that arrives quickly and it is the reason shock structure is a bad advertisement for continuum mechanics.

The thickness computed above is a few mean free paths. At Mach 1.1 it is thirty-four; at Mach 1.4 it is nine; at Mach 2 it is four. A shock computed from the Navier–Stokes equations is a solution of the wrong equations, drawn at the one place they do not hold.

The profile’s shape turns out to be close to what kinetic theory gives, which is a piece of luck rather than a justification, and its thickness is not — measured shock thicknesses are roughly twice the Navier–Stokes prediction at moderate Mach number. The equations are being used a long way outside the separation of scales they were derived under.

That failure is drawn on the figure above rather than corrected, because it is the same failure the Knudsen number describes arriving from a different direction: not a rarefied gas, but an ordinary one with a gradient so steep that the continuum has no room.

The coefficient a computation invents on purpose #

The result that a shock’s thickness depends on the transport coefficients while its entropy does not is usually filed as a curiosity. It is in fact the licence under which almost every compressible computation in existence is performed.

A shock a few hundred nanometres thick cannot be resolved on any grid a person would build, and a scheme that tries to capture it as a discontinuity produces oscillations. The standard remedy, since von Neumann and Richtmyer in 1950, is to add an artificial viscosity — a term proportional to the divergence, exactly the form $\zeta$ takes, with a coefficient chosen so that the shock is spread over two or three cells instead of over two or three hundred nanometres.

The coefficient used is enormous: it corresponds to a bulk viscosity many orders of magnitude above any fluid’s. And the answer is right anyway, for the reason this essay computes. The jump conditions contain no transport coefficient, so the states on either side of the captured shock are the physical ones whatever the artificial dissipation is, provided only that it is confined to the shock and is large enough to resolve it on the grid available. The computation gets the right entropy rise, the right pressure ratio and the right downstream Mach number from a fluid that does not exist.

That is a striking thing to be able to do, and it is worth naming what makes it legitimate. A quantity fixed by conservation laws across a region can be computed correctly by a model that is wrong inside the region, as long as the model conserves the right things and keeps its errors local. The same licence is what lets a boundary layer be replaced by a displacement thickness and a wall by a modelled condition — and it fails in exactly the same way, wherever the interior of the region turns out to affect something outside it.

And the medium where the coefficient is the whole story #

There is a fluid whose behaviour is dominated by the second coefficient, and it is not exotic: water with bubbles in it.

A bubbly liquid is enormously compressible, because compressing the mixture means compressing the gas rather than the liquid, and the bubbles respond with a delay — they have inertia, they oscillate, and they exchange heat with the water around them. Every one of those is a relaxation, and a relaxation under compression is a bulk viscosity. So a bubbly liquid has an effective $\zeta$ vastly larger than either phase’s, and its sound speed falls to tens of metres a second — below both water’s and air’s, which is one of the odder facts in acoustics.

The consequence is used industrially. A bubble curtain — a perforated pipe on the sea bed releasing air — attenuates underwater noise from pile driving by tens of decibels, and it does so by being a region of enormous bulk viscosity and mismatched impedance rather than by absorbing anything in the ordinary sense. The coefficient Stokes set to zero, in the fluid where it is largest, doing a job.

The one flow whose whole cost is the second coefficient #

The cleanest statement of what $\zeta$ is comes from asking which flow charges nothing but it.

A uniform three-dimensional expansion, $\mathbf{u} = A\mathbf{x}$, has a rate of strain that is $A$ times the identity. Its deviatoric part is identically zero, so the shear viscosity charges nothing, and $\Phi = \zeta(3A)^2$ — all of it, at any rate of expansion, in any fluid.

So a fluid obeying Stokes’ hypothesis expanding uniformly dissipates nothing at all. That is a strong claim and it is precisely what the hypothesis asserts. It is false for every gas but the noble ones.

The two-dimensional version is a useful trap and worth recording. An expansion confined to a plane — stretched in two directions, held in the third — is not a pure dilatation: an element doing that has changed shape, so a third of its rate of strain is deviatoric and it pays a shear viscosity too. The difference between the two is invisible in any picture drawn on a page, and it decides whether the deviatoric part is zero or two-thirds of everything.

What the picture cannot show #

Nothing here derives a bulk viscosity. Every value quoted is a measurement, from acoustic attenuation or from Brillouin scattering, and computing one requires the internal degrees of freedom of the molecule and their relaxation times.

The values are frequency-dependent and are quoted at one frequency. For carbon dioxide especially, “the” bulk viscosity varies by orders of magnitude across the audible range.

The shock structure is a weak-shock approximation. The Burgers balance holds for Mach numbers close to one; the profile at Mach 2 is drawn from it anyway, marked, because the alternative — a full Navier–Stokes shock structure — is a solution of equations that are even less valid there.

Nothing here is a dissipation map. The compressible dissipation function has been split and its two halves priced, and no figure draws it over a compressible field — which would need a shock structure resolved rather than captured.

And liquids are a different subject. A liquid’s bulk viscosity is comparable with its shear viscosity and is set by structural relaxation rather than by molecular modes. Water’s is 2.9 times its shear viscosity, which is worth knowing and is not derived here.

What it would take to measure one #

A last practical note, because the difficulty of measuring $\zeta$ is a large part of why it is neglected.

There is no experiment that isolates it. Every measurement is indirect: acoustic attenuation gives the combination $\tfrac43\mu + \zeta + k(1/c_v - 1/c_p)$, and $\zeta$ is extracted by measuring the shear viscosity and the conductivity separately and subtracting. When $\zeta$ is comparable with the other two, as in air, the subtraction is of similar-sized numbers and the result carries the combined error of three measurements.

Brillouin light scattering does better, because the linewidth it measures depends on the same diffusivity and can be taken at a much higher frequency, but it has the same structure — a combination, minus the parts that are known.

So the bulk viscosity of air is quoted with an uncertainty of tens of per cent, which is worse than any other transport property of any common fluid by an enormous margin. Nobody who needs a number to two figures uses it, which is a self-reinforcing reason for its absence from engineering practice.

The exception is computational aeroacoustics, where a shock’s thickness must be resolved and the resolution required goes directly as the diffusivity. There the 32 per cent is a 32 per cent difference in grid spacing in one direction, which is money, and the coefficient gets used.

Who found it, and when #

Stokes made the assumption in 1845. Kirchhoff computed the classical absorption of sound in 1868 with $\zeta$ omitted, and the discrepancy with measurement was noticed almost at once. Tisza proposed the relaxational explanation in 1942 and Kneser had measured the effect in carbon dioxide a decade earlier; the modern kinetic-theory account is Wang Chang and Uhlenbeck’s, from the 1950s.

The surprising connection is with a coefficient in a completely different theory that turns out to be the same object. In cosmology, the bulk viscosity of a fluid is the only transport coefficient compatible with an isotropic, homogeneous universe — a uniform expansion has no shear anywhere, by symmetry, so a shear viscosity has nothing to act on. The one coefficient that does nothing in almost every flow on Earth is the only one that can do anything to the expansion of the universe, and for exactly the reason set out above: it is the coefficient that charges for a change of volume rather than a change of shape.

Where the ladder goes next #

Below this rung is the price of a gradient, which is the incompressible form of the function this essay completes, and what a signal travels at, which is the acoustics the absorption is applied to.

Beside it is what a shock costs, which computes the entropy this essay shows the transport coefficients cannot touch, and where a fluid stops being one, which is the limit the shock’s own thickness runs into.