The viscosity that does not care how much gas there is

Every intuition about a gas says that there is less of it when the pressure is lower, and that everything it does should be weaker. Its pressure is lower, its density is lower, its refractive index is closer to one, it conducts less sound. Its viscosity is exactly the same.

Maxwell obtained this in 1860 and wrote that the result “is certainly very unexpected”. It was unexpected enough that he built an apparatus — a stack of glass discs oscillating in a sealed chamber, whose damping he measured while pumping the chamber down — and confirmed it in 1866. That sequence is worth more than the result. A theory that had produced only agreements would have been a summary of what was already known; a theory that produced a claim its author disbelieved, and that survived being tested, was a theory about the world.

Why a gas has a viscosity at all #

The picture of viscosity as friction between sliding layers has nothing to grip in a gas. There is no contact between layers, no surfaces, no roughness. What there is, is molecules crossing.

The definition is read off one picture and says nothing about mechanism. Fluid between two plates, one moving: the velocity varies linearly across the gap, and the force per unit area needed to keep the top plate going is the viscosity times that gradient. The same picture describes honey, water and air, and the mechanisms in the three are completely different — which is exactly why the definition can be shared and the value cannot be guessed from it.

The mechanism in a gas is transport. Consider a surface parallel to the flow, somewhere in the gap. Molecules cross it in both directions at the same rate, because the gas is not going anywhere on average. But a molecule crossing downward came from a region moving slightly faster and carries that extra momentum with it; a molecule crossing upward came from a slower region and carries a deficit. The net effect is a transfer of forward momentum from the fast side to the slow side, which is exactly a force between them.

Pressure is the rate of arrival of perpendicular momentum at a wall, which is the argument two rungs down this ladder. Viscosity is the same argument turned sideways: a rate of arrival of parallel momentum at an imaginary surface inside the gas rather than at a wall. And it needs one extra ingredient the pressure argument does not — how far the molecule came from, because that is what decides how much parallel momentum it is carrying when it arrives.

How far it came from is the free path.

Putting the three together:

$$\eta \approx \tfrac13 \rho \bar v \lambda,$$

with $\rho$ the mass density, $\bar v$ the mean speed and $\lambda$ the free path.

The cancellation #

The free path is not an independent quantity. A molecule travels until it meets another, and how far that is depends on how many others there are:

$$\lambda = \frac{1}{\sqrt2\,n\sigma},$$

with $n$ the number density and $\sigma$ the collision cross-section.

Now substitute. The density $\rho = nm$ is proportional to $n$, and the free path is proportional to $1/n$, so

$$\eta \approx \frac{1}{3}\,nm\,\bar v\,\frac{1}{\sqrt2 n\sigma} = \frac{m\bar v}{3\sqrt2\,\sigma},$$

and the number density has disappeared. Nothing in the answer depends on how much gas there is. Halving the pressure halves the number of carriers and doubles the distance each one carries its cargo, and the product is untouched.

That cancellation is not a coincidence of algebra. It says something structural: a gas transports momentum at a rate that depends only on what its molecules are, not on how many of them there are, because thinning a gas makes each carrier proportionally more effective. The same argument applies to thermal conductivity, with energy in place of momentum, and gives the same independence — which is why a vacuum flask needs a hard vacuum to work, not merely a low pressure. At a hundredth of an atmosphere it insulates no better than at one.

There is a practical consequence that is worth stating because it surprises people who build things. A rough vacuum does not reduce gas damping. A pendulum, a galvanometer needle, a microbalance or a quartz resonator in a chamber pumped to a thousandth of an atmosphere is damped by the residual gas exactly as strongly as it was in air, and the improvement that a rotary pump appears to be buying is entirely in the buoyancy and the convection rather than in the drag. Reducing gas damping requires going all the way to the Knudsen regime, where the free path exceeds the apparatus, and that is a difference of six or seven orders of magnitude in pressure rather than three. The whole design of vacuum systems for delicate instruments turns on this one flat line.

The factor that is missing #

The elementary derivation is a third too low, and it is worth being clear that this is not experimental error.

The elementary argument gives every molecule the mean speed, and the molecules do not have the mean speed: they have a broad distribution, and the fast ones both travel further between collisions and carry more momentum when they arrive. Weighting the transport by the actual distribution rather than by an average is one of three repairs the elementary derivation needs, and the three together supply the factor of about 1.5 by which it is wrong.

The three defects are all of the same kind. The elementary argument assumes every molecule travels exactly one free path; that it travels perpendicular to the shear; and that it arrives carrying the mean momentum of where it started. In reality the free path is exponentially distributed about its mean, the directions are isotropic, and a molecule’s last collision partially randomises what it carries. Doing the transport calculation properly — the Chapman–Enskog treatment, which solves the Boltzmann equation to first order in the gradient — replaces the coefficient $1/3$ by $5\pi/32$ times it, a factor of 1.473, and gives

$$\eta = \frac{5}{16}\frac{\sqrt{\pi m k T}}{\sigma}.$$

For nitrogen at 300 K that is 17.9 μPa·s against a measured 17.9. For helium 19.3 against 19.9, for argon 21.7 against 22.7. The remaining few per cent is the hard-sphere assumption: real molecules are not hard spheres, their effective cross-section depends on how hard they hit, and the departure grows with temperature.

The important part is that the correction is a pure number. It does not touch the density independence, which follows from the structure of the argument rather than from any coefficient in it, and no amount of care in the transport calculation would produce a viscosity that depends on pressure.

That temperature behaviour is the sharpest qualitative test of the whole picture. In a liquid, viscosity is about molecules escaping from the cages their neighbours make, so heating helps and the viscosity falls steeply — honey at 40 °C is a tenth of honey at 20 °C. In a gas, viscosity is about carriers crossing a surface, so heating makes them faster and it rises. The two mechanisms have opposite signs, and a measurement of the sign alone distinguishes them.

One number in that formula is worth reading backwards. Everything on the right-hand side except the cross-section is known independently — the mass from chemistry, the temperature from a thermometer, the constants from elsewhere — so a measurement of a gas’s viscosity is a measurement of $\sigma$, and therefore of a molecule’s size. That is how molecular diameters were first obtained, by Loschmidt in 1865, from viscosity together with the density of the liquefied gas; the numbers came out in the region of a few tenths of a nanometre, at a time when the existence of molecules was still a hypothesis. A quantity nobody could see was extracted from the damping of a swinging disc.

Three cargoes, one walk, and the ratios between them #

The claim that viscosity, thermal conduction and diffusion are one mechanism carrying three different things is testable, because it predicts the ratios between them rather than merely asserting a family resemblance.

Each coefficient comes out as roughly a third of the mean speed times the free path, multiplied by whatever density of cargo there is: mass density for momentum, heat capacity per unit volume for energy, and nothing at all for the molecules themselves. So the kinematic viscosity, the thermal diffusivity and the self-diffusion coefficient should all be about $\tfrac13\bar v\lambda$, and their ratios should be pure numbers near one with no gas properties in them.

They are. The ratio of kinematic viscosity to thermal diffusivity is 0.67 for argon and about 0.71 for air, against a kinetic-theory prediction of two thirds for a monatomic gas — and the small excess for air is exactly the correction for a diatomic molecule, whose internal modes carry energy but not momentum. The ratio of kinematic viscosity to self-diffusivity is likewise near one.

Those numbers are the strongest quantitative evidence for the picture, stronger than the viscosity value itself, because the cross-section cancels out of them. A wrong molecular diameter would shift all three coefficients together and leave the ratios alone, so a ratio near two thirds is a statement about the mechanism rather than about any parameter that could have been tuned.

It also explains a fact about air that is otherwise a coincidence. That momentum and heat diffuse through it at nearly the same rate means that a boundary layer and a thermal layer over a surface have nearly the same thickness, which is why the same correlations serve for both in engineering and why a hand held near a hot object feels the heat and the draught arriving together.

Where the picture fails #

The cancellation depends on there being a next layer of gas for a molecule to hand its momentum to. When the free path reaches the size of the container, there is not.

The same kind of cancellation appears in the speed of sound, and it fails in the same place. Drop the pressure of a gas by three orders of magnitude and the speed of sound does not change, because it depends on the ratio of pressure to density and both fall together. Below the pressure at which the free path reaches the size of the apparatus, a gas stops being a continuum and neither quantity means what it meant — the sound stops propagating and the viscosity stops being a property of the gas.

At room temperature in a vessel a centimetre across, that happens at about 0.7 pascals for nitrogen — seven millionths of an atmosphere. Below it the molecules travel from wall to wall without meeting one another, momentum is carried directly from one wall to the other, and the drag becomes proportional to the pressure after all. This is the Knudsen regime, and the crossover is set by the ratio of the free path to the apparatus rather than by any property of the gas alone. The same gas is a continuum in a room and a collection of independent particles in a micrometre-wide channel.

A random walk’s mean squared displacement grows linearly with the number of steps, and that single fact underlies three coefficients at once. Viscosity, diffusion and thermal conduction are three names for the same walk carrying three different cargoes — momentum, molecular identity and energy — and the ratios between them are pure numbers of order one that a proper transport theory predicts. That the three are one mechanism is the strongest thing the kinetic picture says.

The gauge that only works where the line bends #

The flat line has an inversion that turns it into an instrument, and it is the reason a vacuum laboratory owns several different gauges rather than one.

Measuring a pressure below a millibar with a mechanical gauge is hopeless — the force on a diaphragm is too small — so what is measured instead is something the gas does. The obvious candidate is heat transport: run a current through a thin wire in the vacuum, let it come to a temperature at which what the gas carries away balances what the current puts in, and read the wire’s resistance. The cooler the wire, the more gas there is.

Except that above the Knudsen crossover there is no such relation, and this essay is why. Thermal conductivity, like viscosity, is independent of pressure while the free path is short compared with the apparatus. A hot wire in a chamber at a tenth of an atmosphere loses heat at exactly the rate it loses heat at one atmosphere, so the gauge reads nothing at all over the entire upper part of its range.

It becomes a gauge only below the crossover. Once the free path exceeds the distance from wire to wall, each molecule carries heat directly across, the rate becomes proportional to how many molecules there are, and the wire’s temperature finally depends on the pressure. So this kind of gauge is useful from roughly a hundred pascals down to a fraction of one — the decades in which the flat line is bending — and is blind above and below.

The same logic run on momentum instead of heat gives the most accurate gauge in that range: a small steel ball, magnetically levitated and spun up to a few hundred revolutions a second, left to slow down. In the free-molecular regime its deceleration is proportional to the pressure, with a constant of proportionality that can be computed rather than calibrated. It is a viscometer used as a barometer, and it works precisely where the viscosity of this essay’s title has stopped being constant.

What a constant viscosity does to a Reynolds number #

One consequence of the flat line reaches far outside vacuum work, because the quantity that decides whether a flow is smooth or turbulent has a density in it and a viscosity in it, and only one of them moves.

The ratio of inertial to viscous effects in a flow is $\rho v L/\eta$. Take an aircraft to altitude and $\rho$ falls by a factor of four between sea level and eleven kilometres, while $\eta$ changes only through the temperature — falling by perhaps fifteen per cent, in the opposite direction to intuition, because a colder gas has slower carriers. So the ratio falls with altitude almost in proportion to the density.

That is why a wing at altitude is operating in a different regime from the same wing near the ground at the same speed, and why aerodynamic data taken in a wind tunnel has to be matched on this ratio rather than on speed. It is also why flight in a very thin atmosphere is hard in a way that is not about lift alone: a vehicle in the Martian atmosphere, at about one per cent of Earth’s density, is working two orders of magnitude down this scale, in a regime where viscous effects dominate the flow over a blade and the aerodynamics of a propeller has more in common with an insect’s wing than with an aircraft’s.

Where the model stops #

A hard sphere is not a molecule. The whole treatment uses a single number $\sigma$ for the collision cross-section, and a real molecular interaction is a potential with a soft repulsive core and an attractive tail. That is why the measured temperature dependence is nearer $T^{0.7}$ than $T^{0.5}$ for most gases, and why the Sutherland correction — an empirical patch with one extra constant — is still in engineering use.

The gas is assumed dilute. Everything here treats collisions as isolated two-body events, which needs the molecules to be far apart compared with their size. At high pressure they are not, three-body collisions matter, and the viscosity acquires a density dependence after all — which is the beginning of the theory of dense fluids and is far harder.

The gas is assumed to be at a single temperature and to be one substance. A mixture has a viscosity that is not the average of its components’ and can exceed both of them, because the transport is done by whichever species crosses the surface, weighted by how much momentum it carries and how far it gets. That is a real effect and it is used analytically: the viscosity of a binary mixture depends on its composition in a way that is monotonic and measurable, so a viscometer can be a composition sensor.

And the gradient is assumed gentle. The Chapman–Enskog result is the first term of an expansion in the ratio of the free path to the scale over which the flow changes. In a shock front, where the flow changes over a distance comparable with the free path, the expansion fails and there is no viscosity coefficient to speak of — the fluid description itself has stopped applying, which belongs to the collection that owns compressible flow.

The same counting applied to a different question is where the kinetic picture hands over. Everything about transport is settled by the free path and the mean speed; everything about storage is settled by how many ways a molecule has of holding energy, and quantum mechanics decides which of those are available at a given temperature. Two different ladders out of one bag of molecules, and the free path appears in only one of them.

What the pictures cannot show #

The hero figure draws a line and the line is the argument. What no figure here can show is the cancellation happening — the density falling and the free path rising in exact compensation — because both are properties of the gas at a given pressure and the figure draws only their product. A pair of curves, one falling and one rising, would show the two factors and hide the point, which is that their product is what viscosity is.

Nor can any figure show what a viscosity is microscopically. It is a rate of transfer of a quantity across an imaginary surface that nothing marks, by carriers that are not tracked, in a direction that is not the direction of anything moving. Every drawing of it is a drawing of a consequence.

Where this ladder ends #

This is the sixth and last rung of the kinetic-theory anchor, and the anchor is closed with it. The ladder ran from the speeds in a still room to pressure as a rate of arrival, to why the air thins with height and why that is the same law, to how far a molecule gets, to the same arithmetic at nineteen orders of magnitude smaller cross-section, and now to what the free path is for.

What remains to be said about a bag of molecules is not on this ladder. Diffusion is its own anchor and the random walk underneath it is drawn there. The counting of accessible states is entropy’s. How much energy a molecule can hold is equipartition’s, and which of its modes are available at a given temperature is a quantum question. The Boltzmann equation itself — from which the coefficient in this essay comes, and from which the arrow of time notoriously does not — is a subject rather than a rung. The anchor is closed because the next essay against it would be one of those essays wearing a different name, not because the subject has run out.

The habit worth carrying away is the one Maxwell’s six years demonstrate. A theory earns its standing on the predictions its author did not want, and the density independence of viscosity is the cleanest example physics has: a result that follows in three lines from a picture, contradicts everyone’s intuition including its discoverer’s, and turns out to be exactly right for six decades of pressure.