A gas's viscosity ignores its pressure until the free path reaches the walls

What a flow is set out the continuum — a fluid treated as a field with a velocity at every point, though it is made of molecules — and, in a short section on where viscosity comes from, said two things about a gas that are hard to guess from the continuum alone. Its viscosity grows with temperature, the opposite of a liquid’s. And it is nearly independent of pressure, because doubling the density doubles the number of molecules carrying momentum between layers and halves the distance each carries it. Both were described as experimental facts that the molecular picture explains.

The second is stranger than it sounds and more limited than it sounds, and both halves are worth computing. Stranger, because it says a gas at a hundredth of an atmosphere drags on a moving plate exactly as hard as one at an atmosphere, which runs against every intuition about thin air. Limited, because it cannot go on for ever: at low enough pressure there are no molecules at all, and no drag. Somewhere the cancellation stops, and where it stops is the subject of this essay.

The cancellation #

The argument is Maxwell’s, and its one line is all this essay needs from it: the momentum a gas carries across a gradient goes as the number of molecules per unit volume times the distance each carries it, and since the free path grows as the density falls, the product does not change. Maxwell found the conclusion surprising enough to test, and in 1866 his swinging discs confirmed it from an atmosphere to a sixtieth of one. What the argument quietly assumes is the subject here. It lets each molecule carry its momentum a free path, and so it needs room for a free path — a space the molecule can cross while colliding many times. Halve the pressure and the free path doubles; keep halving it and any fixed gap, a millimetre or a micrometre, eventually holds less than one.

The kinetic equation, across every pressure #

The cancellation holds only while a molecule collides many times crossing the space the measurement shears. Between two plates a distance $H$ apart that means a free path much shorter than $H$, and the question is what happens as it is not. That needs the kinetic equation rather than viscosity, and the same tool that computed a gas flowing least at one free path across answers it: the linearised BGK equation, with the walls re-emitting every molecule at their own velocity and temperature.

Let the two plates slide past each other at a relative speed $U$. Molecules leaving each wall carry its velocity; on the way across they collide and relax towards the local mean. Integrated along each molecule’s straight path, the kinetic equation becomes an integral equation for the mean velocity across the gap, with Abramowitz’s functions as its kernel and the walls’ velocities as its source, and the shear stress — the same at every height, since nothing else acts on the gas — follows from the same paths. In units of $pU/v_0$ the stress is $1/\sqrt\pi$ for a gas so rarefied its molecules never collide, and $1/\delta$ for a continuum, where $\delta = Hp/\mu v_0$ is the rarefaction parameter, the gap in free paths up to a factor of $2/\sqrt\pi$. Their ratio, the stress over Navier–Stokes’s $\mu U/H$, is what an instrument shearing the gas between plates and believing Navier–Stokes would report as the viscosity.

Five decades, and then a straight line #

The first figure is the answer. Down to a rarefaction of about twenty — a gap of twenty-odd free paths — the gas transmits more than nine-tenths of its full viscous stress. At $\delta = 2$ it transmits half. Below one, the stress is the free-molecular one, proportional to the density: $0.098$ of the continuum value at $\delta = 0.2$, against $\delta/\sqrt\pi = 0.113$ for a gas with no collisions at all.

In pressures, for air at room temperature, the instrument’s gap sets everything. Plates a millimetre apart see air’s viscosity within five per cent from an atmosphere down to about three hundred pascals — two and a half decades of pressure, Maxwell’s whole range and more — and half of it at 15.3 pascals, where the free path is half a millimetre; below that, the apparent viscosity falls in proportion to the pressure. Plates ten micrometres apart reach half at 1.5 kilopascals, inside Maxwell’s range: his discs, a few millimetres apart, were safely in the continuum, but a micro-machined resonator with a ten-micrometre air gap is not, and its damping changes with the barometer. Plates ten centimetres apart keep the full viscosity to a tenth of a pascal.

The engineering rule that follows is simple and widely broken. A device that relies on a gas film’s viscous damping — a micro-machined accelerometer, a hard-disc slider, a resonator in a sealed package — has a damping that is a property of the gas only if its gap is many free paths at the lowest pressure it will meet. Sealed at a tenth of an atmosphere, a two-micrometre gap is about three free paths and its damping is a little over half of what the viscosity table says; the designers of such devices carry an “effective viscosity” correction for exactly this curve, and the curve here is the one that correction approximates.

The viscosity does not fall; the gas stops being able to use it. At low pressure the molecules cross the gap without colliding, each carrying straight from one wall to the other the momentum it got there, and the stress becomes a count of how many make the crossing per second, which is proportional to the density. The viscosity, a property of collisions, has stopped being the relevant quantity, as it did for the flow rate in where a fluid stops being one.

The profile, and its slip #

The velocity profile shows the same thing in space. In a continuum it is a straight line from one wall’s speed to the other’s. In a rarefied gas it is still nearly straight in the middle, with a gentler slope, and it does not reach the walls’ speeds: the gas next to each wall slides relative to it, by about a free path’s worth of the gradient. At $\delta = 1$ the gas at the upper wall moves at half the wall’s speed; at $\delta = 0.1$, at a seventh of it, because almost every molecule there has just arrived from the other wall.

Here Couette flow differs from pressure-driven flow in a way worth noticing. Navier–Stokes with a first-order slip condition at each wall gives the stress as $\mu U/(H + 2\zeta\lambda)$, and between plates it is within 2.7 per cent of the kinetic solve at $\delta = 1$, where the gap is about a free path. For the flow a pressure gradient drives through the same channel, the same slip formula is nearly a quarter low at that rarefaction. A shear flow’s profile is a straight line, which the Knudsen layers at the walls bend only slightly; a pressure flow’s is a parabola whose curvature the layers interfere with more. So the slip formula is a better description of a viscometer than of a microchannel, and the viscosity measured by shearing is trustworthy further into the rarefied range than the one inferred from flow.

A gauge that reads the fall #

The fall that spoils a viscometer makes a vacuum gauge. Below the pressure where the stress becomes free-molecular, the drag a gas exerts on a moving surface is proportional to its pressure, with a coefficient set by the molecules’ mass and temperature alone. A steel ball levitated magnetically and spun up, then left to coast, slows at a rate proportional to the gas pressure round it: the spinning-rotor gauge, which reads pressures from about a pascal down to a ten-thousandth of one and is the secondary standard for that range. It works because its gap — the distance to the chamber’s walls — is many free paths below a pascal, and stops working at the top of its range for exactly the reason this essay computes: above about a pascal the molecules collide on the way, the drag becomes viscous, and viscous drag ignores the pressure.

The same transition, for heat rather than momentum, makes the Pirani gauge, which reads pressure from the heat a hot wire loses: a gas’s thermal conductivity is independent of its pressure for the same reason its viscosity is, until the free path reaches the wire’s surroundings. The two transport coefficients are the same calculation with a different thing carried, which is why every gas’s Prandtl number sits near three-quarters.

One number for every gas: pressure times gap #

The kinetic solve has a single parameter, the rarefaction $\delta = Hp/\mu v_0$, and the stress falls to half its continuum value at $\delta = 2.00$ — within two per cent of the 2.03 the slip formula gives, which is why that formula serves a viscometer so well. So the pressure at which a gap of width $H$ transmits half the viscous stress is fixed by one product,

$$pH = 2.00\,\mu v_0,$$

a property of the gas and its temperature alone. For air at room temperature it is 15.3 millipascal-metres: 15 pascals across a millimetre, 1.5 kilopascals across ten micrometres, 0.15 pascals across ten centimetres. For helium, whose light molecules are fast and whose viscosity is no lower, it is 44 — a helium-filled gap holds its viscosity only to three times the pressure an air-filled one does. Hydrogen’s product is 28, argon’s 16 and sulphur hexafluoride’s 5.6: the heavy, slow molecule keeps a gap in the continuum to a third of air’s pressure. Nine-tenths of the stress is kept down to $\delta = 18.3$, nine times higher.

That product is the design number for anything that relies on a gas film’s damping. A micro-machined accelerometer with a two-micrometre gap sealed at a tenth of an atmosphere has $pH$ of 20 millipascal-metres, barely above air’s half-stress value, so its damping is a little over half of what a viscosity table promises; back-filled with sulphur hexafluoride at the same pressure it would sit at nearly four times that gas’s product and keep about four-fifths. A hard-disc slider flying on ten nanometres of air at an atmosphere has $pH$ of one millipascal-metre, deep in the free-molecular range, and its bearing is designed with a kinetic film equation for exactly this reason. Choosing the fill gas moves the whole curve; choosing the pressure moves the device along it.

A smooth wall moves the curve #

Every number above assumes the walls are diffuse: each molecule that strikes one forgets its velocity and leaves with the wall’s. Real surfaces are partly specular. A fraction of the molecules reflects as a ball off a mirror, keeping their tangential velocity, and those transfer no momentum at all. The fraction that is accommodated, $\alpha$, is close to one for most engineering surfaces in ordinary air and can fall to 0.8 or below for clean, smooth ones and light gases.

Near the continuum the effect is a longer slip. Maxwell’s estimate scales the slip coefficient by $(2 - \alpha)/\alpha$, half as long again at $\alpha = 0.8$ and three times at $\alpha = 0.5$. In the slip formula, which tracks the kinetic solve to a few per cent down to a free path, the half-stress product rises in the same proportion: across a millimetre of air the stress halves at 15 pascals on a diffuse wall, about 23 on a wall that accommodates eight molecules in ten, and about 46 at half. So the accommodation enters a viscometer’s reading exactly where the rarefaction does, and the two cannot be separated by one measurement at one pressure. They can be separated by the shape: rarefaction moves along the curve with $pH$, accommodation shifts the whole curve sideways, and a sweep in pressure across the half-stress point reads both.

The free-molecular end is affected more simply. With no collisions in the gap, a molecule specularly reflected at one wall carries its old velocity to the other, and the stress is the diffuse value multiplied by $\alpha/(2 - \alpha)$ when both walls share the same accommodation. A spinning-rotor gauge, whose rotor is the one surface that matters, carries its accommodation as a single calibration factor, and that factor is the gauge’s main correction.

Where the continuum ends, for a stress #

That last condition is the one the kinetic solve makes quantitative, and its answer is a statement about the measurement as much as the gas. The same air at the same pressure is a continuum between plates ten centimetres apart and a free-molecular gas between plates ten micrometres apart; the threshold is the gap in free paths, and nothing about air alone. For stress the threshold is lower than the flow-rate threshold of where a fluid stops being one: a shear between plates keeps nine-tenths of its continuum value to twenty free paths and half to two, where a pressure-driven flow is wrong by a few per cent long before that. What “of order one” is worth is different for every quantity, and here it is a factor of ten between the stress and the flow.

The practical edge is the satellite’s. At two hundred kilometres the free path is a few hundred metres, a molecule that strikes a spacecraft has met no other within that distance, and the drag is a free-molecular count of the molecules met, proportional to the density with no viscosity anywhere in it. An airliner at ten kilometres, with a free path of a few tenths of a micrometre, is a continuum to its rivets.

The convention, and what the model leaves out #

The rarefaction $\delta = Hp/\mu v_0$, with $v_0 = \sqrt{2RT}$, is the gap measured in units close to the free path: $\delta = \sqrt\pi/2\, \mathrm{Kn}^{-1}$ with $\mathrm{Kn} = \lambda/H$ and $\lambda = (\mu/p)\sqrt{\pi RT/2}$. The “apparent viscosity” is the stress over $U/H$, the number a Navier–Stokes instrument reports; it is not a property of the gas. The model is the BGK equation, linearised, with diffuse walls: every molecule striking a wall forgets its velocity and leaves at the wall’s. A smooth surface reflecting some molecules specularly would slip more, and its curve would lie lower at every rarefaction. The plates are infinite and the flow steady and isothermal; a real viscometer’s oscillating disc has edges and a frequency, and at high enough frequency the gas’s own relaxation enters, which is the bulk viscosity’s territory rather than this one.

Who measured it #

Maxwell derived the independence in 1860 and measured it in 1866. Kundt and Warburg found the departure from it in 1875, pumping Maxwell’s kind of apparatus far lower and finding the damping fall: they explained it as slip at the walls and estimated the slip length as a free path, which Maxwell then put on a kinetic footing in 1879, with the accommodation coefficient that bears his name. The kinetic Couette problem was solved across all rarefactions with the BGK model by Willis and by Cercignani in the early 1960s, and the spinning-rotor gauge, conceived by Beams in the 1960s, was turned into a standard by Fremerey in the 1980s.

How it was checked #

At a rarefaction of a thousandth the stress is the free-molecular $1/\sqrt\pi$ to nine parts in ten thousand, the remainder the first few collisions. At $\delta = 20$, solved on 300 and 600 cells and extrapolated, the stress returns a slip coefficient of 1.0161 against the BGK model’s 1.016 — the same number the pressure-driven channel recovered from a flow rate, here recovered from a stress. Doubling the cells at $\delta = 1$ moves the extrapolated stress by eight parts in a hundred million. The velocity profile between plates moving oppositely is antisymmetric to rounding. And the solver refuses a gas with no density, a rarefaction beyond its grid’s reach, an odd number of cells and a gap of zero.

Still open: the gas that is not at one temperature #

Every number here is for a gas at uniform temperature, and between two plates that is a choice: shear heats a gas, and in a narrow, rarefied gap the heat cannot leave as conduction says, for the same reason the momentum could not. The next calculation adds the energy equation to the kinetic Couette problem — walls at the same temperature, the gas heated by its own shear — and asks how the temperature jump at each wall grows with rarefaction beside the velocity slip, and whether the viscous heating that a continuum puts in the middle of the gap moves to the walls when the molecules cross it without colliding.