Where a fluid stops being one

A gas is not a fluid. It is a very large number of molecules that spend almost all of their time not touching each other, and the reason it can be treated as a fluid at all is that within any volume worth drawing there are enough of them, colliding often enough, that averages over the volume behave like the smooth fields of continuum mechanics. What a flow is sets that out: the continuum is a modelling decision and not a fact about matter.

The decision has a size attached — the same kind of decision as the one that lets a flow be described by a velocity at every point rather than by particles — and the size is the mean free path — how far a molecule gets between collisions. Compare it with the size of whatever the gas is in and the ratio is the Knudsen number,

$$\mathrm{Kn} = \frac{\lambda}{H}.$$

At $\mathrm{Kn} = 1$ a molecule crosses the whole channel without meeting another one, and the picture of a fluid with a velocity at every point is not merely inaccurate — there is nothing left of it. So the number appears to carry its own threshold, more obviously than any other group in the subject.

The ladder everybody draws #

The standard account puts four rungs on the axis. Below $\mathrm{Kn} = 0.01$ the flow is a continuum. From there to 0.1 it is a continuum with slip at the wall. From 0.1 to 10 it is “transition”, which is an admission rather than a description. Above 10 it is free molecular, and the molecules are treated as not meeting one another at all.

Those boundaries are quoted as though they were results. They are a summary of results, and the summary has lost the only part that mattered.

What is exactly computable, and what it says #

Poiseuille flow between flat plates with Maxwell’s slip condition has a closed form. The wall velocity is $\sigma_v\lambda$ times the velocity gradient there, with $\sigma_v = (2-\sigma)/\sigma$ and $\sigma$ the fraction of molecules that leave the surface having forgotten where they came from. Carrying that through the ordinary integration gives

$$\frac{Q}{Q_0} = 1 + 6\sigma_v\mathrm{Kn}\quad\text{(plates)},\qquad \frac{Q}{Q_0} = 1 + 4\sigma_v\mathrm{Kn}\quad\text{(a tube)},$$

with the Knudsen number formed on the gap and on the radius respectively. Both are exact solutions of the Navier–Stokes equations with a stated wall condition — no expansion, no fit, nothing imported.

An ordinary no-slip calculation returns $Q_0$ and the flow is $Q$, so its relative error is $6\mathrm{Kn}/(1 + 6\mathrm{Kn})$. Setting that to $\varepsilon$ and inverting:

$$\mathrm{Kn} = \frac{\varepsilon}{6(1-\varepsilon)},$$

so one per cent falls at $\mathrm{Kn} = 1/594$, five per cent at $1/114$, ten at $1/54$.

Three orders of magnitude separate the number from its threshold, and the reason is entirely ordinary: the group is a ratio of two lengths, the observable is smooth in it, and the onset is the tolerance divided by the slope. That is the commonest kind of threshold in the subject and the commonest way to misread one.

The equations are fine — it is the wall #

The important half of this is what fails first. Nothing goes wrong with the Navier–Stokes equations at a Knudsen number of a thousandth. What goes wrong is the boundary condition: the claim that the gas immediately at the wall is at rest relative to it.

That claim is an idealisation of a collision process, and it is exact only in the limit of an infinitely dense gas. At finite density the molecules arriving at the wall carry momentum from a free path away, where the gas is moving, so the layer immediately at the surface moves too — by an amount proportional to $\lambda$ times the shear rate, which is Maxwell’s condition.

The same structure appears whenever a wall is replaced by an averaged condition: a porous surface presents a slip length of $\sqrt{K}$ with $K$ the permeability, and the interior flow never needs to be resolved. Here the length is the free path and the interior is the molecular one, and the arithmetic is the same.

Two things go wrong, and they are usually confused #

The slip term is a correction, and there are two separate questions about when a correction stops being one.

The first-order term reaches the size of the term it corrects — the slip doubles the flow — at $6\sigma_v\mathrm{Kn} = 1$, that is $\mathrm{Kn} = 1/6$. Past that the “correction” is the larger part of the answer.

The second-order term reaches the first at $\mathrm{Kn} = 1/2$. That is a different and worse statement: an expansion whose second term has caught its first has stopped converging usefully whatever its coefficients are, and the coefficients here are not even agreed — Cercignani’s, Deissler’s and Hsia’s second-order constants differ, because the second order is a modelling choice rather than a derivation.

The gap that no theory covers #

Beyond the slip regime the honest position is stranger than the ladder suggests. The transition rung is not a place where a third theory applies. It is a place where two exact theories both exist and disagree, and the truth is outside both of them.

Write a long tube’s mass flux in the reduced form the kinetic literature uses — a flow rate $G$ scaled so it stays finite at both ends, against a rarefaction parameter $\delta = (\sqrt\pi/2)/\mathrm{Kn}$. Then:

$$G_{\text{slip}} = \frac{\delta}{4} + 1.016,\qquad G_{\text{free}} = \frac{8}{3\sqrt\pi} = 1.5045.$$

The first is Navier–Stokes with a slip wall; the second is Knudsen’s own free-molecule integration, in which molecules cross the tube without meeting each other and the flow rate is set by the geometry alone. Neither contains the other’s physics, both are exact in their own limit, and they cross at $\delta = 1.954$ — in the middle of the transition regime, where neither of them holds.

The minimum is the one number in this essay that is measured rather than computed. It has to be: neither asymptote can produce a minimum, because one of them is monotone and the other is constant. Knudsen found it in 1909 and it is still called a paradox, which is a name for the fact that the ladder has no rung for a place where the flow rate is smaller than either limit allows.

What it means for something that is actually built #

The interesting sizes are not laboratory curiosities. The free path scales as $1/p$, so pressure and size trade off exactly, and four ordinary objects land on four different rungs.

The four cases below span the ladder, and the same arithmetic that places them places every other regime boundary this collection draws — the Reynolds number’s included, since which length goes in it is the same question asked of a different group.

A one-micron MEMS channel at atmospheric pressure has $\mathrm{Kn} = 0.067$, and an ordinary no-slip calculation of its flow rate is 29 per cent low. That is not a correction; it is the difference between a device that works and one that does not.

A ten-nanometre pore in shale at two hundred bar has $\mathrm{Kn} = 0.041$ and a 20 per cent error, which is worth a great deal of money and is why gas-shale reservoir models carry a slip term that oil-reservoir models do not.

The three-nanometre gap under a disc read head is at $\mathrm{Kn} = 24$: free molecular, with the continuum answer wrong by more than ninety-nine per cent. The lubrication theory used to design those bearings is a modified Reynolds equation with a rarefaction factor in it, and the factor is larger than the thing it multiplies.

A one-metre capsule at ninety kilometres has $\mathrm{Kn} = 0.02$, which is why re-entry aerodynamics has a slip-flow regime between the continuum and free-molecular ones and why the transition through it is the part of the trajectory nobody can compute cheaply.

The one number that keeps the model alive #

There is a reason the slip condition is worth having at all, and it is not that it extends the continuum equations by a decade of Knudsen number. It is that it costs one number.

Everything molecular about the wall — the accommodation, the free path, the distribution of velocities in the arriving flux — is compressed into a single length, and the interior of the flow is solved exactly as it always was. No mesh reaches into the molecular scale, no distribution function is carried, and the whole computational apparatus of continuum fluid mechanics survives unchanged. That is a very large return on a boundary condition.

It also fails in a way worth naming. The compression works because the wall region is thin compared with the channel, so the flow there can be summarised rather than resolved. When the free path is comparable with the channel, the “wall region” is the channel, there is nothing left to summarise, and the single number stops being a summary of anything. That is the same failure the second-order term announces, arriving from the other direction — one says the expansion has stopped converging, the other says the separation of scales it was built on has gone.

The effect with no continuum counterpart at all #

Everything above is a correction: a term the continuum model was missing a coefficient for, restored by a boundary condition. There is a rarefied effect of a different kind, and it is worth having because the continuum equations do not merely get it wrong — they have no place to put it.

Run a temperature gradient along a wall, with no pressure difference anywhere, and the gas next to the wall creeps from the cold end towards the hot one. The mechanism is the slip argument with temperature in place of velocity: molecules arriving at any point on the surface come from about a free path away in every direction, and those from the hot side arrive faster than those from the cold, so the flux is unbalanced and the layer drifts. The drift velocity is proportional to the free path times the fractional temperature gradient, and it exists with the gas perfectly at rest in the continuum sense.

It is in the same 1879 appendix as the slip condition, and it is the half of Maxwell’s result that gets dropped.

Its most striking consequence is that two vessels at different temperatures, connected by a small enough tube, do not come to the same pressure. In the free-molecular limit the steady state with no net flow is

$$\frac{p_1}{p_2} = \sqrt{\frac{T_1}{T_2}},$$

so a hot vessel sits at a higher pressure than a cold one for ever. That is thermal transpiration, Reynolds measured it in the 1870s, and it is not a transient: it is the equilibrium of a system that the continuum description says should have equalised.

Two things follow that are worth carrying.

It is a pump with no moving parts. Cascade a series of narrow channels with alternating hot and cold ends and each stage adds its square root, so a Knudsen pump compresses gas using only a temperature difference — no valves, no bearings, nothing to wear — which is why the idea has come back for micro-scale vacuum systems where a mechanical pump cannot be built at all.

And it is what spins a radiometer. The vanes of a Crookes radiometer are not pushed by light pressure, which would turn them the other way and is far too weak; they are driven by thermal creep along the temperature gradient at the edges of each vane. The tell is that the device works only in a middle range of pressure — too dense and there is no free path to creep over, too rarefied and there is no gas to creep — which is exactly a statement that its mechanism lives at intermediate Knudsen number and nowhere else.

Which length, and the factor of two nobody quotes #

Every threshold above is quoted for a particular choice of $H$, and the choice is never stated. $\lambda/H$ and $\lambda/(H/2)$ differ by two; a tube quoted on its radius and on its diameter differ by two; and every published rung boundary is given to one significant figure. A factor of two on a boundary quoted to one figure is the whole boundary.

The solver behind these figures takes the length explicitly and every figure prints which one it used, for that reason. It is the same discipline the site applies to the length in the Reynolds number and to the depth in the Froude one, and the same failure is available: two people can agree on a threshold, agree on a measurement, and disagree by a factor of two about whether the threshold has been crossed.

There is a second ambiguity, quieter than the first. The free path itself is not a single agreed quantity: the hard-sphere value computed from a molecular diameter and the value computed from the measured viscosity differ by about ten per cent for air and by more for anything polar. Every number here uses the viscosity route, because a molecule has no diameter to measure and the viscosity does. That is a choice, it is stated on the figures, and it moves every threshold in this essay by a tenth — which is small beside the factor of two above and is not small beside the two significant figures a rung boundary is quoted to.

What the picture cannot show #

The accommodation coefficient is imported. $\sigma$ is a property of a surface and its adsorbed layer, not of the gas, and it ranges from about 0.8 to 1 for engineering surfaces and can be much lower for clean, smooth ones. Every threshold here is quoted at full accommodation; halving $\sigma_v$ moves them all by a factor of three, and no calculation in this essay can tell anybody what their own surface does.

The second-order coefficient is a choice. The figure that draws it says so. Nothing here derives it, and the published values disagree by tens of per cent — which is precisely why the second-order term is used to say where the expansion stops rather than to extend it.

Nothing here solves the Boltzmann equation. The transition regime is drawn as a gap between two asymptotes, with a measured point in it, and that is the honest shape of the knowledge. A linearised kinetic solve would fill it in, and it would be a different kind of calculation from anything else in this collection.

And the flows are fully developed and isothermal. Thermal creep is described above and computed nowhere: no figure here carries a temperature gradient along a wall, and the transpiration ratio is quoted from the free-molecular limit rather than solved for at any finite Knudsen number.

Who found it, and when #

Maxwell wrote the slip condition in 1879, in an appendix to a paper on the stresses in rarefied gases, and gave the coefficient in terms of the accommodation he had to invent to state it. Knudsen measured the tube flow rates between 1909 and 1911 and found the minimum, which was not expected and has never stopped being surprising. Cercignani and others put the second-order terms on a kinetic footing in the 1960s, by which point the applications had moved from vacuum physics to spacecraft.

The surprising connection is with the porous wall, which is a completely different physical situation with identical mathematics. In both cases an interior nobody wants to resolve — molecular here, a network of pores there — is replaced by a single length in a boundary condition, and in both cases the length is small and the improvement is enormous: replacing no-slip with a slip length beats a plain wall by a factor of hundreds. The interior does not need to be understood. It needs to be given one number.

Where the ladder goes next #

Above this rung is the transition regime itself, and getting into it honestly means solving a kinetic equation rather than a fluid one — a different kind of calculation, with a distribution function over velocities where this collection has a velocity field. The figures here draw the gap and mark a measurement in it, which is as far as a solver of the Navier–Stokes equations can go without pretending.

Beside it sits the porous wall, which is the same trick with a different length, and the boundary conditions a flow must be told, which is where the failure in this essay actually lives. Below it is what a flow is — the decision to treat matter as a continuum, made in the first essay of this collection and unpicked here.