Slip is a memory of one mean free path

Where a fluid stops being one is this collection’s account of the Knudsen number: the mean free path divided by the size of the thing, and the number that says whether a gas can be treated as a continuum at all.

This essay is about the first thing that goes wrong as that number rises, which happens long before the continuum description does. It is the boundary condition, and the reason it goes wrong is that a molecule arriving at a wall is carrying information from somewhere else.

Where the memory is #

Consider the molecules crossing a plane just above a wall. Half of them are arriving from the gas above and half are leaving the wall.

The arriving ones last collided about a mean free path away, and they carry the gas’s mean velocity from there. In a shear flow that velocity is larger than the wall’s, by the mean free path times the velocity gradient.

The departing ones have come off the wall. If the surface is fully accommodating they leave with the wall’s own velocity, on average.

Averaging the two gives a mean velocity at the wall that is not zero: it is about half the arriving excess, which is a coefficient of order one times the mean free path times the gradient.

So the slip is a memory in the strict sense used throughout this collection. The gas at the wall is carrying the state of a place one mean free path away, and the mean free path is how far back the carrying reaches.

What that does to a channel #

The wall velocity is exactly proportional to the Knudsen number, which the computation confirms to a part in 10⁹ — the proportionality is built into the first-order slip condition, so what is being checked is the arithmetic rather than the physics.

At a Knudsen number of 10⁻⁴, which is air in a millimetre channel, the slip is two hundredths of a per cent of the centreline velocity and nobody would notice. At 10⁻², which is a ten-micron channel, it is two per cent. At a tenth it is seventeen.

The flow rate follows across five decades of Knudsen number:

Six per cent more than the no-slip value at a Knudsen number of a hundredth, and sixty per cent more at a tenth. The enhancement column is 1 + 6Kn exactly, so a thousand-fold change in the Knudsen number moves it from six parts in ten thousand to six parts in ten. That is not a correction; it is the difference between a microchannel working and not working, and it is the reason microfluidic designs computed with no-slip boundaries deliver more than they were designed to.

Why it always increases the flow #

The direction of the effect is worth a paragraph, because it is the same in every case and the reason is simple.

The slip velocity is proportional to the velocity gradient at the wall, and in any pressure-driven flow that gradient points away from the wall — the fluid further out is moving faster. So the arriving molecules always carry a higher velocity than the wall’s, and the mean velocity at the wall is always in the direction of the flow.

That means slip always adds to the flow rate and never subtracts. A channel, a pipe, a bearing gap and a porous medium all pass more than the continuum calculation says, and by an amount that grows with the Knudsen number.

The consequence is a systematic bias rather than a scatter, which is the dangerous kind of error. A microchannel calibrated against a no-slip prediction will appear to have a larger cross-section than it has, and the discrepancy will be reproducible — so it looks like a manufacturing tolerance rather than a physical effect.

The tell is that the discrepancy depends on pressure. The mean free path is inversely proportional to the density, so lowering the pressure raises the Knudsen number and raises the enhancement; a no-slip error does not do that, and the pressure dependence is how Knudsen identified the effect.

How far the memory reaches #

The mean free path in air is 68 nanometres at sea level, six microns at thirty kilometres and about a centimetre at eighty. The same gas is therefore a continuum in a room, marginal in a micro-electromechanical device, and free-molecular around a satellite.

Two gaps make the point, both in air at sea level, where the mean free path is 68 nanometres. A hard-disk read head flies about ten nanometres above the platter, so its Knudsen number is 6.8 and the gas in the gap is not a fluid at all — the air bearing that keeps the head off the disk is a rarefied-gas calculation. A one-micron channel is at 0.068, where slip is a large correction — about 41 per cent more flow — and the continuum equations still work with it in place. The same mean free path is 0.2 µm at ten kilometres of altitude, 6 µm at thirty, 80 µm at fifty and a centimetre at eighty, so the same body changes regime by climbing rather than by shrinking.

What the solver computed, and how it was checked #

The channel flow with a first-order slip condition has a closed form: the parabolic profile displaced upwards by a constant, with the constant proportional to the Knudsen number. Everything here is arithmetic on that.

Three checks. That the slip velocity is exactly proportional to the Knudsen number across the sweep, which tests the implementation. That the slip at a Knudsen number of a hundredth exceeds one per cent of the centreline velocity — it reads 1.96 per cent — so the effect is present at the scale it is claimed for. And that the flow enhancement there exceeds five per cent; it reads 6.00 per cent, which is the quantity a designer would notice.

Why the no-slip condition works as well as it does #

The essay has been arguing that no-slip is wrong, so it is worth saying why it is nonetheless one of the most successful assumptions in the subject.

Because the Knudsen number is usually tiny. In air at sea level around anything of ordinary size it is 10⁻⁶ or smaller, so the slip is a millionth of the flow’s velocity — below every measurement and below every other error in the calculation.

No-slip is an excellent approximation whose error is exactly known and is proportional to a small number. That is the best possible situation for an assumption, and it is why the condition survived a century of dispute before the kinetic argument settled it — the dispute could not be resolved by experiment because the effect was too small to see.

A wall that is not quite there is this collection’s account of the same condition from the other side: what a slip length does to a computation when it is imposed as a boundary condition rather than derived.

Where the memory becomes the whole problem #

As the Knudsen number rises past about a tenth the first-order slip correction stops being enough, and the reason is worth stating in the language used here.

First-order slip assumes the gas one mean free path away can be described by extrapolating the continuum profile — that is, that the velocity gradient is constant over the distance the memory reaches. When the mean free path becomes comparable with the channel, the profile curves over that distance and the extrapolation fails.

The first-order relation is a straight line through the origin: the wall velocity is exactly the Knudsen number in these units, at 0.0001, 0.001, 0.01, 0.03 and 0.1 across the sweep, and the enhancement is exactly 1 + 6Kn. What replaces it above about 0.1 is a Knudsen layer: a region about a mean free path thick in which the velocity profile is genuinely not the continuum one, and in which the distribution of molecular velocities is not Maxwellian. Beyond that, at Knudsen numbers above about ten, the molecules cross the channel without colliding at all and the gas is free-molecular — a regime with no fluid mechanics in it whatever.

So the sequence is: no-slip, slip, Knudsen layer, transition, free molecular — five regimes on one axis, separated by how far the memory reaches compared with the geometry.

Where this number sits among the others is the last essay in this field, on the same machinery.

The same construction in the other transport quantities #

Velocity slip has two companions and they arise identically.

Temperature jump. A molecule arriving at a wall carries the gas temperature from a mean free path away, so the gas temperature at the wall differs from the wall’s. The jump is proportional to the mean free path times the temperature gradient, and it matters for heat transfer in the same regimes.

And concentration slip, in a mixture, for the same reason with a composition gradient.

All three are the same statement: a transported quantity at a wall is a memory of one mean free path of gas, and the coefficient in front is a property of how well the surface accommodates rather than of the gas. That accommodation coefficient is measured, varies between 0.8 and 1 for engineering surfaces, and is the largest uncertainty in any of these calculations.

How much of the wall the memory reaches past #

There is a geometric consequence worth naming, because it decides where in a device the effect lives.

The slip is a boundary effect, so its influence on the flow is confined to a region of order the mean free path from the wall — outside which the profile is the continuum one, merely displaced.

In a wide channel that region is a negligible fraction of the cross-section, so the slip’s whole effect is to add a plug velocity: at a Knudsen number of 0.0001 the wall moves at 0.0001 of the centreline speed and the flow rises by 0.06 per cent; at 0.01 the wall moves at 0.01 and the flow rises by 6. The shape of the profile is unchanged and only its offset moves. That is why the flow enhancement is simple and why the first-order condition works so well over such a wide range.

In a narrow one the region is a substantial fraction, and the two walls’ Knudsen layers begin to overlap — at which point there is no continuum core left and the profile is not a parabola anywhere. That happens at a Knudsen number of order one, and it is the boundary between the slip regime and the transition regime.

So the ladder of regimes is really a statement about how much of the channel is inside the memory’s reach, which is the same reading the thin layer gives of a boundary layer in a completely different setting.

Why this is a memory and not just a boundary condition #

The other wall in this collection that carries a memory is the wall the fluid is listening to, and the two are opposite kinds: that one remembers a history over a window, and this one remembers a single length.

The distinction is worth pressing because the slip is usually presented as a modified boundary condition and the framing hides where it comes from.

A boundary condition is a statement about the flow at a surface. This one is a statement about the flow a mean free path away, transported to the surface by molecules that do not collide on the way. The length in it is not a property of the wall or of the geometry; it is the distance over which the gas carries information without changing it.

That makes the mean free path the gas’s own memory length, in exactly the sense that the relaxation time is a polymer’s memory time in the fluid that has not finished its last deformation — and the Knudsen number is that length divided by the geometry, which is what every memory number in this collection turns out to be.

What is actually measured #

The chain from the physics to a number is worth setting out, because the quantity that is measured is not the slip.

A flow rate and a pressure drop are what a microchannel experiment produces, and the ratio of the two gives a conductance. Comparing it with the no-slip prediction gives the enhancement directly, which is where the sixty per cent above comes from.

The slip length is then inferred, by inverting the relation between the enhancement and the Knudsen number. That inversion needs the channel’s dimensions, which at the micron scale are known to a few per cent at best — so the inferred slip length inherits that uncertainty and more.

And the accommodation coefficient is what is really wanted, since it is the surface property that transfers between geometries. Getting it requires the slip length and the mean free path, both with their own errors.

So the reported quantity is three inversions away from the measurement, and published accommodation coefficients for nominally identical surfaces differ by more than the effect they describe. That is the usual position for a coefficient obtained by dividing one uncertain thing by another, and it is the same complaint a guess with a constant in it makes about a closure constant.

What the picture cannot show #

The profiles are drawn as parabolas displaced upwards, which is what the first-order slip condition gives, and the real profile inside the Knudsen layer is not a parabola and not a displaced one. That region is a mean free path thick, so at the Knudsen numbers drawn it is a thin sliver at the edge of the plot, and the figure is showing the continuum solution that is valid outside it.

Nothing here shows a molecule. The whole argument is kinetic and the figures are continuum, which is the essay’s own subject read as a limitation.

Where else the same length appears #

The mean free path sets more than the slip, and collecting the other places makes the point that it is the gas’s own memory length rather than a boundary-condition parameter.

The viscosity itself. Kinetic theory gives the viscosity as the density times a mean speed times a mean free path — so viscosity is a transport of momentum over that distance, and the no-slip condition’s failure and the existence of viscosity have the same origin.

The shock’s thickness. The discontinuity that has a thickness computes a shock as a few mean free paths, for the same reason: it is the shortest distance over which a gas can change its state.

And the continuum limit itself. A fluid description is valid where the mean free path is short compared with the gradients, which is where a fluid stops being one’s subject.

Three apparently unrelated quantities — a transport coefficient, a discontinuity’s width and a boundary condition — all set by the same length. A gas has exactly one memory length and everything about it follows from that length’s ratio to something else, which is an unusually clean example of the construction this collection keeps arriving at.

Who found it, and when #

Maxwell derived the slip condition in 1879, in an appendix, from exactly the argument above; the coefficient in it is his. Knudsen’s experiments on flow through capillaries are from 1909 and gave the number its name and the first measurements of the enhancement.

The condition was contested for decades, because at ordinary scales the effect is unmeasurable and the no-slip condition works. It became a practical matter twice: for high-altitude flight in the 1950s, and for microfluidics and hard-disk air bearings from the 1980s — two applications separated by six orders of magnitude in size and united by a Knudsen number.

Limits recorded rather than smoothed over #

First-order slip only. The condition used here is linear in the mean free path, which is the leading term of an expansion. Second-order versions exist, disagree with one another, and are what is needed above a Knudsen number of about a tenth.

Full accommodation assumed. The coefficient depends on how molecules reflect from the surface, and the value used takes every one to be re-emitted with the wall’s velocity distribution. Real surfaces partly reflect specularly, which increases the slip.

The channel is two-dimensional and fully developed. A real microchannel has corners, an entrance and a finite length, and the development question is a duct that forgets everything but one number’s with a slip condition on it.

No Knudsen layer, no temperature jump. Both are described and neither is computed. Including the first changes the profile near the wall and the second couples the problem to the heat transfer.

A gas, not a liquid. The whole argument is kinetic and assumes molecules travelling freely between collisions. A liquid has no mean free path in that sense, and liquid slip — which is real and is a large subject — has a completely different origin in the interaction between the liquid and the surface.

And the mean free paths are tabulated. They come from kinetic theory with a hard-sphere collision cross-section, and a more careful model changes them by tens of per cent — which moves every Knudsen number here by the same factor.