A length you can only measure by squeezing

Every calculation with a wall in it has been told that the fluid at the wall moves with the wall. The air a wing does not carry is that statement read correctly; a drop that a no-slip wall would never let spread is that statement read to its breaking point. Neither essay asked the prior question, which is how anybody knows.

The answer is not that it follows from anything. It does not. The Navier–Stokes equations are second order in the velocity and therefore need a condition at each boundary; how many, and of what kind, is decided by the order of the equations and not by their content. The equations are perfectly happy with a wall that slips, and Navier wrote the alternative down in 1823 — a wall velocity proportional to the wall shear rate, with a length as the constant:

$$u_{\text{wall}} = b\,\frac{\partial u}{\partial n}.$$

No slip is the case $b = 0$. It is a choice among admissible conditions, it was contested for the better part of a century, and what settled it was measurement.

No ordinary flow can see it #

The difficulty is that a slip length has to be compared with something, and almost every flow compares it with a length set by the apparatus.

Poiseuille flow through a round pipe with Navier slip at the wall carries a flow rate larger by a factor $1 + 8b/a$, where $a$ is the pipe’s radius. A slip length of one nanometre in a pipe of one millimetre is eight parts in a million — which is smaller than the uncertainty in the radius, smaller than the temperature drift in the viscosity, and very much smaller than the deviation caused by the pipe not being exactly round. Widening the pipe makes it worse. Narrowing it helps, and narrowing a pipe to the point where the ratio is respectable means a bore of micrometres, at which point the pipe cannot be measured either.

Every other candidate has the same shape. A rotating-cylinder viscometer divides the slip length by the gap between the cylinders, which is millimetres. A falling-ball viscometer divides it by the ball’s radius. A flow through a packed bed divides it by the pore size, which is the smallest of these and is also the least well known. In each case the instrument’s own size is the denominator, and instruments are large.

There is a general principle underneath that list and it is worth stating, because it explains why the question stayed open for a century and a half rather than for a decade. A boundary condition is a statement about a surface, and a surface has no thickness; any flow measurement is a statement about a volume. To convert one into the other the experiment must form a ratio of the wall’s length to the flow’s, and the flow’s length is whatever the apparatus was built out of. Improving the instrument improves the numerator of the measurement and the denominator of the ratio at the same time, so precision does not help: a pipe-flow measurement good to one part in $10^6$ bounds the slip length at a nanometre only if the pipe’s radius is known to one part in $10^6$, which nobody has ever managed and which would in any case be defeated by the pipe’s own roughness.

The way out is not a better measurement of a large ratio. It is an experiment in which the small length is not a property of the hardware. There is exactly one such experiment in fluid mechanics, and this collection has already spent an essay complaining about the property that makes it one.

The one experiment whose small length is chosen #

A squeeze film is the exception, and the reason is precisely the property that makes it a nuisance everywhere else.

The gap in a drainage experiment is not a property of the apparatus. It is a coordinate: the experimenter drives one surface towards the other and measures the force as a function of separation, so the small length in the ratio is under control and can be taken down to tens of nanometres.

Solving Reynolds’ equation for a sphere approaching a plane with Navier slip $b$ on both surfaces is a short calculation. The film thickness is $h = h_0 + r^2/2R$, the flux with slip is $q = -(h^3/12\mu)(1 + 6b/h)\,dp/dr$, and continuity fixes $q = Vr/2$ without any reference to the pressure. Integrating the pressure over the disc and changing variable from radius to film thickness gives

$$F = 12\pi\mu V R^2 \int_{h_0}^{\infty}\frac{(u - h_0)\,du}{u^2\,(u + 6b)},$$

which partial fractions exactly. The result is the no-slip force times a correction factor:

$$F = \frac{6\pi\mu V R^2}{h_0}\,f^*, \qquad f^* = \frac{h_0}{3b}\left[\left(1 + \frac{h_0}{6b}\right)\ln\!\left(1 + \frac{6b}{h_0}\right) - 1\right].$$

That expression and a Simpson quadrature of the integral it came from agree to $9\times10^{-13}$ across five decades of the ratio between them, which is the check that the partial fractions were done correctly rather than a check on the physics.

The shortfall is twice the ratio, and that is the whole instrument #

Expanded for a slip length small against the gap, the correction is

$$f^* = 1 - \frac{2b}{h_0} + \frac{(6b/h_0)^2}{6} - \cdots,$$

so the fraction of the drainage force that slip removes is twice the slip length over the gap.

The arithmetic of that is worth following, because it is what makes the experiment possible rather than merely conceivable. A three-millimetre sphere driven at ten microns a second through a liquid of 0.1 pascal seconds feels 170 micronewtons at a gap of a micron and 1.7 millinewtons at a hundred nanometres. A one-nanometre slip length removes two per cent of the second of those — 34 micronewtons, on a force that a surface-force apparatus measures to better than a micronewton.

The inverse cube that makes a squeeze film stubborn is the lever that makes a molecular length visible, and the two are the same statement. A quantity that grows without limit as a gap closes is a quantity whose fractional changes are measurable at small gaps, and there is nothing else in fluid mechanics with that property at a wall.

How close the surfaces have to come #

Turning the expansion into a requirement gives the design rule for the apparatus, and it has a pleasant property: it does not depend on the answer.

Solving $1 - f^* = 0.05$ exactly gives $h = 37.05\,b$, against the 40 that the $2b/h$ expansion predicts; the difference is the second term of the series, and it is in the experimenter’s favour. So resolving a one-nanometre slip length at five per cent means holding two surfaces thirty-seven nanometres apart and knowing that distance.

That is a demanding requirement and it is not an impossible one. It is a description of a surface-force apparatus — crossed mica cylinders, separation measured by optical interference to a tenth of a nanometre — and of an atomic-force microscope with a colloidal sphere glued to its cantilever. Both existed before anybody used them for this, and both had been built to measure surface forces rather than flows.

The proportionality is the reason the experiment is honest. An apparatus with a fixed smallest gap has a fixed smallest slip length it can see, stated in advance, independent of what is in the liquid; there is no regime in which the instrument becomes more sensitive because the answer is small. A null result is therefore a bound rather than a failure, and the bound is the gap divided by thirty-seven.

Two practical difficulties sit behind that clean statement and both are about the gap rather than about the force. The first is that the zero of separation has to be established, and it cannot be established by bringing the surfaces into contact, because the essay below this one is about why they will not come into contact. It is established optically, from interference fringes whose order is counted in air before the liquid goes in, and any drift in that datum enters the answer as a slip length directly.

The second is that the approach speed has to be known and steady, since the correction is a ratio of two forces at the same speed and any acceleration of the drive adds an inertial term with the wrong dependence on the gap. This is why the measurement is usually run as a sinusoidal small oscillation superimposed on a slow approach, with the force read at the drive frequency: the film’s response is then a complex number whose real part carries the elasticity of the apparatus and whose imaginary part carries the drainage, and the two are separated by their phase rather than by subtraction. That is the same decomposition the gas film one field over is drawn as, and it is the reason a modern drainage experiment reports a damping rather than a force.

What slip does to the singularity #

At the other end the correction does something that connects the measurement straight back to the drainage problem it is made with.

For a gap much smaller than the slip length the correction becomes $f^* \to (h_0/3b)\left[\ln(6b/h_0) - 1\right]$, and the force becomes

$$F \to \frac{2\pi\mu V R^2}{b}\left[\ln\frac{6b}{h_0} - 1\right].$$

The inverse cube has gone. What is left diverges only as a logarithm of the gap, which is to say it barely diverges at all: closing from a nanometre to a picometre multiplies it by less than two. Computed at a gap a hundredth of the slip length, the scaled force is 12.3047 against $\ln(6b/h) - 1 = 12.3047$.

That is the same trade a moving contact line makes, in the same direction, for the same reason. A singularity produced by insisting that a fluid be motionless at a wall while something forces it to move is removed by letting the wall slip, and what replaces it is a logarithm with a microscopic length inside it. In both problems the macroscopic answer survives and the microscopic one does not — which is why a slip length has to be extracted from the regime where it is small, and why the far side of this curve is useless as a measurement even though it is where the effect is largest.

The layer that pretends to be a wall #

The first careful drainage measurements on smooth hydrophobic surfaces reported slip lengths of hundreds of nanometres to microns, and they were not wrong about the force. They were wrong about what was at the wall.

A layer of thickness $d$ and viscosity $\mu_g$ beneath a liquid of viscosity $\mu$ shears far more easily than the liquid does, so the liquid above it moves as though the wall were sliding. The apparent slip length is $b = d(\mu/\mu_g - 1)$, and for water over air the multiplier is fifty-four. Twenty nanometres of dissolved gas collected at a hydrophobic surface — which is thinner than anything an interference measurement of the gap would notice — is indistinguishable, in the force alone, from a micron of genuine slip.

Distinguishing them needs a second measurement rather than a better one. A true Navier slip length is a property of the liquid–solid pair: it does not depend on how long the surfaces have been in contact, on whether the liquid has been degassed, or on the pressure. A gas layer depends on all three. The modern bounds on slip for wetting liquids come from experiments that varied those parameters and found nothing moving, not from experiments that measured the force more precisely.

What the number turns out to be #

For a wetting liquid on a smooth solid — water on clean glass or mica, an alkane on the same — the answer is that the slip length is under two nanometres, which is a few molecular diameters and is consistent with zero. That is not a dramatic result and it is the one that matters, because it is what licenses every no-slip calculation made anywhere else.

What makes it a real statement rather than an assumption restated is that the same technique finds slip where slip is expected. A gas at a gap comparable with its mean free path slips, and the length it slips by is one mean free path. A surface patterned so that the liquid sits partly on gas slips, and the length is a logarithm of the pattern’s solid fraction. A porous wall behaves as though it slipped, by the square root of its permeability. The instrument is not blind; it is reporting a small number because the number is small.

The bound also has a consequence for every calculation that uses the condition, and it is a comforting one. Two nanometres against the thinnest boundary layer a wing carries — 1.7 millimetres per metre of chord, from the essay that measured it — is a part in a million. Against the viscous sublayer of a turbulent pipe flow, which is tens of microns, it is a part in ten thousand. Against the gap in a journal bearing it is a part in ten thousand again. No flow drawn here is within three orders of magnitude of caring, which is the reason the condition can be applied without comment everywhere except at the four places where it cannot: a moving contact line, a rarefied gas, a patterned surface and a polymer melt.

It is worth being precise about what those four have in common, because “exotic” is not the answer. In each of them the material at the wall is not the material in the bulk. At a contact line there are three phases meeting and the liquid’s thickness goes to zero; in a rarefied gas the wall’s influence reaches a mean free path into the fluid; on a patterned surface part of the wall is not a wall at all; and in a melt the molecules at the surface are a different population from the entangled ones above. None of the four is a case of an ordinary liquid failing to stick to an ordinary solid, and no such case has been found.

What the picture cannot show #

The surfaces are smooth, and a roughness of the size of the gap destroys the measurement. At a thirty-nanometre gap a one-nanometre asperity is three per cent of the separation, and since the force goes as the inverse cube, the error it introduces is comparable with the whole signal. This is why mica — which cleaves atomically flat over square millimetres — did the early work, and why extending the technique to engineering surfaces is not a matter of better electronics.

The slip is Navier’s, which is a linear relation. A slip length that depended on the shear rate would make the correction above wrong in a way no single number could absorb, and for the polymer melts of the next section it does exactly that.

The liquid is a continuum at thirty nanometres, and only just. Thirty nanometres is about a hundred water molecules, and the first two or three layers against a solid are demonstrably ordered differently from the bulk. A measured “slip length” of a nanometre and a measured “viscosity change in the first nanometre” are not distinguishable by this experiment, and the second is at least as likely.

And the two surfaces are given the same slip length. They are usually not the same material; the correction for one slipping wall and one sticking wall is a different function, and using the symmetric one on an asymmetric pair biases the answer low.

Who found it, and when #

Navier proposed the slip condition in 1823 and Stokes assembled the experimental case against it in 1845 — the evidence then being that pipe flows obeyed Poiseuille’s law, which as noted above bounds the slip length only at the level of microns. Vinogradova derived the drainage correction used here in 1995, and it is her expression that turned a surface-force apparatus into a rheometer for the boundary condition. The careful work that retired the large reported slip lengths ran through the 2000s and was substantially a story about dissolved gas.

The century in between is more interesting than a gap in a chronology suggests, because the question was actively contested and the contest was decided by an argument rather than by a measurement. Coulomb had already run the experiment that ought to have settled it, in 1784: he oscillated a disc in water, coated it with tallow so that the water would not wet it, and found the damping unchanged. That is a genuine null result and it was read as proof of no slip — but the disc was centimetres across, so what it actually bounded was the slip length against a centimetre, and it would have been just as unchanged by a micron of slip. Helmholtz and Piotrowski found slip in the 1860s with a similar apparatus and were not believed; Whetham found none in the 1890s and was. None of these experiments could have distinguished the answers that were being argued about, and the argument was settled by the accumulating success of no-slip calculations instead.

That is a specific and avoidable failure, and it is worth naming as one. An experiment whose sensitivity nobody has computed can only confirm what it was expected to confirm, because a null result and an insufficient instrument produce identical data. The quantity that would have stopped a century of this is the one plotted above — the gap a stated signal needs — and it takes two lines from the expansion of a correction factor that would not be written down for another hundred and thirty years.

The surprising connection is that the argument runs both ways and the same film settles both. This essay uses the drainage force to measure a wall’s condition. A sphere bouncing off a floor uses the same force, at the same gaps, and needs a cut-off length put in by hand — and that cut-off is the same physics this experiment is measuring. The instrument and the phenomenon are one calculation, and the reason a bounce has a threshold near ten is, in part, that the wall does not slip enough to soften the film.

Still open: whether the first nanometre is slip or a different viscosity #

A drainage experiment measures one thing — the force against the gap — and two different pictures of the wall produce almost the same curve.

The first is Navier slip with $b$ of a nanometre. The second is no slip at all, with a layer a nanometre thick in which the liquid’s viscosity is several times the bulk value, which is what molecular simulations of water against mica show and what the ordering of the first molecular layers would suggest. Both remove a fixed amount of flux from the gap; to first order in the small length both give a force deficit linear in it, and the two are separated only by the second term of the expansion — at a gap of thirty nanometres, a difference of well under a per cent.

The calculation that would separate them is the same Reynolds equation solved with a stratified viscosity $\mu(z)$ rather than a slip condition, carried to second order in the layer thickness, with the two predictions written as functions of the gap and subtracted. Whether the difference between them ever exceeds the experiment’s own scatter — and if so, at what gap and for which liquid — would say whether the last nanometre at a wall is a question this method can answer at all, or a question it has been quietly converting into a slip length for thirty years.