The heat of a squeeze leaves across the gap

A damper that turns into a spring found that a film of air squeezed between two discs is one of two machines depending on a single number. At a low squeeze number the gas leaves the gap sideways and the film is a damper, as the last of the oil is; at a high one the gas has no time to leave, is compressed where it stands, and the film is a spring. The crossing sat at a squeeze number of 6.23, and six real devices, all of them air, spread across five decades either side of it.

That calculation held the gas at the wall temperature. It said so, and it named the question it left: compressing a gas heats it, and if the heat cannot reach the walls in the time of one squeeze the gas is compressed adiabatically, which makes it stiffer by the ratio of specific heats. The estimate it offered was a thermal squeeze number “in the same shape” as the viscous one, with the thermal diffusivity in place of the viscosity, so that the two numbers differ by the Prandtl number — 0.71 for air — and the film should turn adiabatic at nearly the same frequency as it turns stiff. On that estimate the whole right-hand side of the earlier essay’s device list, the levitator and the read head, is about forty per cent stiffer than drawn.

The estimate is wrong, and not by a small factor. It sends the heat out the same way as the gas — along the disc, a distance of the disc’s radius. Nothing obliges it to go that way.

Two ways out, and heat takes the short one #

Gas leaving a squeeze film has to go sideways, because the walls are solid. Heat has no such restriction. The walls are metal or silicon, with conductivities a thousand times the air’s and heat capacities to match, so for the gas they are a fixed temperature a half-gap away on either side. A disc a centimetre across with a twenty-micron gap has its walls five hundred times closer than its rim. Heat crosses the gap.

That changes the number that decides it. Across the gap the relevant time is the gap’s own diffusion time, $h^2/\kappa$, and the thermal number is that time times the frequency:

$$\Omega = \frac{\omega h^2}{\kappa}.$$

There is no radius in it. The squeeze number, by contrast, is built on the radius squared over the gap squared, so the ratio of the two is fixed by the device’s geometry:

$$\frac{\Omega}{\sigma} = \left(\frac{h}{a}\right)^2 \frac{p\,h^2}{12\,\mu\,\kappa}.$$

The first factor is the square of an aspect ratio, and for a thin film it is tiny. For the levitator of the earlier essay it is four millionths, and the second factor, about nine thousand for a twenty-micron gap of air, brings the ratio to 0.035. The Prandtl estimate said 0.71.

Once the heat is known to go across, the calculation is a single one-dimensional problem, and it has a closed form. The pressure is uniform across a thin film, so the temperature fluctuation in the gap is driven by the pressure alone and conducted to walls that do not change temperature. The figure draws its amplitude across the gap at five thermal numbers. At 0.3 the walls reach everywhere and the gas stays at their temperature; the film is isothermal, as assumed. At a thousand they reach only a thin layer at each side, and the middle of the gap is compressed with no heat lost at all. Between, the profile is a rounded arch that lags the pressure by a fraction of a cycle, and that lag is where the interesting part of the problem lives.

The gap’s own polytrope #

Averaging that temperature across the gap turns the gas’s response into a single complex number. The density of the gas follows the pressure with a polytropic index n: one when the gas is isothermal, γ = 1.4 when it is adiabatic. With heat conducted to the walls the index is

$$n = \left[1 - \frac{\gamma - 1}{\gamma}\left(1 - \frac{\tanh k}{k}\right)\right]^{-1}, \qquad k = \tfrac12\sqrt{i\Omega},$$

and it has an imaginary part, because the temperature lags the pressure. That imaginary part is a loss. Heat flows from the compressed gas to the walls through a finite temperature difference, and heat flowing down a temperature difference generates entropy — a dissipation with no viscosity in it anywhere.

The real part of n climbs from one to γ, half-way there at a thermal number of 18.1. The imaginary part peaks at 0.157 at a thermal number of 13.5, just before it. The two are the same event seen from two sides: the gas is lossiest exactly where it is partly isothermal and partly adiabatic, because that is where the heat has a period’s worth of time to cross and does not quite make it. The same shape runs through the viscosity nobody uses, where a molecular relaxation absorbs sound most strongly at the frequency matched to its relaxation time; here the relaxation is conduction and the time is the gap’s.

Where the devices sit #

The film itself needs only one change. Reynolds’ equation for the gas conserves mass, and mass depends on density, so replacing the isothermal density with the complex polytrope replaces the squeeze number σ with σ/n and scales the pressure by n. The force on the disc becomes n times the isothermal film’s force at the complex squeeze number σ/n. Its slow limit is still Stefan’s law exactly, whatever n is, because a gas that leaves before it is compressed does not care how it would have been compressed. Its fast limit is a gas trapped at the gap’s own index: π times n rather than π.

The figure places the six air films of the earlier essay by both numbers. The dashed diagonal is the Prandtl estimate, on which the two numbers move together. None of the devices is near it. The pneumatic damper and the levitator share a disc and a gap and sit 0.035 of the way along; the micromirror and the proof mass, a few microns thick and a millimetre across, sit a factor of twenty-five below that; the read head, fifteen nanometres of gap under half a millimetre of slider, sits at a thermal number of three parts in ten million, eleven decades below the diagonal. The horizontal line is the thermal crossing. Five of the six films never come within two decades of it. Only the levitator at a twenty-micron gap comes within a decade, at a thermal number of 2.37.

So the stiff devices, which the old estimate made forty per cent stiffer, are isothermal. The read head’s film is the most thoroughly trapped gas in the list and the most thoroughly isothermal: a fifteen-nanometre gap conducts heat across in about ten picoseconds, far faster than any disc spins, while its gas takes long enough to leave along the slider that it is hardly leaving at all. Trapped and isothermal are not in tension. They are set by two different lengths.

That is the reverse of the most famous thermal mistake in the subject. Newton computed the speed of sound as if the air stayed at constant temperature, and came out about fifteen per cent slow, because in a sound wave the compressed regions are half a wavelength from the rarefied ones and heat cannot cross that distance in a period; Laplace’s adiabatic correction is the square root of γ. A squeeze film is Newton’s gas rather than Laplace’s for the same reason turned round: the heat has only half a gap to go. The same competition decides whether a long pipe can hold its gas at the wall’s temperature, and there too the answer comes from comparing a transverse diffusion time with an axial one, not from comparing two diffusivities.

The one film that reaches it #

The levitator is the exception, and it is worth understanding why. It has the widest gap of the six and the highest frequency, and the thermal number grows as both. At twenty kilohertz and twenty microns the heat has about a third of a cycle’s worth of time to cross the gap, and the gas’s index is 1.0098 with an imaginary part of 0.055. That barely moves the stiffness, which changes by less than one per cent. It moves the damping by 27%.

The figure follows the levitator’s own disc and gap as the frequency is raised, and plots the work the disc does on the film per cycle. The isothermal film has the single loss band of the earlier essay, peaking near a squeeze number of six where the gas is half leaving and half trapped. With conduction there is a second band, peaking near a squeeze number of 335, where the thermal number reaches twelve — fifty times higher in frequency than the first, rather than beside it. By a squeeze number of a thousand the conducting film dissipates about four times what the isothermal film does, because the isothermal film’s loss has fallen away as its gas becomes trapped while the conducting film’s has only just arrived.

That settles the question the earlier essay left open about the loss. A film that is neither isothermal nor adiabatic does dissipate through conduction, and the loss curve does have a second bump. But the second bump belongs to the gap’s diffusion time, not the disc’s drainage time, and on any real geometry the two are separated by the factor on the aspect ratio. A gas film has two bad bands, and for the levitator they are a factor of fifty apart in frequency.

A stiffer gas makes a softer film #

The obvious expectation is that turning the gas adiabatic stiffens the film by up to γ everywhere. That is true in only one of the two regimes, and the other is the more surprising half of the calculation.

Holding the levitator at twenty kilohertz on its one-centimetre disc and opening its gap sweeps both numbers at once: the squeeze number falls as the gap squared, since a wider gap drains faster, and the thermal number rises as the gap squared, since heat has further to go. At a five-micron gap the film is trapped and isothermal, and nothing changes. Through twenty to sixty microns the film crosses its thermal band and the damping rises, by up to 37% at a gap of thirty-eight microns. Beyond that the film has become soft — the gas escapes most of each squeeze — and its gas has become adiabatic, and the stiffness falls, towards 1/γ of the isothermal film’s rather than rising towards γ times it.

The reason is in how a soft film makes its stiffness. A film that is mostly draining stores energy only in the small part of the gas that is compressed before it can leave, and the pressure that drives the gas out is what compresses it. A stiffer gas is compressed less by the same pressure, so less of it is held back and more of it leaves; the pressure builds to the same draining value and the stored part shrinks by the gas’s own stiffness. In the mathematics this is the force being n times the film at σ/n: at small σ the stiffness goes as $\sigma^2$, so dividing σ by n and multiplying by n leaves a factor 1/n. Only the trapped film, whose force saturates at nπ, gets the full factor n.

So the one correction the earlier essay was confident of — that an adiabatic film is γ stiffer — holds only for a film that is both trapped and adiabatic, which needs a thermal number past about twenty and a squeeze number past about six at the same time. That takes a gap wide enough for heat to be slow and a disc wide enough for gas to be slower still. For a twenty-micron gap in air it takes a frequency above about 150 kHz. None of the six devices is there.

What it means for a levitator #

Near-field acoustic levitators are the one class of gas film that routinely lives at the thermal crossing, and the calculation says what to expect of them. Their load capacity comes from the trapped film’s stiffness and its rectified mean pressure, and neither moves by more than a per cent at the gaps they use. Their efficiency is another matter. The work a transducer does on the film is lost as heat, and at twenty microns and twenty kilohertz a quarter more of it is lost than the isothermal theory predicts, through conduction into the plate and the levitated object. At a thirty-to-forty-micron gap, which heavier objects push towards, the excess is more than a third.

That loss has one feature nothing viscous has: it relies on the walls being a heat sink. What decides whether a wall holds its surface temperature through a cycle is its thermal effusivity, the square root of conductivity, density and heat capacity, against the gas’s. Silicon’s is about 2,800 times air’s, glass’s about 260 times and a polymer such as acrylic’s about 100 times, so even a levitated plastic disc sees its surface temperature swing by only about one per cent of the gas’s. The fixed-wall model holds for every object a levitator is used to carry, and the extra loss goes into the plate and the object as the calculation says.

A ball that bounces in water and not in oil and a torn film that still pulls found the liquid film’s outward leg governed by a pressure floor rather than the film equation; the gas film’s own limit is thermal, and it is set by a length the film equation throws away the moment it averages across the gap.

What was checked #

With the gap held isothermal the film must be exactly the closed form of the earlier essay, and it is to the last digit at four squeeze numbers across the crossing. With conduction its slow limit must still be Stefan’s law, independent of the gas’s index, and it is to one part in a hundred thousand at a squeeze number of a thousandth; its fast limit with an adiabatic gas must approach γπ as $1 - \sqrt{2\gamma/\sigma}$, and at a squeeze number of 900 it sits at 94.4% of γπ, matching that approach to four figures. The gap-averaged temperature, which is the whole of the thermal model, was checked against a direct quadrature of its profile, to five parts in a million million. The calculation is also offered a negative thermal number, a zero squeeze number and a tolerance of zero, and it refuses all three.

What the model leaves out #

Walls at a fixed temperature. To about one per cent for a polymer and far better for glass, metal or silicon, as the section on levitators estimated from the ratio of effusivities. The exact calculation with a finite wall is a matched pair of diffusion problems and was not done here; on those ratios it could not change any number above by more than that per cent.

Slip. The read head’s film is below a micron and stops being a continuum long before anything thermal matters. Rarefaction also brings a temperature jump at the wall, which moves the gas further towards isothermal at small gaps rather than away from it.

Small amplitude. Everything here is linear in the gap’s excursion. The mean pressure that holds a levitated object up is quadratic in it, and how conduction changes that rectified pressure is a second-order calculation. At the levitator’s own thermal number the index is within one per cent of one, so the answer cannot be large there.

Parallel rigid discs. As in the earlier essay: a real transducer bends, and an oscillation with somewhere to go is the reminder that the edges of an oscillating film are where it streams.

Who worked it out #

Kirchhoff in 1868 gave the thermal boundary layer at an oscillating wall its modern form, in the same paper that added heat conduction to the absorption of sound in tubes; the gap-averaged factor tanh k / k is the same function that appears in the thermoacoustics of narrow channels, which Rott worked out in 1969. The compressible squeeze film is Langlois’s, from 1962, and Griffin, Richardson and Yamanami’s; in the micro-electromechanical literature the isothermal assumption is standard and usually justified in a sentence by exactly the across-the-gap argument above. What is added here is the full complex index folded into the film, and what it does to the six devices that the earlier essay placed on the squeeze-number axis: nothing to five of them, and a quarter more loss to the sixth.

Still open: the gas and the pressure as a thermal handle #

Every figure is air at one atmosphere, and the geometry was the only thing that moved the thermal number. It is not the only handle. The thermal diffusivity of a gas falls as its pressure rises and is several times smaller for a heavy gas than for air, while the viscosity barely changes with either, so at a fixed disc, gap and frequency the ratio of the two numbers grows as the square of the pressure. A levitator run in a pressurised cell, or with a heavy gas such as sulphur hexafluoride, could be moved onto its own thermal loss peak without touching the mechanism — though a heavy polyatomic gas also has a ratio of specific heats close to one, which shrinks the whole thermal effect as it moves it. The next calculation carries γ, κ and μ as properties of the gas and the pressure as a variable, and asks which gas and which pressure put a given levitator where it stores most and wastes least.