A damper that turns into a spring

A wet plate on a table is hard to lift because the oil under it has to be fed inwards through a very thin gap, and the arithmetic of that ends in a sentence worth quoting back: the force is proportional to the approach speed, so the film is a damper rather than a spring — it resists motion and stores nothing.

That sentence is true, and it is true for a reason that was never stated. A liquid film has exactly one way out of the gap, which is sideways, and the resistance to going sideways is what the whole calculation is about. A gas has a second way out. It can stay where it is and be compressed.

Which of the two happens is not a property of the gas. It is a race between the time the film is given and the time it needs to escape, and when the race is written down it produces a single number that decides which kind of machine the film is.

The number is two times divided by one another #

Reynolds’ equation for a film that can be compressed carries the density along with the gap:

$$\frac{\partial (p h)}{\partial t} = \nabla\!\cdot\!\left(\frac{p h^3}{12\mu}\,\nabla p\right).$$

Scale the pressure on ambient, the gap on its mean, the radius on the disc and the time on the forcing frequency, and every dimensional quantity collects into one group multiplying the left-hand side:

$$\sigma = \frac{12\mu\omega a^2}{p_a h_0^2}.$$

That is the squeeze number, and like every other such group it is one time over another. The denominator is the period of the forcing. The numerator is the time a pressure disturbance needs to diffuse out to the rim of the disc, which for a compressible film is a genuine diffusion — the gas’s own pressure gradient drives it out, and the gas’s own compressibility supplies the capacity that makes the process diffusive rather than instantaneous.

The two lengths in it enter very differently. The radius is squared, so a disc twice as wide gives the gas four times as far to go. The gap is squared and in the denominator, so halving a clearance quadruples the number. And the frequency is first order, which is the reason the same physical film is a different machine at a hundred hertz and at twenty kilohertz.

What the group leaves out is as informative as what it contains. There is no density in it, and therefore no Reynolds number: a film a hundred times longer than it is thick has no room for inertia at any frequency a machine will reach, and the gas’s mass appears nowhere in the equation that governs it. What appears instead is the gas’s compressibility, and in an ideal gas at fixed temperature that is simply one over the pressure — which is why the ambient pressure sits in the denominator of the squeeze number and why pumping a chamber down raises it.

That last consequence is not a curiosity. A micro-machined device sealed at a thousandth of an atmosphere has a squeeze number a thousand times larger than the same device in the open, so evacuating a package does not merely reduce a film’s damping in proportion to the pressure. It moves the film across the transition. A damper at atmosphere is a spring in a vacuum can, and the spring’s stiffness falls only as the square root of the pressure while its damping falls as the pressure itself.

The other absentee is the gas’s identity. Viscosity and ambient pressure are the only two properties in the group, so helium and air at the same pressure differ by their viscosity ratio alone — about ten per cent — and a film filled with a heavy fluorocarbon vapour behaves like air at a lower frequency rather than like something new.

Two limits, and both of them are machines already in use #

With a small gap swing, $h = h_0(1 + \varepsilon\cos\omega t)$, the terms of first order in $\varepsilon$ leave a linear equation for the pressure swing alone:

$$\nabla^2 p_1 = i\sigma\,(p_1 + 1),$$

which is Bessel’s equation with a complex argument. Its solution is

$$p_1(R) = \frac{J_0(\beta R)}{J_0(\beta)} - 1,\qquad \beta = \sqrt{\sigma}\,e^{-i\pi/4},$$

and the force follows in closed form because the integral of $J_0$ against $R$ is another Bessel function. There is no fitting anywhere and no numerical solve; what comes out is one complex number per squeeze number, and its real and imaginary parts are a stiffness and a damping.

Both limits of that expression are results already established. As the squeeze number goes to zero the force amplitude approaches $-i\pi\sigma/8$ — pure damping, no stiffness — which is Stefan’s law to the last factor, recovered here to six figures at $\sigma = 0.01$ by an expression that was never told about it. As the squeeze number goes to infinity the force amplitude approaches $-\pi$ — pure stiffness, no damping — which is $p h$ held constant, an isothermal gas squeezed in a closed box.

The approach to that second limit is worth having as a rate rather than as a fact, because a statement that the stiffness rises towards π would be satisfied by any increasing curve. The shortfall falls as $\sqrt{2}/\sqrt{\sigma}$ and the damping falls as $\pi\sqrt{2}/\sqrt{\sigma} - \pi/\sigma$, both with coefficients that come out of the Bessel ratio and are put in nowhere. At $\sigma = 3000$ the computed coefficients are 1.41427 and 3.1517, against $\sqrt{2}$ and π.

So the film has not become a different object. It has been asked a different question. Nothing about the gas changed between the two ends of that curve, and nothing about the geometry did either.

The gas in the middle never learns there is an outside #

The stiffness and the damping are integrals over the disc, and what they are integrals of says more than either number.

At a squeeze number of a thousand the pressure under nearly the whole disc rises and falls as though the disc had no edge. Only a ring near the rim, whose width is of order $a/\sqrt{\sigma}$, knows that there is an atmosphere outside; inside that ring the gas is simply being compressed, and its pressure is whatever $p h = \text{constant}$ says it is.

That is the same structure as every other oscillating thin layer. A rim ring of width $a/\sqrt{\sigma}$ is a diffusion length in disguise — it is how far the pressure signal gets in one cycle — and it is the same object as the layer a shaking wall makes, with pressure in place of velocity and compressibility in place of inertia. The reason the fast film is stiff is that the layer which could have relieved it has become too thin to matter.

The slow film has the opposite anatomy. The ring has swallowed the disc, the pressure has time to reach the rim everywhere, and the profile is the parabola Stefan’s law integrates.

The worst place to put a film is between its two jobs #

A film that stores nothing and a film that resists nothing both waste very little. Between them is a squeeze number at which the gas is compressed a good deal and also escapes a good deal, and that is where the energy goes.

Those two numbers have no obvious reason to coincide. The crossover is where the real and imaginary parts of one complex number are equal; the dissipation peak is where the imaginary part times the squeeze number is largest. They land at 6.23 and 6.31, and the near-coincidence is the practical content of the whole curve: a film cannot be made stiff and lossy at the same time, and the transition between the two is exactly where it is worst at both.

A squeeze-film damper is designed to sit well to the left of that peak, so that its damping is the incompressible one and can be computed from Stefan’s law without any of this. A gas bearing is designed to sit far to the right, where the stiffness is nearly π and the loss is a few per cent. The band in between is the one a designer is told to avoid, and this is what is being avoided.

The shape of the peak deserves a second look, because it is not symmetric and the asymmetry is useful. Approached from the left the work rises as the squeeze number, linearly, since the damping is Stefan’s and the loss is simply proportional to how hard the gas is being pushed out. Approached from the right it falls as the inverse square root, because the damping does. So a film that drifts upward in squeeze number — by running faster, or by closing its gap as it wears — climbs into the peak quickly and leaves it slowly, and a device designed on the right-hand slope has a much larger tolerance than one designed on the left.

A resonator that cannot ring #

The damping is worth a number rather than a shape, because a shape does not say how bad it is.

Resolving the force on the gap’s rate of change gives a dashpot constant $c = p_a a^2 D/(h_0\omega)$, where $D$ is the dimensionless damping plotted above. For the micromirror tabulated above — half a millimetre of radius, two microns of gap, a kilohertz — that is 0.65 newton-seconds per metre. A silicon plate a millimetre square and ten microns thick weighs twenty-three nanograms, so its quality factor $m\omega/c$ is about two parts in ten thousand.

A quality factor of $2\times10^{-4}$ is not a lightly damped resonator, a heavily damped one, or a critically damped one. It is a device that cannot oscillate at all: released, it creeps back to its rest position over thousands of periods, and driven, it follows the drive with no resonance anywhere. Every micro-machined mirror, comb drive and gyroscope has this problem, and it is the reason such parts are sealed in evacuated packages rather than merely protected from dust.

What evacuating buys is less than it looks, and the curve above says why. Dropping the ambient pressure by a factor of a thousand raises the squeeze number by a thousand, which moves the film from 0.84 to 840 — from the left of the transition to the far right. The dimensionless damping there has fallen by a factor of about thirty, and the dimensional damping by the pressure ratio as well, so the quality factor rises by some thirty thousand and reaches about one half. Still not a resonator.

Getting further means going to pressures at which the gap is no longer full of a continuum, because the mean free path at a millibar is 68 microns against a two-micron gap. The device is then in the free-molecular regime, its damping is set by molecules crossing the gap without meeting one another, and none of the arithmetic here applies. The route from an overdamped micromirror to a usable one leaves this calculation entirely, which is a more useful thing to know than a correction factor would have been.

Six devices, one gas, five decades #

The list below is the argument against the film’s behaviour being a property of the fluid, because every entry in it is air.

The levitator and the damper in that picture have the same disc radius and the same gap. They differ only in frequency, by a factor of two hundred, and they are on opposite sides of the transition: $\sigma = 0.34$ against $\sigma = 67.3$. Running the same levitator down to a five-micron gap takes it to 1077, where the stiffness is 3.006 of a possible 3.1416 and the damping has fallen to 0.132.

The read head is the extreme, at $\sigma = 7.5\times10^4$, and it is the reason a hard disc’s slider flies rather than floats: the film beneath it is very nearly an ideal spring, and a spring is what a head positioned to a few nanometres needs. The damping there is 0.016 — which is also why such a slider needs its damping supplied by something other than the air it flies on.

Reading the same list as dimensional numbers rather than dimensionless ones makes the design conversation short. The pneumatic damper’s dashpot constant is 106 newton-seconds per metre and its stiffness 3.7 kilonewtons per metre, so it is overwhelmingly a damper. The levitator’s stiffness is 1.3 meganewtons per metre and its dashpot constant two newton-seconds per metre, so it is overwhelmingly a spring. Both are air, both are a centimetre across, and both have a twenty-micron gap; the entire difference between a component that absorbs energy and one that stores it is a factor of two hundred in frequency.

That is the lever a designer actually has, and it is not the obvious one. The gap enters as its square and looks like the strongest handle, but a gap is usually fixed by manufacture and by the travel the device needs. The frequency enters only to the first power and is usually fixed by what the device is for. What is left is the ambient pressure, which enters to the first power, is free to change, and is the one quantity in the group that has nothing to do with the mechanism at all.

Where the gas stops being a gas #

The read head is also where this calculation stops, and it stops for a reason that has nothing to do with compressibility.

The mean free path of air at atmospheric pressure is 68 nanometres. A read head flies at about fifteen, so its Knudsen number is above four: the gas in that gap is not a continuum at all, the viscosity in Reynolds’ equation is not a property of the bulk gas, and everything rarefaction implies applies with full force. The practical form of that correction is a slip length at each wall, which multiplies the effective $h^3$ by a factor greater than one and therefore lowers the squeeze number — the gas escapes more easily than the no-slip equation believes, and the slip is a memory of one mean free path.

So the extreme right-hand end of the curve is drawn for a gas that is not there. Everything from the micromirror to the levitator is a continuum with room to spare, and the read head is in the picture as a limit rather than as a computed case.

A cycle whose mean is not zero #

Everything so far is first order in the gap swing, and first-order quantities average to nothing over a cycle. The second order does not, and it does something a liquid film cannot do at all.

For a film trapped hard enough that no gas leaves, the pressure is exactly $p_a h_0/h$ at every instant. Put a symmetric gap swing through that and the mean is not $p_a$:

$$\langle p \rangle = \frac{p_a}{2\pi}\int_0^{2\pi}\frac{d\theta}{1 + \varepsilon\cos\theta} = \frac{p_a}{\sqrt{1-\varepsilon^2}}.$$

The pressure rises further on the squeeze than it falls on the release, because it goes as one over the gap, and a symmetric motion of the gap is an asymmetric motion of the pressure.

Fifteen per cent of an atmosphere is fifteen and a half kilopascals, which on a disc twenty millimetres across is about five newtons. A surface vibrating by five microns holds up half a kilogramme with nothing touching it, and the energy for that comes from the vibration rather than from any supply of gas. There is no bearing feed, no pump, and no contact.

This is the same second-order effect that arrives from two other directions. Steady streaming is a mean flow left behind by an oscillation with no mean, and the drift in a wave that has none is a mean displacement left behind by an orbit that nearly closes. All three are products of two first-order quantities that happen to be in phase, all three are invisible to any calculation that stops at first order, and all three are the reason a zero-mean forcing is not the same thing as no forcing.

What holds the plate up is the gas’s refusal to average #

A lift that grows with the excursion ratio is a lift with a restoring slope in it, and that is what makes levitation work without anything controlling it.

Two hundred pascals — a sheet of paper’s worth — floats eighty microns up. Twenty kilopascals, which is two tonnes on a square metre, floats nine microns up on a surface that is moving five. The excursion ratio there is 0.55, the film is being squeezed to under half its mean gap twice per cycle, and the arithmetic above is being asked well outside the small-amplitude expansion that produced the stiffness curve.

That is the honest limitation of the pairing. The linear analysis says when the gas is trapped; the trapped-gas rectification says what a trapped film lifts. Neither is a solution of the full nonlinear compressible film equation, which is what an ultrasonic levitator actually is, and the combination is a bracket rather than a calculation.

What the picture cannot show #

The film is isothermal, and a film at twenty kilohertz is not. Compressing a gas heats it, and whether that heat has time to reach the walls is a second ratio of times — a thermal squeeze number, built on the gas’s thermal diffusivity rather than its kinematic viscosity. For air the two diffusivities are within thirty per cent of one another, so the two numbers are close and the film crosses from isothermal to adiabatic near where it crosses from soft to stiff. An adiabatic trapped film is stiffer than this one by the ratio of specific heats, 1.4, and nothing above carries that factor.

The discs are flat, parallel and rigid. A levitator’s plate is none of the three: it is driven by a transducer that bends it, so the gap is a function of radius as well as of time, and the mode shape decides the load capacity as much as the amplitude does.

Nothing here is a continuum below about a micron, as the Knudsen figure above says, and the correction goes the way that makes a film less stiff rather than more.

And the small-amplitude expansion and the trapped limit are two different calculations. The stiffness curve is linear in the excursion and the lift is quadratic in it; there is no single expression above that carries both, and the region where a levitator works is the region where neither is comfortable.

Who found it, and when #

Langlois wrote down the compressible squeeze film and its squeeze number in 1962, working on the gas bearings that were then being proposed for gyroscopes; the Bessel solution for a circular disc and the recognition that the two limits are Stefan’s law and a trapped spring came out of the same body of work. The rectified mean is older than that as a phenomenon and younger as an explanation: plates floating above vibrating surfaces were a curiosity in the 1960s, and near-field acoustic levitation became a technique with a theory in the 1990s.

The surprising connection is not with a bearing at all. The squeeze number is a Womersley number for pressure. In an oscillating pipe one number decides whether the velocity profile has time to become the steady one; above about ten the core moves as a plug and only a ring near the wall shears. Here one number decides whether the pressure profile has time to become the steady one; above about ten the core is compressed as a block and only a ring near the rim relieves. Both are a diffusion length compared with a geometric one, both put the interesting behaviour in a ring of width $a/\sqrt{\text{number}}$, and neither has anything to do with the particular quantity being diffused.

Still open: what an adiabatic film crosses over at #

The thermal ratio named above is the loose end, and it is a calculation rather than a difficulty.

The film equation as written assumes the gas stays at the wall temperature, which holds when heat diffuses out of the gap faster than the gap is squeezed. That is a squeeze number built on the thermal diffusivity — $\sigma_T = 12 k \omega a^2 / (\rho c_p \ldots)$ in the same shape — and for air its ratio to the viscous one is the Prandtl number, 0.71. The two crossings therefore sit within a factor of one and a half of each other, which is close enough that no film in the table above is clearly on one side of both.

What that costs is a factor of γ in the stiffness of every device to the right of the transition, and the interesting part is what it does to the loss. An isothermal film dissipates only through the gas it pushes out of the gap. A film that is neither isothermal nor adiabatic dissipates through heat conduction as well, at a rate that peaks where the thermal number is of order one — so the loss curve drawn above should have a second bump on it, out of a mechanism with no viscosity in it at all. Computing where that bump sits, and whether it is large enough to matter for a levitator’s efficiency, would decide whether a gas film has one bad band or two.