A levitator lifts by its amplitude, not its gas

The gas sets a squeeze film’s loss, not its stiffness filled one levitator — a disc ten millimetres in radius vibrating at 20 kHz over a 20-micrometre gap — with six gases in turn, and found that its stiffness hardly changed while its loss nearly doubled from helium to xenon. Everything there was linear, as it was for the oil film of the last of the oil and the bouncing ball of a ball that bounces in water and not in oil: a small vibration, a small response, a complex stiffness. But a levitator does not hold its load up with stiffness. It holds it up with the mean of the film’s pressure over a cycle, which is above ambient only because the pressure rises further when the gap closes than it falls when the gap opens. That mean is second order in the amplitude, and a linear film cannot see it.

A damper that turns into a spring computed the mean for the simplest case, a film trapped hard enough that no gas leaves and cool enough that its temperature never moves: the pressure is $p_a h_0/h$ at every instant, and its mean over a symmetric swing is $p_a/\sqrt{1 - \varepsilon^2}$. The question left was whether the heat changes that. A gas compressed faster than it can shed its heat follows an adiabat, $p \propto h^{-\gamma}$, whose curvature is steeper, and the natural expectation is that a gas which keeps its heat — xenon, slow to conduct and heavy — lifts more than one which loses it.

The film, solved to every order #

At 20 kHz the levitator’s squeeze number is between fifty and ninety for all six gases: the gas has nowhere near enough time to flow out of the gap in a cycle, so a column of it is trapped, with a fixed mass over each patch of the disc. What is not fixed is its temperature. The walls are metal and hold their own; the gas between them is compressed and expanded twenty thousand times a second, heats as it is squeezed, cools as it is released, and loses heat to the walls across a gap only twenty micrometres wide. The heat of a squeeze leaves across the gap, not along it, and the number that decides how well is the thermal number $\Omega = \omega h^2/\kappa$, the cycle against the time heat takes to diffuse across the gap.

The calculation follows the temperature across the gap through the whole cycle, with nothing linearised. In a coordinate that marks each layer of gas by the fraction of the column’s mass below it, the energy equation is

$$\frac{\partial\theta}{\partial\tau} = \frac{\gamma - 1}{\gamma}\,\frac{\theta}{P}\,\frac{dP}{d\tau} + \frac{P}{\Omega}\,\frac{\partial^2 \ln\theta}{\partial s^2},$$

with $\theta$ the temperature over the walls’, $P$ the pressure over ambient, $\tau$ the phase of the cycle and $s$ the mass fraction; the walls hold $\theta = 1$. The first term is compression, the second conduction, and conduction in mass coordinates carries a factor of the pressure because a denser gas has more of its mass within a given distance of the wall. The gap closes the system: the column’s thickness is its mass times the gas constant times its mean temperature over the pressure, so $P = \bar\theta/\eta$ with $\eta = 1 + \varepsilon\cos\tau$ the gap over its mean. The equation is marched in time, implicitly in the conduction and iterating on the pressure at each step, from a gas at rest until a cycle repeats to ten decimal places.

Within a few per cent, whatever the gas #

The result is the first figure, and it is not what the linear film led anyone to expect. At one atmosphere the six gases sit at thermal numbers from 0.3 for helium to 16 for sulphur hexafluoride, and squeezed by a quarter of the gap — a five-micrometre vibration in the twenty-micrometre gap — they lift

The isothermal film’s excess is 3.28 per cent of an atmosphere, 3.3 kilopascals, about a newton on the disc. Five of the six gases are within half a per cent of it; xenon, the slowest conductor among the monatomic gases, gains four per cent. Sulphur hexafluoride, the slowest of all to conduct, gains almost nothing, because a molecule with that many internal modes has a ratio of specific heats of 1.1 and its adiabat is barely steeper than its isotherm.

The gas does not set the lift. It sets the loss — the previous calculation’s finding, which this one confirms by an independent route below — and the lift belongs to the amplitude. A levitator designer choosing a gas is choosing how much power the film wastes, not how much it carries.

Why the heat costs loss and not lift #

The reason is visible in the shape of the two effects. The loss of a cycle is the area of the loop the pressure traces against the gap, and a gas that lags its own temperature opens that loop at first order in the phase lag: the loss is proportional to the imaginary part of the film’s polytropic index, which the previous calculation found to be 0.0095 for helium and 0.205 for xenon at one atmosphere, a factor of twenty-one. The lift is the mean pressure, which depends on the curvature of the pressure against the gap, and the phase lag enters that only at second order: the real part of the index, which sets the curvature, is 1.0002 for helium and 1.107 for xenon. A gas halfway between isothermal and adiabatic is a lossy gas long before it is a stiffer one.

The loss figure is also the solution’s best check. Squeezed by only a twentieth of the gap, the work done on the gas per cycle, divided by $\pi\varepsilon^2$, must reproduce the imaginary part of the polytropic index that the linear film computes in closed form from the temperature profile across the gap. For all six gases it does, to 1.3 parts in a thousand — a nonlinear marching calculation in a mass coordinate agreeing with a linear analytic one in a space coordinate, neither of which knows about the other.

A film too slow to cool #

Raise the thermal number — a thicker gap, a faster vibration, a higher pressure, a heavier gas — and the film moves towards adiabatic. The lift then does rise: 1.06 times the isothermal film’s at $\Omega = 10$, 1.15 at 30, 1.20 at 100, 1.21 at 300 for a monatomic gas. But it rises towards a ceiling far below the one the adiabat promises.

A gas squeezed from ambient along a single adiabat would have a mean pressure $\langle\eta^{-\gamma}\rangle$, 2.27 times the isothermal excess for a monatomic gas at this amplitude. That is not the right comparison, because a trapped gas that is adiabatic within each cycle still exchanges heat over many: it settles at whatever mean temperature makes the heat it loses to the walls balance the work done on it. If that mean were the walls’ own, its pressure would follow $\eta^{-\gamma}/\langle\eta^{1-\gamma}\rangle$, and the excess would be 1.69 times the isothermal. The solution gives 1.21 at $\Omega = 300$, and 1.22 at a thousand. Something is holding the mean pressure down.

The walls pump heat out of the gas #

What holds it down is that the gas’s middle settles cooler than the walls. Over a cycle the gas is hot when it is compressed and cold when it is released, and if conduction were symmetric those two halves would cancel and the gas would settle at the walls’ mean. Conduction is not symmetric. The conductivity of a gas does not depend on its density — a dense gas has more molecules carrying heat and shorter free paths for each, and the two cancel, the same kinetic arithmetic that fixes every gas’s Prandtl number near three-quarters — but the compressed gas has more of its mass packed close to the walls. Heat flows out of the compressed, hot gas more readily, per unit of its mass, than it flows back into the expanded, cold gas. Over a cycle the walls take out more heat than they put in, and the gas’s mean temperature falls until the extra outflow is balanced by the cycle’s dissipation.

A one-per-cent cooling sounds small, and on the mean pressure it is not, because the excess being computed is itself only a few per cent: a column of gas whose mean temperature is 1.2 per cent low has a mean pressure 1.2 per cent low, which is a third of the whole excess at this amplitude. The rectified heat flow takes back most of what the rectified pressure would have given. It is the thermal cousin of the rectification the lift itself is — a symmetric swing producing an asymmetric response — and of the steady streaming an oscillating boundary layer drives: all three are means of products that a linear calculation sets to zero.

The same asymmetric conduction is the working principle of thermoacoustic engines and refrigerators, in which a standing sound wave in a stack of plates pumps heat along them; here the plates are the levitator’s own walls and the pumping is across the gap rather than along it. Nothing about it requires a resonance. It requires only a gas oscillating in pressure against a solid that holds its temperature, which every squeeze film does.

Why the cooling stops at about a per cent #

The middle’s temperature deficit grows with the thermal number and then stops: 1.2 per cent at a hundred, 1.3 at three hundred, and about the same at a thousand. The pumping happens in the thin layers next to each wall where the gas does exchange heat within a cycle, and those layers thin as $1/\sqrt\Omega$. The cycle’s dissipation, which is also made in those layers, thins in the same proportion. The deficit is the ratio of the two, so it tends to a constant set by the ratio of specific heats and the amplitude, not by how slowly the gas conducts. A gas a thousand times too slow to cool is pumped cooler by the same per cent as one a hundred times too slow.

That fixes the shape of the lift curve at large thermal numbers too. With the deficit constant, the mean pressure approaches the adiabat’s excess minus a fixed share of the gas’s mean temperature, and both are reached together; the monatomic film’s lift flattens near 1.22 times the isothermal film’s and stays there. For a gas with a ratio of specific heats of 1.4 the ceiling is lower, and for one of 1.1 it is barely above one. The table of six gases is therefore not a point on a curve that a heavier gas would climb: the whole climb is a fifth, and it needs thermal numbers a hundred times what the levitator has.

The amplitude’s own law #

What the amplitude does is simpler and larger. The isothermal excess is $1/\sqrt{1-\varepsilon^2} - 1$, which is $\varepsilon^2/2$ for a small swing and $3\varepsilon^4/8$ more at the next order: doubling the vibration from a quarter of the gap to a half multiplies the lift by 4.7, from 3.3 per cent of an atmosphere to 15.5. The gas multiplies it by between 1.00 and 1.04. A levitator that needs more lift needs a larger swing in a smaller gap, and the excursion ratio is the whole design; the gas is a matter of efficiency. That is the same division of labour the drift in a wave that has none shows for a different mean — the size of a second-order mean is set by the square of the first-order motion, and the medium supplies a coefficient of order one.

What it means for a levitator #

For the levitator itself the consequence is a clean separation of the two design choices. The amplitude and the ambient pressure set the lift: the excess scales as the square of the amplitude and in proportion to the pressure, so three atmospheres lift three times as much. The gas sets the loss. At one atmosphere helium lifts the disc with 1.04 newtons at a loss tangent of 0.19, xenon with 1.09 at 0.36 — five per cent more lift for nearly twice the loss, counted with the viscous loss of the gas squeezed at the edge of the disc, which the trapped column leaves out. The stiffness frontier the previous calculation drew is a lift frontier too, with the same gases on it in the same order, because lift and stiffness both track the real part of the index and neither moves much.

Raising the pressure is the one handle that raises the lift substantially, and it comes with its own limit. The squeeze number falls as the pressure rises, so at ten atmospheres the gas in this levitator is no longer trapped — its squeeze number is under ten — and leaks out at the disc’s edge within a cycle, which a trapped-column calculation cannot describe and which a torn film still pulls showed can change the film’s character entirely.

The convention, and what the trapped column leaves out #

Every lift is the cycle-mean pressure over ambient divided by the isothermal film’s, at the same amplitude, so that the gas’s effect is read separately from the amplitude’s. The amplitude is a quarter of the mean gap throughout except where stated. The thermal number is formed on the mean gap and the ambient density. The walls are isothermal: the disc and the stage are metal, whose thermal effusivity is thousands of times a gas’s, so the wall temperature cannot follow the gas.

The column is trapped, which is the squeeze number’s high limit; at the levitator’s fifty to ninety, the outer few per cent of the disc’s radius leaks, lifts less and dissipates more, and the figures here are for the centre of the disc. The gas is ideal and its conductivity independent of density — kinetic theory’s result for a dilute gas, correct at one atmosphere and above. The film is one-dimensional across the gap, so the disc’s radial structure enters only through the choice to trap the column. And the load is taken as rigid: a levitated plate that moved with the film, as a real one can, would change the amplitude it sees.

Who found the lift, and the heat #

Salbu’s work at IBM in the 1960s put the squeeze-film bearing on an engineering footing and computed its load capacity from the isothermal Reynolds equation; Langlois had analysed the isothermal squeeze film a few years earlier. That the lift of an ultrasonic levitator is a rectified, second-order effect, and that its value depends on the gas’s polytropic behaviour in the gap, was argued again when near-field acoustic levitation was taken up for handling silicon wafers and other delicate parts in the 1990s and 2000s, by Hashimoto and co-workers and by Minikes and Bucher, who compared isothermal and adiabatic film models of the force. The heat-pumping that settles the gas’s middle below the walls is the mechanism of Rott’s thermoacoustic theory of the 1960s and 1970s, applied here to a gap rather than a stack.

How it was checked #

At a thermal number of a thousandth the film reproduces the isothermal excess $1/\sqrt{1 - \varepsilon^2} - 1$ to a part in a billion. Over a settled cycle the work done on the gas equals the heat conducted out through both walls, computed from the temperature gradients there, to three parts in ten thousand — the check that found an early version of the solver iterating too few times on the pressure, when the imbalance was two per cent. Halving the time step and refining the grid by half again changes the lift by four parts in a hundred thousand and the loss by one in a thousand. Squeezed by a twentieth of the gap, the loss matches the linear film’s $\pi\varepsilon^2\, \mathrm{Im}\,n$ for all six gases to 1.3 parts in a thousand. And the solver refuses an excursion past nine-tenths of the gap, a negative thermal number and a gas with no heat of compression.

Still open: the film that leaks at its edge #

Every number here is for the trapped centre of the disc, and a real levitator’s film is trapped only there. The next calculation solves the compressible Reynolds equation across the disc’s radius with the same temperature across the gap at each radius — a two-dimensional problem in radius and mass fraction, marched in time — and asks how much of the lift the leaking rim gives up at the levitator’s squeeze numbers, whether the rim’s lateral flow carries the cooled gas out or draws fresh gas in at the walls’ temperature and so undoes the heat-pumping, and how the lift then depends on the pressure once the squeeze number has fallen to the tens.