A cushion that changes its physics

The cushion that is there after all finds a term in ground effect that genuinely is about carried air. A wing descending onto a plane borrows more mass than it does in the open, because the fluid in the gap has to leave sideways through a narrowing passage, and that borrowed mass grows without limit as the gap closes. It is inertia, not a spring, and a pilot feels it as a wing reluctant to settle.

It also names what it leaves out, and names it precisely. A small enough gap fills with the layers the fluid’s viscosity makes, and then the inertial force is replaced by a viscous one — the squeeze film every bearing runs on — with a different power of the gap and a different power of the speed. Which one governs is a Reynolds number of the gap flow, it says, and computing where the crossover sits is a calculation with a real number in it.

The number turns out to be twenty-four, and the gap it gives for a wing is a third of a millimetre.

Two forces from one gap #

Both forces come from the same geometry: a flat plate of chord $c$, parallel to a plane, closing on it at speed $V$ across a gap $h$ much smaller than the chord. The fluid in the gap has nowhere to go but sideways, and mass conservation fixes how fast it goes. At a distance $x$ from the middle of the plate, the fluid crossing that station per second is the plate’s speed times the width inside it, so the mean outflow speed is $\bar u = Vx/h$ — zero in the middle, fastest at the edges, and faster everywhere the thinner the gap.

What that outflow costs depends on which of the fluid’s two properties is doing the resisting.

If viscosity resists, the outflow is a Poiseuille profile across the gap and the pressure that drives it satisfies Reynolds’ lubrication equation. For a plate that is a parabola across the chord,

$$p(x) - p_{\text{edge}} = \frac{6\mu V}{h^3}\left(\frac{c^2}{4} - x^2\right), \qquad F_\mu = \frac{\mu V c^3}{h^3},$$

and the force per unit span is its integral. It goes as the first power of the speed, because viscous stress is linear in the rate of strain, and as the inverse cube of the gap, because the same flow through a thinner passage needs a steeper shear and a longer run of it.

If inertia resists, the outflow carries kinetic energy, and it is the kinetic energy that costs. Summed across the whole chord — both halves, since the fluid leaves by both edges — the energy in the gap is $\rho V^2c^3/24h$, so the plate is behaving as though it carried an extra mass

$$m_a = \frac{\rho c^3}{12h}$$

per unit span, on top of the free-air value a plate borrows in the open. The force that mass exerts is not the rate of change of its momentum. A body whose fluid carries $\tfrac12 m_a V^2$ with $m_a$ depending on position obeys Lagrange’s equation, and at constant speed the fluid pushes back with exactly half of $V^2\,|dm_a/dh|$:

$$F_\rho = \tfrac{1}{2}V^2\left|\frac{dm_a}{dh}\right| = \frac{\rho V^2 c^3}{24\,h^2}.$$

The half has a plain meaning. Closing the gap at a steady speed makes the fluid’s energy grow, and the plate supplies that growth by doing work against the fluid; the work per unit distance is the force, and it is half the rate at which the mass itself changes because the energy is half the mass times the square of the speed.

A Reynolds number of twenty-four #

Divide one force by the other and everything about the plate cancels:

$$\frac{F_\rho}{F_\mu} = \frac{\rho V h}{24\,\mu} = \frac{\mathrm{Re}_h}{24}.$$

The chord has gone, the cubes have gone, and what is left is the Reynolds number of the gap — the closing speed times the gap over the kinematic viscosity — divided by twenty-four. The two forces are equal where that Reynolds number is twenty-four, which is at the gap

$$h^{*} = \frac{24\,\nu}{V}.$$

A kinematic viscosity over a speed, and nothing else. No chord, no load, no mass. For air at twenty degrees closing at a metre a second it is 0.361 millimetres, and there the two forces on a ten-centimetre plate are each 384 newtons per metre of span. Above that gap the inertial force is the larger and grows only as the square of the closeness; below it the viscous force is larger and grows as the cube.

That a factor of twenty-four appears rather than one is worth a sentence. The viscous force carries the six of Reynolds’ parabola and the inertial one the twelve of the added mass and the two of Lagrange’s half, and the ratio of their numerical factors is what shifts the crossover away from a gap Reynolds number of one. A scaling argument would have put the handover at order one and been wrong by more than an order of magnitude in the gap — which is what an order-one estimate is worth when the constants are computed rather than assumed.

Air, water and oil #

Because the crossover is a kinematic viscosity over a speed, three fluids and three speeds put it in three completely different worlds.

A wing settling onto a runway at a metre a second has a crossover gap of a third of a millimetre, and at three metres a second of 0.12. No wing has ever flown at a gap that small. So the cushion a pilot feels in the flare is the inertial one, at every height it can be felt at, and the viscous squeeze film under a wing is a force that exists only in the last fraction of a millimetre before the tyres have already touched. The claim that ground effect’s cushion is a lubricating film of air is not merely an exaggeration of its size; it is the wrong physics at every gap in question.

A plate settling through water at ten centimetres a second hands over at a quarter of a millimetre, and at a metre a second at twenty-four micrometres. Water’s kinematic viscosity is fifteen times smaller than air’s, so for the same speed its inertial range extends fifteen times closer to the wall.

A microscope slide pressed onto a film of light oil at a millimetre a second has a crossover gap of 2.4 metres. Every gap a slide is ever at is fifteen hundred times smaller than that, and the cushion under it is entirely viscous — which is why a slide lowered onto a drop of oil settles slowly and never quite stops settling, and why a gauge block wrung onto a flat is so hard to lift straight off.

The inertial term, solved rather than summed #

The inertial force rests on an added mass, and the added mass so far has been the sum of two limits: the plate’s free-air value $\rho\pi c^2/4$, exact far from the wall, and the thin-gap term $\rho c^3/12h$, exact close to it. That sum has both ends right and solves nothing in between. The potential flow of a plate near a plane can be solved directly, and it is worth doing, because the sum is what the force above was built from.

The solution represents the plate as a thin ellipse covered in source panels, adds a mirror image of every panel beneath the wall so that no flow crosses it, solves for the panel strengths that make the plate move rigidly, and reads the kinetic energy off the surface. The panel method converges at first order in the number of panels, so each value is computed at two resolutions and extrapolated; the isolated circle it is checked against comes out at π to four parts in ten thousand.

The sum of the two limits is high everywhere, by at most 10.3 per cent, and it converges on the solution as the gap closes. At a twentieth of a chord the thin-gap term carries 96.9 per cent of the extra mass the solution finds, and at 0.035 chords the sum is within two per cent. So the inertial force $\rho V^2c^3/24h^2$ is not an estimate that happens to have the right shape. It is the leading term of the exact potential flow, and it is accurate to a few per cent at precisely the gaps where the crossover argument is made.

One plate, three films #

A crossover of two forces is a statement about the forces. What a plate actually does when it closes on a plane depends on how the force changes as it goes, and the two films change in ways that disagree about the ending.

Through the inertial film alone, the plate lands. Near the plane the borrowed mass dominates the plate’s own, and the equation of motion with $m_a \propto 1/h$ and the Lagrangian force gives a closing speed proportional to the square root of the gap. A speed that falls as $\sqrt h$ still closes the gap in a finite time, since the integral of $1/\sqrt h$ converges, and the plate reaches the plane after 0.108 seconds, arriving at 0.13 millimetres a second from a start of two hundred. The fluid has absorbed almost all of the plate’s kinetic energy into the outflow, and what arrives at the plane is a plate barely moving.

Through the viscous film alone, it never lands. Once the viscous force governs, the plate settles at the terminal speed that balances its load against $\mu Vc^3/h^3$, which is proportional to the cube of the gap. Then $1/h^2$ grows linearly in time and the gap closes as $\sqrt{\mu c^3/2Wt}$ — the straight pale line on the figure, which the computed curve lies along within two per cent after the first seconds. After ten thousand seconds the plate is 1.6 micrometres clear, and after a year it would still be nearly thirty nanometres clear.

With both films present, summed, the plate follows the inertial curve down to a gap of about a quarter of a millimetre and then stops following it. It is caught by the film the inertial cushion handed it to. In reality it is caught earlier still, by the roughness of the two surfaces: the viscous film’s promise that the gap never closes is a promise about perfectly flat plates, and a real plate lands on its highest asperities — the same roughness that, below a certain size, a flowing layer cannot feel and that here, at the bottom of a squeeze film, decides where the landing is.

Speed against gap #

Drawn as speed against gap, the two films are two straight lines on logarithmic axes with slopes of one half and three, and the handover is where the plate’s own trajectory crosses the line on which the gap Reynolds number is twenty-four. That line is not a property of the plate. It is $V = 24\nu/h$ — the same crossover drawn the other way round — and a plate of any size, mass or load crosses it wherever its own speed happens to fall on it.

The slope of three is the reason a viscous cushion is so much more effective at the end than an inertial one. Halving the gap under a viscous film cuts the closing speed by eight; halving it under an inertial film cuts the speed by only 1.4. A viscous film does its work in the last micrometres, and does it so thoroughly that it never finishes; an inertial film does its work over the last centimetres, and lets the plate through.

The same film under water #

In water at a tenth of the speed the lines have moved and their crossing has hardly moved at all: 0.241 millimetres against 0.361 in air. Water is eight hundred times denser and fifty-five times more viscous than air, and the crossover gap depends only on the ratio of the two, which differs by a factor of fifteen — and the tenfold drop in speed undoes most of that. The forces at the crossover are nineteen times larger, because both scale with the density or the viscosity themselves rather than their ratio.

That is the practical content of the crossover being a kinematic viscosity over a speed. Two squeeze films in fluids nothing like each other can hand over at the same gap, and whether a given cushion is inertial or viscous is never a question about the fluid alone.

The same film, run backwards, is called suction #

There is a familiar object that is a squeeze film run in reverse, and it belongs in this field because it is routinely given the wrong name.

A wet glass set down on a smooth table sticks to it. Lifting it straight up takes a surprising pull, and the pull is described, almost universally, as suction. Pulling two wet plates of glass apart is harder still, and the explanation offered is the same. But nothing sucks, and the gap under the glass is not sealed: water can flow into it from the edge. What resists the lift is the film’s viscosity. To open the gap at a speed $V$, liquid has to be drawn in from the rim through a gap $h$ thin enough that the viscous force $\mu Vc^3/h^3$ is enormous, and it is that force — the lubrication force with its sign reversed — that the hand is working against.

The test that separates the two accounts is time. A seal holding a pressure difference would hold it indefinitely; a viscous film yields to any steady pull, slowly, at the rate its cubic law allows. A wet glass that resists a sharp jerk comes away under a gentle steady lift, and two wet plates slide apart under a small sideways force that no seal would permit. Stefan measured exactly this adhesion in 1874 and gave the viscous law for it, and the relation is still quoted as Stefan’s adhesion. What is not usually quoted is that it is the same squeeze film this essay has been following towards a wall, with the gap opening instead of closing.

What the two limits leave out #

Neither is accurate at the crossover. Each force is exact in its own limit, and the handover gap is where they are equal, which is exactly where neither limit applies. The flow at a gap Reynolds number near twenty-four has both inertia and viscosity in it, and the true force there is not their sum; the summed film in the figures is a construction with both ends right, called that where it is drawn.

The gap is parallel and the plate is rigid. A plate that tilts as it closes has a gap varying along its chord, and the cubic law then concentrates the force at the narrow end. A flexible plate deforms under the pressure the film makes, which is the elastohydrodynamic problem of every gear tooth and is not here.

The air in the gap is incompressible and continuous. A gas film thin enough is compressed by its own squeeze pressure, which is how a gas bearing stores energy and gives some of it back; and a gap approaching air’s mean free path of about seventy nanometres is where a fluid stops being one and slips at the walls. The viscous film’s last nanometres in the touchdown figure are beyond both.

And the plane has no roughness. The viscous film never closes on flat plates, and no plates are flat. The distance at which a real landing becomes contact is set by the heights of the asperities, which is a property of the surfaces rather than of the fluid.

The forces are checked against arithmetic that does not share their derivation. The viscous force is recomputed by integrating Reynolds’ parabola across the chord by quadrature and agrees to a part in a million; the ratio of the two forces equals the gap Reynolds number over twenty-four at every gap tried, to machine precision; and the viscous touchdown’s late gap agrees with $\sqrt{\mu c^3/2Wt}$ to two per cent, while the inertial touchdown reaches the plane in finite time at under five per cent of its starting speed.

Three old results in one gap #

The viscous squeeze law is Reynolds’, from his 1886 paper on lubrication, and Stefan had the adhesion form of it twelve years earlier. The added mass of a body moving through an ideal fluid is older than both: Green computed it for an ellipsoid in 1833 and Stokes for a sphere in 1843, and the method of images that puts a wall into a potential flow is Kelvin’s and Lamb’s.

None of those authors was thinking about the others’ problem, and the crossover between them is a comparison nobody needed to make until somebody asked what a landing wing’s cushion is made of. The answer has the shape this field keeps finding. A phrase — a cushion of air — attaches to a real sensation, a real mechanism explains the sensation, and the phrase is then quietly given to a different mechanism with a more respectable name, lubrication, which is correct physics for a bearing and has nothing to do with the wing.

Still open: what the boundary’s own condition does to the borrowed mass #

Every calculation here has been against a solid wall, whose one instruction to the flow is that nothing passes through it. The borrowed mass grew because of that instruction, and a boundary that gives a different instruction should give a different answer.

That is the borrowed mass the boundary decides: the same plate closing on a free surface, or accelerating inside an open-jet wind tunnel rather than a closed one, where the boundary holds its pressure rather than its position — and where the borrowed mass falls instead of growing, by amounts that reverse the sign of every interference computed here.

Beside it is the lubrication problem proper, in which the squeeze film is not a nuisance at the end of a landing but the whole purpose of the device: a bearing carries its load on a film held between surfaces by exactly the cubic law above, and the gap it chooses is set by the same competition between speed and viscosity that put the crossover at twenty-four.