The last of the oil

A wet plate on a wet table is hard to lift straight up and easy to slide sideways. That asymmetry is the whole of this essay, and the explanation almost everybody gives for it — surface tension — is wrong by three orders of magnitude and predicts the wrong dependence on speed.

What actually holds the plate down is that lifting it requires liquid to flow inwards, from the rim towards the middle, through a gap that is a fraction of a millimetre. The film has to be fed, the feeding is a viscous flow through a very thin channel, and thin channels are extraordinarily reluctant.

The same equation with the wall moving the other way #

A sliding bearing generates pressure because its gap is narrowing along the direction of sliding. A squeeze film has no sliding at all, and generates pressure because its gap is narrowing in time.

Reynolds’ equation carries both. For two circular discs of radius $a$ approaching at speed $|\dot h|$, with no sliding, it reduces to

$$\frac{1}{r}\frac{d}{dr}\left(r h^3 \frac{dp}{dr}\right) = 12\mu\dot h,$$

which integrates directly, with zero pressure at the rim, to

$$p(r) = \frac{3\mu|\dot h|}{h^3}\left(a^2 - r^2\right), \qquad F = \frac{3\pi\mu|\dot h|a^4}{2h^3}.$$

That is Stefan’s law, and every term in it is worth a moment. The force is proportional to the approach speed, so the film is a damper rather than a spring — it resists motion and stores nothing. It goes as the fourth power of the radius, so a disc twice as wide is sixteen times as stubborn. And it goes as the inverse cube of the gap, which is where everything interesting comes from.

No useful output at all #

A sliding bearing turns a little dissipation into a large load capacity, and the load is the point. A squeeze film has no output whatever. Nothing is sliding, nothing is being carried, no work is being done against anything but the fluid itself.

So every joule that goes into a squeeze film becomes heat in the film, and the accounting is as stark as this collection gets. The two routes to that number are set out in the essay that established them: integrate the dissipation function over the film, or multiply the force by the approach speed. They agree to eight figures, and neither knows about the other — one is a volume integral of a squared shear rate at every radius, the other is a single product.

The gap that will not close #

The inverse cube is what makes the film interesting rather than merely stiff. Put a constant load $W$ on the disc and ask how the gap falls. Setting $F = W$ and separating gives

$$\frac{1}{h^2} = \frac{1}{h_0^2} + \frac{4Wt}{3\pi\mu a^4},$$

so the gap falls as $t^{-1/2}$ and reaches zero only as $t \to \infty$. Not slowly. Never.

The energy version of the same statement is sharper. The work to close from $h_0$ to $h_1$ at a constant approach speed is

$$W = \frac{3\pi\mu|\dot h| a^4}{4}\left(\frac{1}{h_1^2} - \frac{1}{h_0^2}\right),$$

which diverges as $h_1 \to 0$. There is no finite amount of energy that removes the last of the oil, at any speed, with any machine.

Why a bearing survives being started #

That divergence is not a curiosity; it is the reason rotating machinery exists in the form it does.

A journal bearing under load is a squeeze film for as long as it is stationary. The shaft is trying to reach the bush, and it approaches by squeezing oil out of the converging gap — and by the law above it does not arrive. A machine left loaded overnight has a film thinner than it had at shutdown and thicker than nothing, and when it is started the sliding action takes over and rebuilds the film before contact occurs.

The number that matters is the time constant, and it is long. A bearing carrying a tonne on a 50 mm journal with a 25 µm clearance takes hours to squeeze to a tenth of that, which is comfortably longer than any shutdown. The design does not have to prevent contact; it has to be slower than the next start-up.

Where the resistance actually is #

The force is a pressure integral over the disc, which invites the picture of a uniform push. It is not uniform, and the dissipation is less uniform still.

The radial velocity in the film is parabolic across the gap and grows linearly with radius, because everything inside a radius has to come out through the circumference at that radius. So the shear rate grows with radius too, and the dissipation — which goes as the square of it — grows as $r^2$. Weighted by the area at each radius, that puts half of the heat outside 84 per cent of the radius: a squeeze film’s bill is paid almost entirely at its rim, where the pressure is nearly zero.

That is the same distribution the pressure-driven channel has, and for the same reason: the fluid that is being deformed hardest is the fluid nearest the exit, and it is nowhere near where the load is being carried.

The number a machine designer actually uses #

The force law has four quantities in it and only one of them is usually adjustable, which makes the design conversation short and worth setting out.

The viscosity is chosen for other reasons — a bearing’s oil grade is decided by the running film, not by the standing one. The approach speed is whatever the machine is doing. The radius is set by the part. What is left is the clearance, and it enters as its cube.

So halving a clearance multiplies a squeeze film’s stiffness by eight, and the time it takes to drain by four. That is an enormous lever and it is used in both directions. A damper wants a small clearance and gets one. A hydraulic component that must respond quickly wants a large one, and the usual failure mode of a valve that has silted up is not that it is blocked but that its clearance has closed and its response time has gone up as the inverse square.

The corresponding number for the pressure is a warning rather than a lever. The peak pressure at the centre is $3\mu|\dot h|a^2/h^3$, which for the disc drawn above is 0.12 megapascals and for the same disc at a micron is 120 megapascals — past the yield stress of mild steel. A squeeze film that is thin enough deforms the surfaces that are squeezing it, and everything above assumes they are rigid.

Soft surfaces make it worse, which is why joints work #

The list of limitations notes that the surfaces are rigid and that the pressures computed become large enough to deform them. Following that through changes the conclusion in a direction worth having: a compliant surface does not let the film escape sooner. It traps it.

The mechanism is a feedback. The pressure peaks at the centre, so a soft surface is pushed away hardest there; the gap therefore becomes flatter and wider rather than lens-shaped; a flatter, wider gap has a larger effective radius, and the force goes as the fourth power of the radius. So the deformation raises the resistance, which raises the pressure, which raises the deformation. A soft contact under load flattens into a broad plateau with a nearly uniform film under it, and the drainage time is longer than the rigid calculation predicts by orders of magnitude rather than by a factor.

That is the reason a rubber sucker holds, why a wet fingertip on glass feels adhesive when a wet coin does not, and why a soft seal is a better seal than a hard one of the same nominal geometry.

The case where it matters most is biological. A synovial joint is a squeeze film between two layers of cartilage, and cartilage is not merely soft: it is a porous solid saturated with fluid. Load it and the fluid inside cannot escape quickly, so it takes up the pressure — carrying the great majority of the load, with the solid matrix taking the rest — and the surface stays separated because the fluid holding it apart is inside the surface as well as between the two.

That support decays as the interstitial fluid seeps out, over minutes to hours, which is exactly the timescale over which a person stands still. Move the joint and the sliding action recharges the film, in precisely the way a bearing’s sliding action recharges its own — so a joint is well lubricated when it is being used and progressively less so when it is held, which is why standing still is more uncomfortable than walking and why stiffness is a symptom of rest rather than of exertion.

The engineering moral runs the other way from the intuition. A designer wanting a film to persist should not make the surfaces harder and flatter; hard flat surfaces are the case this essay computes, and they drain as fast as anything can. Compliance, porosity and a fluid that resists being squeezed out of a matrix all lengthen the same time constant, and evolution found all three.

And the divergence at the end of the essay becomes stronger rather than weaker. If the last of the oil cannot be removed from between two rigid discs with any finite energy, it certainly cannot be removed from between two soft porous ones — where the film is no longer merely a layer between the surfaces but a part of them.

What it is used for on purpose #

Dampers. A squeeze-film damper is a bearing deliberately supplied with a film that is not supposed to carry load, only to resist motion; every gas-turbine rotor has one, and the design parameter is the fourth power of the radius over the cube of the clearance.

Getting things apart. The inverse cube runs both ways. Separating two surfaces requires fluid to flow in, and at small gaps that is just as slow — which is why a suction cup works, why a freshly-lapped optical flat has to be slid off its partner rather than lifted, and why a wet sheet of glass will support its own weight against a wall for minutes.

And things that hammer. A film that is being squeezed hard enough will reach a pressure at which the liquid can no longer take tension on the retreating stroke, and it cavitates. What happens next is a collapse rather than a flow, and it is loud and it damages metal.

Sliding is easy for the same reason #

The asymmetry the essay opened with now has an arithmetic. Lifting the plate requires the film to be fed radially inwards through the gap, and the resistance to that goes as $a^4/h^3$. Sliding the plate requires nothing of the sort: the film simply shears, and the resistance is $\mu U \pi a^2 / h$.

The ratio of the two, for a plate being moved at the same speed either way, is $a^2/h^2$ times a number of order one. For a 40 mm disc on a 0.1 mm film that ratio is about forty thousand.

Sliding a wet plate is four orders of magnitude easier than lifting it, and neither number has anything to do with surface tension. It is worth doing the capillary comparison explicitly, since it is the explanation usually offered: the capillary force on the rim of that disc is about a hundredth of a newton, against seventy-five newtons of viscous resistance at a millimetre a second. Surface tension is present, is real, and is four thousand times too small.

The observational test is the speed dependence. A capillary force does not care how fast the plate is being pulled; a viscous one is proportional to it. Pull the plate away slowly enough and it comes; pull it fast and it does not. That is not what an adhesive does.

What the picture cannot show #

The surfaces are perfectly flat and perfectly smooth. Real ones are neither. Once the gap is comparable with the roughness — which for a good machined surface is a fraction of a micron — the film is no longer a film, contact begins at the asperities, and the divergence above stops being physical. The divergence is the model’s, not the world’s, and where it stops is set by a length this calculation does not contain.

And below that, the liquid stops being a continuum. At a gap of a few nanometres the molecules of a liquid are ordering themselves against the surfaces and the viscosity in Reynolds’ equation is not a property of the bulk fluid any more. Both limits arrive long before the arithmetic does.

The film is isothermal here and is not in a real damper. The dissipation computed above is deposited in a very small volume — a squeeze film at a micron dissipating a kilowatt per square metre is heating a gap smaller than a hair — and a film that heats itself thins itself, which changes the number that produced the heating.

And no air gets in. The whole calculation assumes the film stays full. In practice a rapidly separated film draws air in from the rim, and the force collapses when it does — which is why a wet plate lifted quickly at one corner comes away and lifted flat does not.

The damper that a film makes, and its one awkward property #

A squeeze film is a damper, and the qualification worth adding is that it is a nonlinear one, in a way that changes how a machine behaves rather than only how much.

An ordinary dashpot has a force proportional to velocity with a constant of proportionality. A squeeze film’s constant is $3\pi\mu a^4/2h^3$, and $h$ is not a constant — it is the very displacement being damped. So the damping coefficient rises steeply as the gap closes and falls as it opens, and a rotor supported on one is stiff against large excursions and slack against small ones.

That is usually what is wanted, and it is why squeeze-film dampers are used in preference to mechanical ones in turbomachinery: the damping arrives when it is needed and gets out of the way when it is not. It is also why they are difficult to model, since a coefficient that depends on the amplitude of the motion cannot be put into a linear rotordynamic analysis at all.

The second awkwardness is the one already mentioned: a film being squeezed and released alternately spends half of each cycle in tension. Liquids withstand very little tension, so a real damper cavitates on the retreating stroke, the film goes slack, and the force is not symmetric between the two halves of the cycle. The measured damping of a cavitating film is roughly half the uncavitated prediction, and which half of the cycle is doing the work has moved.

Who found it, and when #

Stefan gave the law in 1874, five years before Reynolds’ bearing paper, and for a while the two results were not recognised as the same equation. Reynolds’ 1886 paper contains both terms and is where the squeeze contribution acquired its modern form.

The surprising connection is with a result about time rather than about force. The gap falling as $t^{-1/2}$ under a constant load is exactly the same law as the growth of a diffusive layer — a Stokes layer, a thermal front, a boundary layer’s thickness with distance. The reason is the same in every case: a process whose rate is proportional to the inverse square of a length, driving that length, gives a square root of time. A film draining under load and vorticity spreading from a wall are the same differential equation, which is why neither of them ever finishes.

Where the ladder goes next #

Beside this rung is the sliding film, which is the same equation driven by geometry rather than by time, and the film that heats itself, which is what the dissipation computed here does to the viscosity it was computed from.

Below it is the price of a gradient, which is where the claim that a squeeze film’s whole input becomes heat is made precise, and above it is the question of what happens when the film is not liquid at all — a gas film in a disc read head is a squeeze film whose gap is smaller than a mean free path.