Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Reynolds equation — where it appears

What is left of the Navier–Stokes equations in a film much thinner than it is long. It relates the pressure in the film to the shape of the gap and the motion of its walls, with no inertia in it anywhere, and it is what every bearing calculation solves.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

20.45 MPa out of a film 25 µm thick. The pressure along a tapered pad, from the closed-form solution of Reynolds' equation, with the same equation's tridiagonal grid solve drawn over it as points. The peak is 20.45 MPa — enough to yield mild steel — and it sits at 69 per cent of the way along rather than in the middle, because the pressure gradient vanishes where the film equals the harmonic mean of its two ends and the harmonic mean is biased towards the thinner one. The pad's own shape is drawn along the top, to a vertical scale of its own. Nothing pumps this oil: the runner drags it into a narrowing gap and the gap does the rest.

Nothing but the shape of the gap

A machine that holds a steel shaft off its bearing with a film of oil twenty-five microns thick has no pump in it, and the pressure it generates would yield mild steel. The mechanism is not the oil and not the speed; it is that the gap narrows.

viscous · Lubrication A film with nothing useful to show for itself. The pressure under two discs squeezed together, as a fraction of the pressure at the centre. It is a parabola, and it holds an enormous load — but nothing is sliding, so there is no output at all and every joule put in becomes heat in the oil. The two routes to that heat share no arithmetic: one integrates the dissipation function over the film, the other multiplies the force by the approach speed.

The last of the oil

A wet plate on a table is held down by nothing but the difficulty of getting air in underneath it. The force required to separate two flat surfaces with a film between them goes as the inverse cube of the gap, which means there is no finite energy that removes the last of the film and no finite time in which it drains away.

viscous · Squeeze film A pure number with no fluid in it. The load a plane slider bearing carries, divided by everything dimensional in it. What is left is a function of the convergence ratio alone — no viscosity, no speed, no size, no material — so the tilt that carries the most load is decided before anything about the machine is known. It is the root of one transcendental equation, at 2.1887048.

The gap that carries the most

A bearing's most consequential dimension is decided by a number with no oil in it, no speed and no size — the root of one transcendental equation, at 2.1887048. And the maximum it sits on is flat enough that the value printed in every handbook is not it.

viscous · Lubrication Where a film stops being a damper and starts being a spring. The stiffness and the damping of a circular gas film against its squeeze number, both in units of ambient pressure times disc area. At the left the stiffness is nothing and the damping is Stefan's incompressible law; at the right the damping has gone and the stiffness has reached π, which is an isothermal gas trapped in the gap with no way out. They cross at σ = 6.23, and neither limit was put in by hand — both fall out of a Bessel function of a complex argument.

A damper that turns into a spring

A film of oil squeezed between two plates resists motion and stores nothing, and that is a property of the oil rather than of the film. Fill the same gap with air and the same equation gives a film that stores and resists nothing — above a squeeze number of six, with nothing changed but the frequency and the gap.

viscous · Squeeze film A film pulled apart asks for more tension than a liquid has. The pressure below ambient in the film under a sphere moving away from a wall, scaled by the tension the liquid can bear, against the distance from the axis in sphere radii, at the contact gap. The lubrication solution (dashed) asks for four times that tension on the axis. A liquid that cannot give it cavitates: a disc of vapour opens where the demand exceeds the floor, and outside it the pressure is exactly the solution it would have had. Here the disc reaches 0.12 sphere radii.

A torn film still pulls

A sphere bouncing off a wall under liquid has to climb back out through the film it squeezed, and the film pulls it back with a suction no real liquid can supply. Let the liquid cavitate and the obvious guess is that the sphere escapes the torn part of the film for free. It does not. The liquid round the vapour disc goes on pulling, and the disc itself holds the full tension over its area, so a film that tears at five times the contact gap saves a third of what the guess says — and the rebound threshold moves by five per cent where the guess said a third. At an atmosphere, in the liquids the threshold was measured in, it barely moves at all.

viscous · Squeeze film Five gas films never meet their thermal crossing. Six air films placed by their squeeze number, which decides whether the gas leaves the gap or is trapped in it, and their thermal number, which decides whether the heat of compression leaves. The dashed diagonal is the estimate that the two differ by the Prandtl number. The devices lie between one and eleven decades below it, because heat leaves across the gap and the gas along the disc. Only the levitator at a twenty-micron gap comes within a decade of the thermal crossing at ωh²/κ ≈ 18.

The heat of a squeeze leaves across the gap

Squeeze a film of air and it warms, and whether that heat reaches the walls in time decides whether the gas is compressed isothermally or adiabatically. The natural estimate has the heat leaving the way the gas does, along the disc, and puts the thermal crossing beside the viscous one. It leaves across the gap instead, a distance hundreds of times shorter, so the crossing sits decades away — and when a film does reach it, what it gains is not stiffness but a second band of loss.

viscous · Squeeze film A layer read as slip gives a slip length that drifts. The slip length a drainage measurement would fit at each gap to the force of a 1-nanometre viscous layer, using the slip formula, for layers three and ten times as viscous and immobile. Far off it is the first-order value, −0.667 nm for κ = 3; closer, it shrinks towards zero — −0.597 nm at 10 nm and −0.525 at 5. A slip length that depends on the gap is the layer's signature, and the only one a drainage curve carries.

A viscous nanometre reads as slip until the gap is ten of it

A drainage experiment turns a molecular length at a wall into a measurable force, and reports it as a slip length. A layer of liquid a nanometre thick and several times as viscous as the bulk changes the same force by the same amount, to first order. Solved with the stratified viscosity, the two pictures part by a per cent only when the gap is eleven to seventeen layers wide, and the part of the difference a measurement can see is a slip length that drifts with the gap. A layer that does not move at all is exactly a wall in a different place.

misconceptions · The no-slip condition Helium wastes least for most stiffnesses; xenon reaches furthest. The film's loss tangent — damping over stiffness — against its stiffness, each gas traced from a tenth of an atmosphere up to the pressure of its greatest stiffness. Raising the pressure buys stiffness and costs loss for every gas. Up to about a hundred newtons per unit relative amplitude helium's film loses least at any stiffness — 0.29 at 50 N against air's 0.4 and xenon's 0.46. Beyond, helium is near its peak of 127 N and the heavy monatomic gases take over: xenon reaches 149 N, and at 120 N loses 0.73 against helium's 0.78.

The gas sets a squeeze film's loss, not its stiffness

A squeeze-film levitator wastes part of every cycle as heat that crosses the gap, and the gas and its pressure decide how much. Fill one levitator with six gases at one atmosphere and its stiffness hardly changes while its loss nearly doubles from helium to xenon. Raise the pressure and every gas stiffens towards a peak near ten atmospheres. Helium wastes least at any stiffness up to a hundred newtons; above that, only the heavy monatomic gases get there.

viscous · Squeeze film Elasticity adds a spring at first order and a slip only at second. The damping's deficit, one minus the damping ratio, and the in-phase force against Λ on logarithmic axes. The in-phase force rises in proportion to Λ — a fitted slope of 1.0000 — and the damping's deficit as its square, 2.0000. A slip length changes the damping at first order and adds no spring at all, so a soft wall and a slipping one are distinct in kind: the soft wall announces itself first in the phase of the force.

A soft wall is read as slip only at second order

A drainage force smaller than Taylor's has been read as the liquid slipping at the wall. A wall that gives under the drainage pressure lowers the force too, and it can be mistaken for slip. Solved together, the flow and the elastic wall say how: the softness first adds a spring to the force, in phase with the motion, and only at second order takes anything from the damping a slip length is read from. When it does, the slip it imitates grows as the square of the frequency and falls as the square of the gap.

misconceptions · The no-slip condition A mica sheet shields the glue's false slip only where the pressed region is narrower than the sheet bends. The slip length, in nanometres, that a dynamic drainage measurement at 100 Hz would read from a wall that does not slip, against the gap: a 3 mm sphere in water over bare glue of plane-strain modulus 1 GPa, and over mica sheets 1, 3 and 5 µm thick on the same glue. At 3 nm the bare glue reads 9.48 nm and the sheets 8.75, 2.68 and 1.06; at 30 nm, 0.1 against 0.099, 0.067 and 0.043.

A mica sheet hides its glue only at the smallest gaps, and changes how the false slip falls

A drainage measurement against a soft wall reads a slip that is not there, falling as the square of the gap. The wall in a surface-force apparatus is not a soft half-space but a stiff mica sheet on a soft glue, and the sheet spreads the drainage pressure over its own bending length before the glue feels it. Where the pressed region is wider than that length the sheet changes nothing; where it is narrower the sheet sinks as one piece, a spring in series with the liquid. That cuts the false slip only by the square of the bending length over the pressed width — threefold at three nanometres for three micrometres of mica — and it turns the slip's fall with the gap from the inverse square towards the inverse, which is the test an experimenter would have used to see it.

misconceptions · The no-slip condition

Named alongside it

The objects these essays reach for when they reach for this one.

Squeeze filmModel limitLubrication filmDampingLubricationMeasurementThe no-slip conditionPressureThin filmCompressibilityDimensionlessDissipation

All concepts