Concept

Lubrication — where it appears

The thin-film limit of the equations of motion, in which inertia is absent and the pressure is set entirely by the shape of the gap. It is what lets a converging film of oil carry a load with no pump anywhere in the machine.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

The one term in ground effect that really is about carried air. The added mass of a plate approaching a plane, as a multiple of its free-air value. In free air a plate borrows ρπc²/4 per unit span — exactly 0.79 for a unit chord in unit density, which is the mass of the circle its chord spans. Near a wall the fluid in the gap must leave sideways through a narrowing passage, so it moves faster than it would in the open and carries more energy: the borrowed mass grows as the inverse of the gap and diverges as the gap closes. At half a chord it is 1.21 times, at a tenth 2.06, at a twentieth 3.12. This is a cushion in the ordinary sense — fluid that has to be got out of the way and resists being — and it is the term the steady argument has none of.

The cushion that is there after all

A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.

misconceptions · Ground cushion

Three exponents for one dimensionless group. The local slope of each error, measured over one decade at a time. The duct's is exactly 1, the long wave's is exactly 2, and the slender body's drifts from 1.900 to 1.733 across the range and never reaches either. The same geometric ratio, in three problems that look alike, and the third one has no exponent at all.

One group, three exponents

Lubrication theory, shallow water and slender-body theory are taught in three places and are one expansion in one group — a ratio of two lengths, with no speed, no viscosity and no fluid in it at all. The error is supposed to be second order. In three problems that look alike it is first order, second order, and an exponent that does not exist.

regimes · Slenderness

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

The cushion changes its physics 0.36 mm from the ground. The two forces on a plate 10 cm across closing on a plane at 1 m/s in air at 20 °C, per metre of span, against the gap on logarithmic axes. The viscous squeeze film, Reynolds' lubrication result μVc³/h³, rises as the cube of the closeness; the inertial one, ρV²c³/24h² from the potential flow's added mass, as the square. They are equal where the gap Reynolds number ρVh/μ is exactly 24, at 0.361 mm, where each is 3.84e+2 N/m. Above that gap the cushion is the fluid's inertia and below it the fluid's viscosity — and at the crossover neither formula is accurate, since it is where one limit hands over to the other rather than a solution of the flow between them.

A cushion that changes its physics

A plate closing on a plane is resisted by the fluid it has to squeeze out, and the resistance is two different forces with two different laws — one from the fluid's inertia and one from its viscosity. They hand over at a gap of twenty-four kinematic viscosities per unit of closing speed, which for a wing in air is a third of a millimetre, and the two films disagree about whether the plate ever lands at all.

misconceptions · Ground cushion $The best efficiency runs from nine-eighths of φ² to one. The best efficiency a peristaltic pump can reach, against the fraction of the channel its wave closes, with its two limits. For a shallow wave it is 9φ²/8, which is small — a wave closing a fifth of the channel is at best 4.5 per cent efficient. As the wave closes the channel the best efficiency tends to one, and its shortfall shrinks in proportion to the remaining gap: about 1.9(1 − φ). Nothing in between is independent of the amplitude.$

The pump that is better the more it squeezes

A waving sheet swims at a cost per metre with no amplitude in it. A waving wall pumping fluid is the same mechanism turned round, and its efficiency is nothing like that: it starts at nine-eighths of the amplitude ratio squared, is exactly 2 − √3 at half closure, and rises towards one as the wave closes the tube — where the pump stops being a wave and becomes a piston.

kinematics · Continuity

Named alongside it

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

Added mass Model limit Asymptotics Efficiency Ground effect Image system Measurement Misconception Optimisation Continuity Convergence Correlation

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