Concept

Lubrication film — where it appears

A layer of fluid so thin against its own extent that the pressure does not vary across it and inertia is absent. It is what carries a bearing's load, and its governing equation is Reynolds' rather than Navier–Stokes'.

Named by 21 essays across 3 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

How hot seven films get. The peak temperature rise inside the film itself, on a logarithmic axis, for seven lubricated contacts with ordinary engineering numbers. A journal bearing runs tens of kelvin above its own housing, which is why lubrication systems are designed around heat rather than around load; a water-lubricated bush runs at a hundredth of a kelvin, because water conducts six hundred times better per unit of viscosity. The air bearing is marked: its gap is smaller than air's mean free path, so its film is not a continuum at all.

The film that heats itself

The oil in a bearing is not at the temperature of the metal around it. It is tens of kelvin hotter, the rise contains no length whatever, and whether the heating runs away or settles down depends on something that is not a property of the oil at all — whether the machine driving it holds the speed or holds the force.

viscous · Viscous heating

Twice as fast is not twice as thick. The film a plate carries out of water, against the speed it is withdrawn at, both logarithmic. The slope is exactly two-thirds, so doubling the speed thickens the film by 58.7 per cent and never by more. The open marks are outside the range the derivation holds in — above a capillary number of about a hundredth the film is no longer thin against the capillary length, and the measured thickness leaves this line.

What a plate takes with it

Pull a plate out of a bath and it comes out wet. How wet is not set by the plate, the bath or how much liquid there is, but by a competition in a region a fraction of a millimetre long that nobody looking at the plate can see — and the film goes as the two-thirds power of the speed, never as the first.

viscous · Coating

A force that depends on the molecular scale only through its logarithm. The force per unit length needed to move a 30° contact line of water at 1 mm/s, out to 1 mm, against the slip length on a logarithmic axis, from a picometre to a tenth of a millimetre. Each tenfold change in the slip length moves the force by the same fixed amount, so eight decades of the most uncertain length in the problem change the answer by a factor of about twenty — and at zero slip length the line keeps rising without end. The dashed curve takes the exact wedge's angle factor with a sharp cutoff; the solid one the thin-film wedge with Navier slip.

The drop a no-slip wall would never let spread

Liquid touching a solid moves with it, and nearly everywhere that is as close to exact as anything in fluid mechanics. At the edge of a spreading drop it cannot be: the stress in the corner rises as one over the distance from the edge, and the force needed to move the edge is infinite. Something slips over a nanometre, and because the answer depends on that length only through its logarithm, a drop spreads at almost the same rate whatever the something is.

misconceptions · The no-slip condition

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 $Below a Stokes number of ten a ball does not come back. The restitution of an impact through a liquid film, as a fraction of the same impact's dry restitution, against Stokes number. Nothing rebounds below the threshold and the recovery above it is a hyperbola: the restitution, as a fraction of its dry value, is one minus the critical Stokes number over the Stokes number, with that critical value 10.18 computed from the film and the dry restitution alone. The measured curve, drawn beside it, is the same expression with ten in place of that number — and ten is what four decades of viscosity and four materials all give.$

A ball that bounces in water and not in oil

A squeeze film cannot be closed with any finite energy, so nothing should ever touch anything. A sphere dropped into a tank nevertheless rebounds, and whether it does is decided by a number near ten that four materials and four decades of viscosity all agree on.

viscous · Squeeze film

A length you can only measure by squeezing

No slip is a boundary condition rather than a law, and nothing in the equations requires it. The evidence that it is right for an ordinary liquid to within a nanometre comes from the one experiment whose small length is chosen rather than given — a squeeze film closed to tens of nanometres.

misconceptions · The no-slip condition

The drip spacing is set by the depth of the layer. The spacing of the fastest-growing wave, in units of 2π capillary lengths, against the depth of the hanging layer in capillary lengths, for water, glycerol and honey. A thin film of any of them drips at √2. Deep water drips at √3. Deep glycerol and honey keep growing past √3, their fastest wave as long as the layer is deep, because viscosity slows short waves more than long ones.

How far apart a ceiling drips

A layer of liquid hanging from a ceiling is heavy fluid over light, and every ripple on it longer than about seventeen millimetres grows. Which ripple grows fastest, and so how far apart the drips form, is usually given as one number. It is at least three, and what chooses between them is not the liquid's surface tension but the depth of the layer.

turbulence · Instability

A film just past its fold lingers where the fold was. The temperature excess at the middle of a sheared film, scaled by the temperature over which the viscosity falls by e, against time in thermal diffusion times, switched on from rest at five stresses either side of the fold. Below the fold the film settles. Just above it the film climbs to the fold's own centre temperature, 1.19, lingers there as if it had found a steady state, and only then runs away — the longer the closer it is to the fold.

A lost steady state still holds the film

Push a self-heating oil film one per cent past the stress at which it can no longer settle and it does not run away at once. It warms to the temperature the vanished steady state would have had, sits there as if nothing were wrong, and only then goes — after a delay that grows as one over the square root of the overload, and during nearly all of which the load could still be taken back.

viscous · Viscous heating

Where a motor's line crosses the film's own characteristic. The stress a self-heating film carries against the speed it lets one wall slide past the other, both scaled, for every steady state (solid): it rises while the film is cool, peaks at the fold and falls as the film heats and thins. A motor's torque falls along a straight line as its speed rises (dashed), and the film runs where the two cross. A stiff drive crosses once. A soft drive, shallower than the falling limb's steepest slope, 0.0738, can cross three times. A drive held at a fixed stress is a flat line, and at the fold's stress it only touches.

A motor turns the runaway into a jump

A self-heating oil film has a fold at fixed stress and none at fixed speed, and a real motor is neither: its torque falls along a line as its speed rises. Put that line across the film's own torque–speed curve and the runaway disappears for any motor at all. What replaces it depends on the line's slope. A drive softer than one fourteenth of the cold oil's resistance jumps to a hot state and keeps a memory of the load; a stiffer one does neither. And the boundary between the two falls exactly where the motor is turning at a quarter of its no-load speed, which no motor near its rated speed can reach.

viscous · Viscous heating

Whether a sloping ceiling drips in place is one ray's growth. How fast a disturbance grows as seen by an observer moving along the ceiling at speed v, for four speeds at which the film carries its disturbances downhill, in the film's own units. On a flat ceiling (V = 0) the growth peaks at a quarter for the observer standing still and falls to zero for one running at ±1.622. Tilting the ceiling slides the whole curve downhill. While the observer at the point disturbed, v = 0, still sees growth, the ceiling drips where the disturbance began; at V = 1.622 that observer sees none, and above it the disturbance grows only while it is carried away.

A sloping ceiling drips downhill, or not at all

A film hanging from a level ceiling drips where its ripples form. Tilt the ceiling and the film flows downhill, carrying its ripples with it, and at some slope they are carried away faster than they spread and the ceiling stops dripping in place. That slope is set by one number, 1.622 — the speed at which a level ceiling's disturbance spreads — and for water it is tiny: a degree for a tenth of a millimetre of film, six for half a millimetre. Past it the ripples still grow, and still drip, but downhill, at a distance that grows with the slope: a ceiling shorter than that delivers its water to the edge.

turbulence · Instability

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

Every rotor lands on the same hot state, later and more smoothly. The centre temperature against time, on a logarithmic axis, for a soft drive five per cent past its fold switched on from rest, with rotors of four inertias. Each climbs, lingers and jumps, and each ends at the hot operating point without overshooting it. The time to pass three e-folds is 7, 16, 101, 948 diffusion times for M = 0, 0.1, 1, 10.

A rotor's inertia slows the jump and cannot make it ring

Give the motor driving a self-heating oil film a rotor that has to spin up, and there are two clocks: the film's diffusion time and the rotor's. Two clocks are what an oscillator is usually made of, and this pair cannot make one. Every eigenvalue stays real at every inertia, because the film and the rotor only ever push each other the same way. What inertia does instead is add its own delay to the film's, and in a real machine, where the rotor is hundreds of times slower, the delay past the fold is almost entirely the rotor's.

viscous · Viscous heating

Below the capillary length, the coat is set by the fibre, not the bath. The coat's thickness in units of ℓ_c Ca^(2/3) against the fibre's radius in capillary lengths, on logarithmic axes: the dynamic meniscus's 1.3376 over the curvature the static meniscus asks of it, 1/b + Z/ℓ_c². Thin fibres take Quéré's 1.3376 b Ca^(2/3), set by their own radius; thick ones take Landau–Levich's 0.9458 ℓ_c Ca^(2/3). The two laws cross at b = 0.71 ℓ_c, where the coat is 0.61 of either; at b = 0.1 ℓ_c it is 0.137 of what the plate's law would give.

A thin fibre coats by its own radius, and beads by it

A plate drawn out of a bath carries a film set by the capillary length, the size at which surface tension and gravity balance. A fibre thinner than that length carries a film set by its own radius instead, often a tenth of what the plate's law promises — and the same radius then decides how quickly that film gathers into beads. The faster the fibre is drawn, the thicker its coat and the shorter the length of it that comes out smooth.

viscous · Coating

An S-shaped set of states and a slow variable that crosses it. The film's steady mean temperature against the housing's, on the cool and hot branches of operating points, with the line on which the housing is in equilibrium with the film, at a derating of 0.22 per unit. Between housing temperatures 1.24 and 1.59 both branches exist. At the derating the essay uses, 0.22, neither meets the equilibrium line there: on the hot branch the housing is always warming, on the cool branch always cooling, so the film runs round the loop drawn over them, jumping at each fold.

A slow sensor makes the bearing cycle

Derate the motor driving a self-heating oil film on the temperature of its housing, which warms over minutes, and a protection meant to hold the bearing cool instead makes it cycle for ever: jump hot, warm the housing until the hot state is lost, drop cool, let the housing cool until the cool state is lost, jump again. The period belongs to the housing, the amplitude belongs to the film, and the derating strength decides only whether the cycle happens at all.

viscous · Viscous heating

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 warm bush stops the bearing hunting

A bearing whose motor is derated on its housing's temperature can hunt, jumping between a cool film and a hot one on the housing's clock. Let the bush warm with the housing, and a second feedback joins the first, of the kind that usually makes things run away. It does the opposite: a warm wall is a stiffer drive, the stiffening closes the S the cycle runs round, and past a bush coupling of about a third the bearing settles, warm and derated, however the protection is set.

viscous · Viscous heating

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

A slow bush leaves the bearing hunting, a sixth faster. The film's mean temperature excess over two cycles of a bearing hunting under a derating of 0.2 with a fifth of the housing's excess reaching the bush, for a bush that follows the housing instantly and one ten housing times slow. Both jump between a cool film and a hot one; the instant bush's period is 0.347 housing times, the slow bush's 0.291. The slow bush holds the walls at the cycle's mean, and the film's jumps are the same size.

A slow bush quickens the hunt it cannot start

A bearing's bush warms with its housing and makes the drive stiffer, and past a coupling of about a third it stops a derated bearing hunting. Give the bush a time of its own and the obvious fear is that the lag reopens the hunt. It cannot: whether a bearing hunts is decided by where its equilibrium sits, which no lag can move, and a lag on a positive link cannot make a settled state oscillate. What a slow bush does change is the hunt itself, which runs a sixth faster.

viscous · Viscous heating

A cut deeper than the film's window flips the film twice a cycle; a shallower one leaves it where it is. A protection that trips at a housing excess of 3.0 and resets at 2.6, against the depth of its cut: left, the film's jumps between its branches per protection cycle; right, the cycle's period in housing times. The film has both branches between 0.6496 and 0.7279 of full drive, so a cut past 0.35 leaves it only the cool one. cut 0.3: never resets; cut 0.34: 0 flips, period 1.85; cut 0.36: 2 flips, period 0.767; cut 0.6: 2 flips, period 0.756.

A trip deeper than the film's window flips the bearing twice a cycle

A smooth derating acts on a bearing's housing temperature as it stands. Most real protections trip instead: they cut the motor at one temperature and restore it at a lower one, a hysteresis of their own laid on top of the film's. Whether the oil then jumps between its cool and hot states is decided by one comparison — the cut's depth against the width of the film's window — and past it the protection's own band sets the rhythm. A trip never lowers the film's peak at all; it only protects the housing, and a deep one buys that with a plunge to the cool branch twice in every cycle.

viscous · Viscous heating

Named alongside it

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

Model limit Bifurcation Stability Thermal runaway Viscosity Viscous heating Dissipation Squeeze film Reynolds equation Thin film Hysteresis Surface tension

All concepts