Threshold — where it appears
Named by 50 essays across 9 fields — each of them below, with the objects they name alongside it.
Whether the droplet turns
The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.
What "of order one" is worth
A dimensionless group is built by comparing two terms, so it is one when the terms are equal — and that is the only thing it says. Where the behaviour actually changes is a separate question with a separate answer, and across fourteen groups on this site the two numbers differ by factors from one to five hundred and ninety-four.
Where a fluid stops being one
The Knudsen number is the mean free path over the size of the thing, and at one a molecule crosses the whole channel between collisions. The continuum equations with a no-slip wall are already one per cent wrong at one part in five hundred and ninety-four, which is a factor nothing about the definition would suggest.
Slow enough to be steady
A wing moving slowly enough is assumed to carry the lift its instantaneous angle asks for. The reduced frequency has two thresholds — one where the apparent-mass and circulatory lifts are equal, and one where the quasi-steady answer stops being right — and they are a hundred and seventy-eight apart.
How small is small enough
Creeping flow drops the inertia terms and gets an exact answer for the drag on a sphere. The Reynolds number at which those terms are equal is one — and at one, Stokes' law is already sixteen per cent low. The honest boundary is 0.054, and the reason it is so far down is the same reason the theory needed repairing in the first place.
The size a drop is allowed
The Bond number sets a drop's weight against the force its own skin can supply, and it is one when they are equal. By then the drop is a bun — it is a per cent from being a ball at Bond number 0.0079, which is a water drop half a millimetre across.
The drop that is not a tear
A falling raindrop is flattened along the direction it is going, by the pressure of the air passing it rather than by its own weight, and the group that decides is the Weber number. The teardrop of every illustration has the wrong symmetry entirely — there is no up in the problem it is drawn for.
Which speed goes in the number
The most quoted threshold in the subject — air is incompressible below Mach 0.3 — is a five per cent tolerance on the density at a stagnation point wearing a physical boundary's clothes. A wing working for its living is at Mach 0.55 over its shoulder while the free stream is still at 0.3.
The balance that is its own error
Geostrophic balance is licensed by the Rossby number being small, and the fractional error in the geostrophic wind is the Rossby number — exactly, not approximately. So the balance everybody uses at Ro of order one is a hundred per cent wrong, and the same quadratic has a hard limit at a quarter that no anticyclone can pass.
The tracer that is not one
Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.
The threshold the walls decide
A layer heated from below convects at a Rayleigh number of 1707.762, and nothing whatever happens at one. The free–free case has a closed form and the two that do not differ from it by a factor of 2.6 — produced by nothing but what the top and bottom surfaces are permitted to do.
Where the parabola goes
A pipe carrying a steady flow has a parabolic profile, and the Womersley number is supposed to say when an oscillating one still does. For the flow rate it is very nearly honest — one per cent at α = 0.91 — and for the phase it is out by a factor of three, because a lag is second order in the number and an amplitude deficit is fourth.
Five numbers, one name
The Richardson number has a threshold at a quarter, and the quarter belongs to one of the five quantities that go by the name. On a single tanh shear layer they run from J to seventeen J, and the largest of them grows without limit as the measurement is extended further from the layer.
How far before the heat arrives
A duct's thermal entry length is quoted everywhere as x/(D·Re·Pr) = 0.05, with no tolerance attached. Working out what it delivers gives a Nusselt number 1.45 per cent above the developed value — and the developed value itself is not a term ratio at all but an eigenvalue, 3.6568, which is also the rate at which the duct forgets its inlet.
A limit nothing reaches
A dimensional argument that succeeds says a variable has dropped out of the answer. The Blasius profile has no Reynolds number in its shape at any Reynolds number; the overlap layer's power-law exponent is still 0.102 at Re_τ of a million and falls as a logarithm, so the limit exists and nothing ever gets there.
The bed that weighs itself
Blow hard enough through a pile of sand and the pressure drop stops rising. It cannot rise — a control volume round the bed says the drop can never exceed the buoyant weight of the solid in it, and at the velocity where the two meet the bed stops being a structure and starts being a fluid.
The number that really is one
Almost every threshold in this subject sits somewhere other than where its dimensionless group is one. The Froude number does not. At Fr = 1 a disturbance stops being able to travel upstream, the specific energy is least and the equations change type — three statements, one number, and no tolerance anywhere in it.
A solid, if it is not given time
The Deborah number is the only such group with no fluid in it at all — two times and nothing else — and it says a material is a solid or a liquid depending on how long anybody watches. What it throws away is that no real material has one time, and the spectrum it replaces changes the answer in kind rather than in degree.
A transition that needs a second number
Fluid between rotating cylinders goes unstable at a Taylor number of 1707.762 — which is the same number, to every digit, as a layer of fluid heated between two rigid walls. It is not an analogy. And two things the number cannot carry decide whether the transition happens at all and what it looks like when it does.
Sufficient, and not necessary
A stratified shear layer whose Richardson number exceeds a quarter everywhere cannot go unstable. That is a theorem with an exact number in it. What it does not say — and what it is constantly read as saying — is that a layer below a quarter will.
A threshold that is also a duration
The cavitation number treats inception as a pressure: below it the liquid tears, above it does not. A five-micron bubble asked to grow in 0.3 microseconds needs 43 bar of tension and the same bubble given a hundred needs 1.05, because it has to make a journey and not merely respond.
One formula for both ends
Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.
The core that does not move
Toothpaste in a tube has a region in the middle that is not being sheared at all, and its edge is at a radius you can write down before solving anything. Four things about that core are exact, and one of them is that the fluid does not flow below a threshold — not slowly, at all.
The speed where the damping is exactly zero
Flutter is an algebraic condition on a quartic: one root crosses the imaginary axis, at one speed, exactly. And the quantity that locates it is so nearly flat there that the standard way of finding it from flight test overshoots by a quarter.
One channel, one flux, two flows
Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.
The carpet an accelerating aeroplane folds
Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.
The wall the suction puts in the curve
A liquid jet pump's lowest pressure is where the entrained stream enters the throat, and when that reaches vapour pressure the pump curve stops being a curve. The flow ratio freezes at a value no lower discharge pressure can move — and raising the motive pressure, the obvious cure, brings the wall closer.
The heat the eddies do not carry
In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.
A boom is aged in the thin air it starts in
The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.
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.
A cavity that cools the water it came from
The usual account takes the vapour pressure as a property of the liquid, looked up once. It is a property of the liquid at whatever temperature the cavity has left it — and making vapour costs latent heat, so a cavity suppresses itself. In cold water that is unmeasurable and in liquid hydrogen it is the dominant term.
The drop falls at a fold
Tate's law says a drop leaves a tube when its weight equals the tension round the rim. No force balance decides it. A tube holds a family of static drops, the family has a largest member, and the drop falls because there is no static shape with more liquid in it — which Tate's balance overestimates by a quarter on a millimetre tube and underestimates on a wide one.
The hill a slow current will not climb
The Taylor–Proudman theorem says a rotating fluid goes round an obstacle rather than over it, as if a solid column stood above it. Conserving potential vorticity turns that limit into a threshold. A current is stopped over a hill once the hill's height against the depth exceeds a fixed multiple of the Rossby number — 2 for a flat-topped hill, 3.13 for a Gaussian one, 16/3 for a cone — and the fluid it holds sits beside the summit rather than on it.
One fast pump or two slow ones
A pumping station with several identical pumps in parallel has a control that a single pump does not: how many of them are running. With no static lift the answer is all of them, always, slowed together. With lift it is a number that falls with the flow and is almost never whole — the flow times the number of pumps, over the square root of the head the system asks. Staging rounds it, and the right moment to round up comes before the running pumps have run out of speed.
A strained vortex holds until it has no shape to hold
A patch of vorticity in a strain has two steady shapes for every strain below 0.150 of its vorticity, a rounder one and a longer one, and none above. The longer one comes apart at the slightest nudge. The rounder one, computed with disturbances of two, three, four and five lobes, never does: it nods and holds right up to the strain at which it ceases to exist. So the existence limit is the real limit, and past it a vortex is not shattered but stretched — lingering first near the shape it has lost, for a time that grows as the fourth root of how close the strain is to the limit.
Leapfrogging rings end by merging, not by parting
Two smoke rings that leapfrog in an ideal fluid do it for ever, because their energy binds them. A real fluid drains the energy, and the obvious guess is that the rings drift out of the bound state and part. They do not: diffusion drains the pair's energy and the energy of every possible pair of free rings together, and the margin that binds them never goes negative. What viscosity does instead is fatten the cores until, as one ring threads the other, the two are close enough to merge. That takes a number of passes in proportion to the Reynolds number — one to three at the few thousand of a laboratory smoke ring.
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.
The force a contact line holds is a range
Capillary rise and the drop on a window are usually drawn with one contact angle, and a contact line with one angle makes a force that is a single number. A real contact line pins, and stops anywhere between a receding and an advancing angle. The force it holds is then a range, as static friction is, and its width is surface tension times the difference of two cosines. A tube holds its column at any height in the range, so which way the meniscus last moved matters more than how patchy the wall is — and a tilted pane holds a drop only as large as that difference allows.
A sliding drop is held harder the faster it goes
A ridge of liquid on a tilted plate starts to slide when its weight beats the difference between its two contact angles' cosines. Once it moves, the angles move too: the front steepens and the back flattens, by a law set in the viscous corners at each edge. So the resistance rises with speed from exactly the static value, and a sliding drop has no kinetic friction lower than its static one — it stops at the tilt it started at. The back edge's angle falls to nothing at a finite speed, and past that no drop slides with a clean back.
A stratified sea blocks a current sooner and traps less
A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.
Drag decides whether a cloud folds, and the air decides at what
A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.
A crown dissolves its nuclei or breaks on them, and fast
A siphon of degassed water runs until its crown's tension reaches the breaking threshold of its largest nucleus. A nucleus lodged at the crown does not stay the size it arrived: it gains or loses gas by diffusion, and across a micron diffusion takes milliseconds, so the crown gives its verdict almost at once. Every stable nucleus loses gas unless the water holds more than half the crown's tension, and past a knee in gas content, dissolved air takes height from a siphon's crown that no amount of further degassing gives back.
A lattice of vortices folds a cloud at the saddles' quarter
A cloud of heavy particles folds in still air above a Stokes number of one, and in converging air above a quarter. Air that turns as well as converges might have settled between them. In a lattice of vortices it settles exactly on the quarter: the vortices fling the particles to the cell walls, but only the saddles where the walls meet can make them cross. Below the quarter nothing folds, and the cloud is gathered onto the walls without limit instead.
The longest thermal entrance belongs to neither metal nor gas
A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.
A crevice keeps the nucleus a free bubble loses
A free bubble at a siphon's crown dissolves in milliseconds, so it cannot be what breaks a siphon that has run for an hour. A pocket of gas in a crack of the hose wall can be, because its meniscus is curved by the wall and need not dissolve. Followed through its equilibria, the pocket escapes at a tension set almost entirely by the crack's mouth. What the wall's wettability decides is slower and more consequential: whether the crack fills, or keeps drawing gas in until the siphon breaks.
Wobbling saddles keep the quarter, and raise it
Heavy particles in a steady lattice of vortices fold onto the cell walls only above a Stokes number of a quarter, the converging saddles' own threshold. Random flows fold at every Stokes number, so it was natural to suspect the quarter of belonging to steady saddles. Set the lattice wobbling and the particles cross from cell to cell, seventy per cent of them in forty time units, and still none folds below a quarter. The threshold belongs to the strongest strain the flow has; moving the saddles only moves particles out of reach of it, and raises the threshold.
A stronger strain folds a cloud only if it lasts
Heavy particles in a lattice of vortices fold onto the walls above a Stokes number of a quarter of the saddles' strain. Let the strain pulse and the threshold moves — down to the quarter of the peak strain when the pulse is slow, back up to the quarter of the mean when it is fast — because a particle responds to the strain it has been in for about four strain times, not to the strain at the instant. In a random strain the same rule makes the folds arrive in bursts, at the excursions that are strong and long at once.
Two small pumps carry two big discs
Split a pumping duty between two pumps of the same head and speed and each one keeps the big pump's impeller diameter, because the head fixes the tip speed. So the pair spins two full-size discs in their casings for the flow one disc used to carry, and the efficiency penalty that a station comparison assumed to be two points is four. That moves the lift at which splitting saves energy from 44 per cent to 65, and below a specific speed of about a half it never does.
Moody's fifth root is true at one size
A model pump is less efficient than the full-size machine, and since 1925 the gap has been bridged with a fifth root of the size ratio. Built from its losses, a pump's inefficiency scales three ways at once — leakage as the inverse size where a clearance cannot shrink, friction as a fifth root until the passages go rough, and a fixed share not at all — so the exponent runs from a third down to a twentieth, and the fifth root is right only near one size. The same arithmetic says a duty should be split by speed, not by width.
A cloud folds where a saddle is strong, not when
When every saddle of a lattice strengthens and weakens together, a cloud of heavy particles folds in one step, at the strain's first strong excursion. Give each saddle its own random strength and the folding spreads out into a steady rate — and falls, because a particle now has to be at a saddle while that saddle is strong. The rate follows the Gaussian tail of the strain, at about a tenth of the time the strain spends past its fold condition, and a longer-lived strain helps only up to a few strain times.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitDimensionlessMeasurementToleranceInertiaStokes numberEigenvalueSurface tensionCavitationRegimeRelaxation timeDrop