Field

Regimes and numbers

Reynolds, Mach, Froude, Strouhal. One number decides whether a flow creeps, separates or shocks, and the same shape behaves differently at each.

Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

One number decides which physics applies

A bacterium and a whale both swim, and they are not doing the same thing at different sizes. The ratio of inertia to viscosity separates them, and crossing it changes the rules rather than the magnitudes.

The Reynolds number, and the length in it

The most useful number in fluid mechanics has an arbitrary quantity buried in it, and quoting one without saying which length was used makes it meaningless. That detail is where most misuse comes from.

Mach number: one number, four different flows. Mach number is speed ÷ speed of sound. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

When air stops being incompressible

Air is a gas and can obviously be squeezed, yet most of aerodynamics treats its density as fixed. The assumption holds until the flow approaches the speed at which pressure information travels — and then everything changes at once.

The Kelvin wake, and the angle it comes out at. The wake behind something moving over deep water. Each curve joins everything radiated at one moment, wherever it has since travelled to. All of it stays inside a wedge whose half-angle is the inverse sine of one third, about nineteen and a half degrees, and that number does not depend on the speed, on gravity, or on the size of the vessel.

The angle that does not care

Every ship on deep water leaves a wake inside a wedge of the same angle. Not roughly the same — the same, for a rowing boat and a supertanker, at any speed either of them can manage, on any planet with any gravity.

The model that cannot be matched

A scale model behaves like the real thing when its dimensionless numbers agree. With one number that is a matter of choosing the tunnel speed. With two it is usually impossible, and every wind-tunnel result ever published has been obtained in spite of that.

Creeping flow, and the same body with inertia. The exact creeping-flow solution beside a solved field at a Reynolds number where inertia matters. The creeping flow is a mirror image of itself front to back — a photograph of it run backwards is a photograph of it — and the field with inertia has a wake, which is what a direction of time looks like.

The world with no inertia

Drop the viscosity and the equations become exactly solvable and wrong about drag. Drop the inertia instead and they become exactly solvable again — and for a cylinder in an unbounded fluid there is no solution at all, which took fifty years to notice and longer to fix.

Where the flow over the section first reaches Mach one. Two curves running towards each other. The falling one is the section's peak suction, solved exactly in incompressible flow and then deepened by the Prandtl–Glauert factor as the free-stream Mach number rises. The rising one is the pressure coefficient at which the local flow would be exactly sonic. Where they cross is the critical Mach number, and above it there is a pocket of supersonic flow on a subsonic aeroplane.

The pocket on top of the wing

An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.

5 quantities, 3 rows, 2 left over. The dimension matrix for the drag on a sphere: one column per quantity, one row per base dimension, and every entry an exponent. Buckingham's theorem is a statement about this matrix and nothing else — the number of independent dimensionless groups is the number of columns minus the rank, computed here by elimination. Nothing about fluids enters until somebody decides which columns to write down.

Counting what matters

Five quantities decide the drag on a sphere, and the experiment that measures it has one curve in it rather than a five-dimensional table. The reason is a rank: the matrix of dimensions has three independent rows, and what is left over is the number of dimensionless groups the answer can possibly depend on.

Particles at St = 1, against the flow that carries them. Particle paths and the streamlines they were released on, in this site's exact cylinder solution. At small Stokes number the two are indistinguishable and the body catches nothing; as the particles get heavier their paths straighten, cross the streamlines, and begin to strike. The paths are integrated with Stokes drag and nothing else — no gravity, no lift, no effect of the particles on the flow.

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.

The Ohnesorge diagram, with the boundaries where they belong. The classical map of jet break-up: the Ohnesorge number against the jet Reynolds number, with the five nozzles placed on it. The three sloping lines are Reitz's transitions in the gas Weber number, and their geometry is computed rather than sketched — a fixed We_g means Oh·Re is fixed, which is a straight line of slope exactly −1 in these coordinates, and a decade in Reynolds number is checked to move each line by exactly one decade. Where they sit is borrowed; that they are straight and parallel is not. A nozzle below and to the right of the last line atomises.

Where a jet stops being a jet

A tap makes drops a few centimetres down, a garden hose makes a stream that carries, a sprayer makes a mist and a diesel injector makes fog. Same liquid, same mechanism, four regimes — and the number that separates them is not the jet's inertia but the surrounding air's.

A million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

Two slow things make a fast one

Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.

Too fast for a profile

A pipe carrying a steady flow has a parabolic profile. Make the pressure oscillate and one number decides whether it still does — and above about ten the core moves as a plug, a quarter of a cycle behind the pressure, with the fastest fluid in a ring near the wall rather than on the axis.

The other layer, and the one number that separates them

A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.

The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

Where the number says, and where it happens. Fourteen dimensionless groups on one logarithmic axis. The open circle on each row is the value at which the two terms the group compares are equal, which is one by the way the group is formed; the filled mark is the value at which the thing a reader cares about first changes by 1%. The bar between them is the distance the folklore phrase "of order one" hides, and it runs from nothing at all to a factor of 594.

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.

How wrong the ordinary answer already is. The error in a no-slip continuum calculation of the flow through a channel, against the Knudsen number, both logarithmic. The rule at Kn = 1 is where a molecule crosses the whole channel between collisions — the value the number is named for. The error is one per cent at Kn = 1/594, five per cent at 1/114 and ten at 1/54, so the continuum regime of the usual classification, which runs to Kn = 0.01, is a region in which the continuum answer is already six per cent out at its far end.

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.

Quasi-steady stops being true a long way before one. The magnitude of Theodorsen's function, which is the factor a quasi-steady lift calculation is wrong by, and the phase the lift lags the motion. Quasi-steady means C = 1, and the amplitude is already one per cent low at k = 0.0061 and fifteen per cent low at k = 0.1 — a reduced frequency at which nobody hesitates to call a flow quasi-steady. The lag is worse: it reaches a degree at k = 0.003, and a flutter calculation is decided by phase rather than by amplitude.

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.

How big before gravity shows. A drop's height over its width against the Bond number, which is the ratio of its weight to the force its own skin can supply. The number is one where those two are equal, and by then the drop is a bun: it is one per cent from a ball at Bo = 0.0079, five per cent at 0.054 and ten at 0.13. Every one of those is below one, and the first is below it by a factor of a hundred and twenty-six.

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.

A falling drop is a bun, not a tear. The shape of a drop pressed on by the air it is falling through, computed by matching the Legendre component of a sphere's own potential-flow pressure distribution against the change in curvature it produces. The result is oblate — flattened along the direction of travel — because the pressure is high at the poles and low round the equator. The teardrop of every illustration has the wrong symmetry entirely: a drop in free fall has no up, and one at terminal speed is being pressed on from in front.

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.

$Which speed the number is formed on. The fractional change in air density at three places on a body, against the free-stream Mach number. At a stagnation point the density rises, and it reaches five per cent at M = 0.314 — which is where the familiar 0.3 comes from, and it is a five per cent tolerance rather than a physical boundary. At the suction peak the density falls instead, and how fast depends on the body: a lightly loaded section is milder than its own nose, and one working at cp₀ = −2 reaches five per cent at M = 0.22 and is at Mach 0.55 over its shoulder while the free stream is at 0.3.$

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.

$The error in the balance is the number itself. The fractional error in the geostrophic wind, against the Rossby number, on logarithmic axes. It is a straight line of slope one through the origin, and that is not an approximation: keeping the centripetal term gives V_g/V = 1 ± Ro exactly, so the error and the number are the same quantity. The geostrophic wind is one per cent right at Ro = 0.01 and a hundred per cent wrong at Ro = 1, which is the value the number is named for and is quoted as the boundary of the approximation.$

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 relaxation time a particle actually has. Two quantities against the particle-to-fluid density ratio. β = 3ρ_f/(2ρ_p + ρ_f) is three for a bubble, one for a neutrally buoyant particle and nearly zero for anything heavy; it is the factor by which the fluid's own acceleration is felt. The other curve is the true relaxation time over the usual formula's, which is one for a heavy droplet, exactly three halves for a neutrally buoyant tracer, and unbounded for a bubble — the usual formula gives a bubble a relaxation time of zero, and therefore no dynamics at all.

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.

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.

The number a duct settles at is an eigenvalue. The local Nusselt number against x⁺ = x/(D·Re·Pr), from a Crank–Nicolson march that contains no eigenvalue anywhere. It settles at 3.6568, which is λ₀²/2 for the Graetz eigenvalue problem — a completely separate calculation. The mark at x⁺ = 0.05 is the entry length every textbook quotes: it delivers a Nusselt number 1.45 per cent above the developed value, which is a perfectly reasonable tolerance and is never the one stated.

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 that exists and is never reached. The exponent of the best power law fitted across the overlap layer, against the friction Reynolds number. A logarithm is the zero-exponent member of that family, so the log law is what this sequence is heading for — and it heads there as 1/ln Re_τ, which is the slowest useful way of approaching anything. The exponent is still 0.102 at Re_τ = 10⁶, and driving it to a hundredth needs a Reynolds number with a hundred and fourteen in its logarithm.

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 one number that really is one. Three quantities against the Froude number. The upper line is the speed of a surface wave travelling downstream and the lower one the speed of the same wave travelling upstream, both in units of the wave speed itself; the second changes sign at Fr = 1 and not near it. That sign change is not a comparison of two term sizes going through unity — it is the moment a signal stops being able to reach upstream at all, so the equations change from elliptic to hyperbolic and the flow stops knowing what is ahead of it. The specific energy, drawn beneath, has its minimum at the same place, and for the same reason.

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.

Storage and loss moduli, for one relaxation time and for a spectrum. The two moduli of a Maxwell fluid and of a Rouse chain, against frequency in units of the longest relaxation time. One relaxation time makes the storage modulus overtake the loss modulus at λω = 1 and then leave it behind without limit. A spectrum of 1000 modes makes them rise together as the square root of frequency and stay a fixed ratio apart, so the material never becomes the solid the single time predicts.

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.

Three bluff bodies, whose Strouhal numbers agree once the wake's width is used. The measured Strouhal number of each body, and Roshko's universal number formed with the wake's width and the speed on the free streamline that bounds it. The raw numbers span a factor of 1.462; the collapsed ones span 1.0011, with a mean of 0.16281. The shedding was never body-dependent — the length in the number was.

The frequency a wake chooses

Bluff bodies shed at Strouhal numbers from 0.145 to 0.212, and the spread is not a fact about shedding — it is a fact about which length went into the number. Change the length to the wake's own width and three bodies agree to a tenth of a per cent. Then let the body move, and the number stops deciding anything at all.

Drop deformation in simple shear, against the capillary number. The shape model's steady deformation for six viscosity ratios. Every curve is linear in the capillary number at small Ca — which is Taylor's result — and every one of them saturates, at 5/2(2λ+3), because the shear's own rotation turns the drop out of the stretching direction. A drop in simple shear cannot be deformed beyond that however hard it is sheared.

The number that cannot break a drop

The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.

Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Long enough to make a wake

A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

Nu/√Re against the Prandtl number, over eight decades. The whole of the flat plate's heat transfer, as one curve. It is not a power law: it goes as Pr^½ at the bottom, where the thermal layer is far thicker than the viscous one, and as Pr^⅓ at the top, where it is buried inside it. The Pr^⅓ everybody quotes is the upper half. The two asymptotes are drawn beside it, and the low one is √(Pr/π) in closed form.

The number that is an answer

Almost every dimensionless group is a hypothesis: somebody sets the speed, the size and the fluid, and the number licenses a model. The Nusselt number is not. It is what the experiment produces, it sits on the left of the equals sign, and a regime diagram drawn on it is a category error.

The marginal Taylor number against the axial wavenumber. The smallest Taylor number at which a disturbance of a given axial wavenumber is neutral. Its minimum is 1707.757 at a wavenumber of 3.1158, which are the critical Rayleigh number and critical wavenumber of a layer of fluid heated between two rigid walls — the same numbers, because in the narrow-gap limit the two problems are the same sixth-order eigenvalue problem. This curve is computed by that essay's own solver.

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.

Nu/Gr^¼ against the Prandtl number, with the exact solution's points on it. The closed form 0.508 Pr^½(20/21 + Pr)^−¼, over eight decades, with Ostrach's exact similarity values marked. The integral method is two to eight per cent high from Pr = 0.7 upwards and 10 per cent low at Pr = 0.01 — which is where the thermal layer is ten times the momentum layer and giving them one thickness stops being an approximation to anything.

A speed nobody imposed

Every regime number in this collection contains a velocity somebody chose. Natural convection has none: a warm plate makes its own flow, and the Grashof number is what is left when the speed is taken out. The Reynolds number of the result — six thousand, on an ordinary radiator — is an output of the solution rather than a setting on an apparatus.

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.

The convergence exponent depends on the gas, which a dimensional exponent cannot. R ∝ (−t)^α for a converging shock, against the ratio of specific heats, for cylindrical and spherical symmetry. Guderley's exact values are marked and the agreement is to four figures. The Sedov blast's two-fifths is drawn beside them: it is the same for every gas, because it comes from dimensions and a conserved energy, and γ is dimensionless.

An exponent dimensions cannot give

A blast wave's radius goes as the two-fifths power of time, and the two-fifths is arithmetic: count the dimensions and it falls out. A shock converging on a point goes as the 0.717 power, and no amount of counting will produce that number — because it depends on the gas, and γ is dimensionless.

The exact solution and its three approximations, at ε = 0.02. The outer solution is excellent everywhere except in a layer of width ε at the left, where it is wrong by a whole unit. The inner solution is excellent inside that layer and wrong everywhere else. The composite is their sum less the part they agree about, and it is within order ε of the exact solution across the whole interval — which is the entire content of matched asymptotics, drawn.

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.

And they are one piece of arithmetic. All three corrections against one over the logarithm of their own large parameter. Each is a straight line through the origin, with its own slope: one third exactly for the enstrophy range, near one for the wall layer, and near ln 10 for Oseen's per-decade measure. Three disappointments in three different fields, written up separately, are the same function.

The three that never converge

Oseen's drag coefficient, the overlap layer's power-law exponent and the enstrophy range's slope are three separate disappointments in three different fields. They are one piece of arithmetic, and the arithmetic says why none of them will ever be reached.

Fourteen pure numbers, and where each came from. Every one of these is dimensionless, exact and quoted as a fact about fluids. None of them comes from dimensional analysis, which says which numbers an answer may depend on and never what any of them is. They come in four kinds — algebra, an integral, a root and an optimum — and the last two are not equally knowable.

Where a pure number comes from

Sixty-four for a round pipe, sixteen twenty-sevenths for a wind turbine, 0.332 for Blasius. Counting dimensions produces none of them — it produces the list of arguments, and the function has to be solved. Which of the four ways it was solved decides how many digits are worth printing.

Exact in the total, free in the profile

A constraint is one number imposed on a function. Four wakes built to carry exactly the same drag differ by a factor of four and a half in their peak deficit, and the general statement behind that is a question about angles: how much of the wanted answer survives being projected off the constraints, and how much does not.

The same data, in a basis the theorem allows just as much. The identical two hundred and forty points, plotted as F/(mu U d) — which is the drag coefficient times the Reynolds number, a perfectly legitimate pi group forming a complete pair with the abscissa. The result is a straight line of slope 0.9985 through five decades with an r-squared of 0.9985, and it contains no physics: the ordinate contains the abscissa.

The groups are not the only groups

Buckingham's theorem fixes how many dimensionless groups an answer can depend on and says nothing about which. Two of the infinitely many legitimate choices are used here on the same data: one manufactures a straight line through five decades out of a constant, and the other erases Stokes' law completely.

Two forces on one cylinder, a quarter of a cycle apart. The inertia and drag terms of Morison's equation over one wave period, at a Keulegan-Carpenter number of ten. The inertia term follows the acceleration and peaks where the velocity is zero; the drag term follows the velocity and peaks where the acceleration is. They are a quarter of a cycle apart and they are different kinds of quantity.

Two forces, and only one of them remembers

Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are two different kinds of memory, one a function of the present acceleration and one a function of the wake left by the previous half cycle.

The curve that lets in more than one answer. The neutral stability curve: the driving needed to sustain a disturbance of a given wavelength. It has a minimum, so at any driving above the minimum there is a whole band of wavelengths that can grow — and the system has to pick one.

The state a machine was started into

Above the critical driving there is not one flow but a band of them, and a container of a given size can hold several. Which one appears is decided by how the apparatus was brought up to speed, and at twice critical there are eight to choose from.

Captured, and not captured. The amplitude of the wake's oscillation over time, for a forcing inside the capture band and one outside it. Inside, the amplitude settles; outside, it beats at the difference between the two frequencies, because the wake is keeping its own time and the forcing is keeping its.

A wake told what to do

Force a shedding wake near its own frequency and it abandons its own and adopts the forcing's. The band over which it will do so is proportional to how hard it is pushed — and near the edge of the band it takes six times as long to make up its mind.

Slip is a memory of one mean free path

A molecule arriving at a wall last collided about a mean free path away and carries the velocity from there. Averaged over arrivals and departures, that leaves the gas at the wall moving — by two per cent of the centreline speed at a Knudsen number of a hundredth, and sixty per cent more flow through a microchannel at a tenth.

$A particle is a low-pass filter. The fraction of a fluctuating flow's velocity that a particle follows, and the phase by which it lags, against the Stokes number. At one the particle follows 71 per cent of the motion and lags by 45 degrees — which is the corner frequency of a first-order filter, arrived at from mechanics rather than from electronics.$

A particle is a low-pass filter

A seeding particle does not report the flow; it reports the flow through its own transfer function. At a Stokes number of one it follows 71 per cent of the motion and lags it by 45 degrees, and a fifty-micron droplet at a kilohertz is following two per cent of what it is supposed to be measuring.

Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present.

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

$A gust's lift keeps falling where a pitching wing's stops at a half. The magnitude of Sears' function, the lift a wing gets flying through a sinusoidal gust as a fraction of the quasi-steady value, against the reduced frequency on a logarithmic axis, beside Theodorsen's function for a wing that pitches or heaves. The two agree at low frequency. Above a reduced frequency of about a tenth they part: Theodorsen's levels off at one half, because a moving wing changes its whole boundary condition at once, while Sears' keeps falling as one over the square root of 2πk, because several wavelengths of gust lie along the chord and cancel.$

The gusts that cancel along the chord

A wing that pitches keeps half its circulatory lift however fast it moves. A wing flying through a gust does not: once the gust is a few chords long, its ups and downs lie along the chord together and cancel, and the lift falls without limit. For an airliner that barely touches the root-mean-square gust load, and cuts the load spectrum at the wing's own torsion frequency to a quarter of the quasi-steady value.

$A wave that travels and a wave that spreads. A harmonic pressure wave's amplitude and its instantaneous value along a tube, over two wavelengths of the inviscid wave, at four Womersley numbers. At α = 15 the wave marches on, a little weaker each wavelength. At α = 5 it is visibly damped. At α = 2 it is nearly gone within a wavelength. At α = 0.5 there is no wave to speak of: the disturbance falls away within a small fraction of the inviscid wavelength, as heat does into a wall.$

The pulse that has to travel

In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.

One number decides which pulse grows. The pressure pulse and the flow pulse at the far end of the tube, each as a multiple of its value at the entrance, against the load's reflection coefficient. A load that reflects pressure with the same sign — a stiffer or narrower continuation — amplifies the pressure pulse and damps the flow pulse. One that reflects it inverted — a wider continuation, or many branches — does the opposite. With no reflection both fall slightly, by the wave's own attenuation. The two curves cross near Γ = 0 and pull apart on either side.

The pulse that grows as it leaves the heart

The pressure pulse measured at the wrist is larger than the pulse in the aorta that drives it, and the flow pulse is smaller. Nothing downstream is pumping. A wave reflected from the end of an elastic tube arrives back in step with the outgoing wave near the end and out of step near the start, and a single number — the reflection coefficient — decides whether it is the pressure or the flow that grows.

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.

The spreading per metre has a best speed; the spreading per second has none. The plate height — how much variance a slug picks up per unit length travelled — against the velocity, for four retention factors. Each curve falls as one over the velocity, because molecular diffusion along the tube matters when the flow is slow, and rises with it, because that is Taylor's dispersion. The minimum is what a chromatograph is run at, and Taylor's effective diffusivity has no such point anywhere.

The best a tube can do is its own radius

Taylor's dispersion coefficient rises without limit with velocity, so there is no best speed to run a tube at. Ask instead how much a slug spreads per metre travelled and an optimum appears at once — and the spreading there is the tube's radius over the square root of three, with no diffusivity and no velocity left in it.

Transport with nothing transported

Taylor's mechanism needs shear and diffusion and nothing else — and in particular it does not need a mean flow. Oscillate a tube about a fixed position and the tracer still spreads along it, by hundreds of times the molecular rate, which is how a patient is ventilated with a tidal volume smaller than the airway it goes down.

A drop fed into a tube, to scale, up to the largest one it can hold. Six members of the family of static drops a tube of radius half a capillary length holds, drawn to scale and hung from the rim, with their volumes in cubic capillary lengths. The drop grows from a shallow cap through a hemisphere to a bulb with a neck. The last is the fold: no static drop on this tube holds more, and it holds 76 per cent of what Tate's law says the tube can carry.

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.

Four tubes to scale, and the level Jurin's law gives each. Water rising in tubes of radius half a capillary length to four, drawn to one scale, with the level of the liquid far outside the tubes at the bottom. The red line in each tube is Jurin's height, 2ℓ²/R. In the narrow tube the surface is nearly flat and sits on it. In the wide ones the liquid gathers in a rim at the wall, the centre hardly rises, and Jurin's level runs through the middle of a surface that is nowhere near it — while still being its exact average.

A law that is exact as an average

Jurin's law gives the height water climbs in a tube as twice the square of the capillary length over the radius. As a statement about the height it is an approximation for narrow tubes. As a statement about the mean height of everything lifted it is exact for every tube — and a wide tube shows what the average was hiding: a rim of water at the wall and almost nothing in the middle.

The first three ways a drop can ring. A drop's first three modes of oscillation, each drawn at the two ends of its swing: the shaded outline and the red one are half a period apart. The fundamental, l = 2, alternates between a lemon and a lentil. The next two ring at 1.94 and 3.00 times its pitch, with three and four lobes. The swings are drawn at a fifth of the radius so they can be seen; the frequencies belong to swings much smaller than that.

A drop rings like a bell

Disturb a drop and it swings between a lemon and a lentil at a pitch set by its surface tension and its size, and it keeps swinging for dozens of cycles because water is nearly frictionless at that scale. The usual story has a falling raindrop's wake ringing it. The wake strikes several times too fast for the note the drop plays.

A slow current divides round the fluid over a hill (δ = 4). A uniform stream, left to right, over a Gaussian hill on the floor of a rotating layer, at δ = h₀/(H·Ro) = 4, where h₀/H is the hill's height against the depth and Ro the Rossby number U/fa; the onset for this shape is δ = 3.134. The dashed circle is the hill's e-folding radius. Water crossing the hill is squashed and, keeping its potential vorticity, spins clockwise; that anticyclone adds to the stream on one side and opposes it on the other. Here it stops the stream: the outlined region, 1.54 square radii, holds fluid that circles for ever and never leaves, and it sits beside the summit rather than on it, on the side where the swirl runs against the current.

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.

Hull speed is not the hump

The hull-speed rule says a displacement vessel meets a wall where its wave is as long as its hull, at a Froude number of 0.4. Michell's thin-ship integral, computed for a standard hull, puts that speed on the steep climb between the last hollow and the main hump, and the hump itself at 0.5. Past the hump the wave resistance grows more slowly than the speed. The hollows sit where the hull's own transverse wave switches off, and a bulb that cancels at one speed multiplies the resistance at another.

The exponent a pulsing tree wants is between two and three. The tree's reflection at its root, averaged over the pulse's ten harmonics with the pulse's own weights, against the branching exponent k in r_parent^k = 2 r_daughter^k — two is area-preserving, three is Murray's law. With a wave speed that does not change with radius the best exponent is 2.15, just above the inviscid match of two; with one that rises as smaller arteries stiffen, 2.58, just above 2.5. Viscosity in the smallest branches pushes the best exponent a little towards Murray's.

Murray's law is not the rule for a pulse

Murray's law sizes a branching vessel for the cheapest steady flow, and at every junction built to it a pressure pulse is partly reflected. The rule that makes a junction transparent to a pulse is a different exponent, set by how the wave speed changes with radius. A tree is not the sum of its junctions, either: a Murray tree six generations deep reflects a third of the pulse at the heart rate, three times what one of its junctions does, and viscosity in the smallest branches means no area rule can make it transparent at every frequency. The best a pulsing tree can do lies between area-preserving and Murray.

What a tilted pane holds is a difference of two cosines. The largest ridge a tilted plate holds, as a cross-section in square capillary lengths, against the tilt, for clean glass (advancing 30°, receding 10°), a plastic (90°, 70°) and a water-repellent coating (115°, 95°): the force balance (cos θᵣ − cos θₐ) ÷ sin α, and, as points, the areas of drops shot from Young–Laplace, which do not use it. Clean glass holds least, not because water sticks to it less but because on a surface it wets well the two cosines are nearly equal.

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.

Murray's law passes the pulse where the pulse is viscous

The rule that makes an arterial junction transparent to the heart's pulse was an exponent, and real arteries do not split evenly: the aorta sheds side branches a fraction of its size and carries on. For a lopsided junction the transparent rule is still an exponent — the same one, exactly, while viscosity is negligible. With viscosity it is a single function of the parent's Womersley number: close to area-preserving in the aorta, and exactly Murray's cube in arteries under a millimetre, where the pulse moves as the steady flow does. The asymmetry matters only in between, and a lopsided tree reflects far less than an even one.

Past its threshold a ridge slides at a speed its angles set. The steady speed of a ridge of liquid one square capillary length in cross-section, as a capillary number, against the plate's tilt, on three surfaces. Below each surface's threshold it is stuck. Past it the speed rises from zero, linearly at first, as far as the tilt allows. On clean glass the curve barely exists: between its threshold and the speed at which its uphill contact line fails there is a sliver of tilt, and a ridge pushed past that cannot slide steadily with a clean trailing edge.

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.

Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

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.

The hull Michell's integral prefers

Michell's integral gives a thin ship's wave resistance as a quadratic in the hull's offsets, so the hull of least resistance at a given speed and displacement is a quadratic minimisation. Allowed only to reshape its waterline, the integral rediscovers the naval architect's oldest rule: fine ends for a slow ship, full ends for a fast one. Allowed to reshape its depth too, it drains the waterline and piles volume at the keel until the hull is not a ship. And it cannot grow a bulb at the bow, because it cannot tell the bow from the stern.

$With a wake, the bulb belongs at the bow. The Wigley hull's wave resistance with a spherical bulb, as a fraction of the bare hull's, against Froude number, with a wake fraction of 0.25: the bulb just ahead of the bow, and the same bulb just behind the stern. Without a wake the two curves are one. With it they separate: the stern's waves are made by slowed water and are weaker, so the bulb's cancelling wave has less to cancel there, and at the design speed of 0.30 the bow bulb leaves 0.443 of the bare resistance and the stern bulb 0.624.$

The stern's wake puts the bulb at the bow

Michell's thin-ship integral cannot tell a ship's bow from its stern: reverse any hull and its wave resistance is unchanged, so the least-resistance hull is symmetric and a bulb is worth as much at the stern as at the bow. Real ships are fuller aft and put their bulbs forward. Let the stern's waves be made by water the hull's own boundary layer has slowed, and both follow: the optimum hull leans aft, and a bulb at the bow cuts the waves by far more than the same bulb at the stern.

Local transparency is the best tree only when the ends absorb. The root's reflection at the heart rate against the reflection at the tree's leaves, for the map tree, for the single exponent that is best at that leaf reflection, and for Murray's. With leaves that reflect nothing the map tree is the best of all, 0.035 against 0.0419: making every junction transparent is then the whole job. As the leaves start to reflect, the best single exponent pulls ahead, because it leaves its junctions slightly mismatched in the way that cancels the echo coming back from the ends; at a leaf reflection of 0.5 it reflects 0.0735 to the map tree's 0.13.

A tree transparent at every junction is not the quietest

The branching rule that lets a heart's pulse through one junction without reflection is set by that junction's Womersley number: area-preserving in the aorta, Murray's in the small arteries. Build a whole arterial tree that way, every junction at its own transparent rule, and it reflects more of the pulse than the best tree built to a single exponent. The single exponent wins by leaving its junctions slightly mismatched, in the way that cancels the echo from the tree's own ends — the trick an antireflection coating plays on light.

The entrance grows with the Péclet number, peaks, and then shrinks. The distance from a wall-temperature step at which the local Nusselt number has come within 5 per cent of its developed value, in diameters, against the Péclet number, for turbulent pipes at Reτ = 500, 1000, 2000 and 5000, with the slug's laminar line L/D = 0.029·Pe. The liquid metals sit on the line; the entrance peaks at 11.6 diameters at Pe ≈ 4700 for Reτ = 1000 and falls to about three and a half diameters for water.

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.

$A wake held at the stern moves the volume aft a third as far. The least-resistance hull's centre of volume, in per cent of the half-length from midships, negative aft, against the design Froude number, holding the Wigley hull's length, draught, depth profile and displacement. With the linear wake it sits 2.8 per cent aft at Fr 0.30; with the shaped wake of the same propeller-disc fraction, 0.86 if uniform in depth, 1.5 if deepest at the waterline, and 0.11 if deepest at the keel, where the sources make the fewest waves.$

A wake held at the stern keeps the bulb and loses the lean

A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.

Viscosity slows every ripple, and the short ones most. The growth rate of a varicose ripple on a liquid jet against its wavenumber times the jet's radius, at Ohnesorge numbers of 0, 0.1, 1 and 10, in units of the capillary time. Every ripple longer than the circumference grows. Viscosity damps each one in proportion to the square of its wavenumber, so the short ones lose most: the fastest moves from ka = 0.697 inviscid to 0.344 at Oh = 1 and 0.123 at Oh = 10, and its rate falls from 0.343 to 0.0114.

Viscosity lets a jet break, later and into bigger drops

A thread of honey falls for metres before it breaks, and a thread of water for centimetres, which suggests that viscosity holds a jet together. It does not: it cannot stop any ripple longer than the jet's circumference from growing. It slows them, the short ones most, so the ripple that wins is longer, it takes the viscous time rather than the capillary one to win, and each drop it makes is bigger. The wavelength grows as the square root of the Ohnesorge number and the drop as its sixth root.

The air pulls on the crests, and short ripples start to grow. The growth rate of a ripple on an inviscid jet against its wavenumber times the radius, at gas Weber numbers of 0, 0.4, 2, 6 and 13 on the diameter. The air flowing over a rippled jet is faster over the crests and its pressure lower there, which pulls them further out. Without it nothing shorter than the circumference grows; at a gas Weber number of 2 ripples up to ka = 1.45 grow, at 13 up to 6.19, and the fastest moves with them.

The air shortens a jet's fastest ripple, and the drops follow

A jet in a vacuum breaks into drops nearly twice its own width, whatever its speed. A jet in air does not, and the reason is a pressure the air puts on its surface: flowing over a rippled jet it is faster over the crests and its pressure lower there, which pulls them further out. That pull lets ripples shorter than the jet's circumference grow, shortens the fastest one, shrinks the drops, and puts a ceiling on how long a fast jet can be — at the gas Weber number where the measured break-up regimes change.

A resistive wall lengthens a slug's thermal entrance by half, and no more. The 5 per cent thermal entrance length of a slug flow, as a multiple of its length with the wall held at a fixed temperature, against the wall Biot number hw·D/kf. A wall that conducts well changes nothing; as the wall's resistance grows the entrance lengthens to a peak of 1.501 times at a Biot number of 2.29, and then shortens slightly to 1.457 times, the fixed-flux value, for a wall that conducts hardly at all.

A steel wall lengthens a sodium entrance by half

A liquid metal's thermal entrance is long because its heat crosses the pipe by conduction, and a steel tube conducts a tenth as well as sodium, so the wall is part of the path. Put it in series with the fluid and the entrance does grow — but by half at the very most, at a wall Biot number near two, which is where sodium in a steel tube happens to sit. A wall that resists more changes the boundary condition from a fixed temperature to a fixed flux, not the length.

Between fixed ends the transparent-junction tree is the quiet one. The magnitude of the pulse reflected at the aortic root at the heart rate, 1.2 Hz, and its first four harmonics, for trees grown from the same aorta to the same 0.5 mm vessels by four branching rules. The Womersley-map tree reflects 0.028 at the heart rate and 0.0046 or less at the harmonics; the best single exponent 0.035; Murray's 0.083; and the area-preserving tree 0.21 at the heart rate though little at the harmonics.

Between fixed ends the transparent tree is the quiet one

Grown from one aorta for six generations, an arterial tree whose every junction passes the pulse reflected more than one built to a single compromise exponent. Grown instead from the same aorta down to the same small vessels, however many generations each rule needs, the order turns over: the junction-by-junction tree is the quietest, and Murray's tree, which looked cheapest for the heart, becomes the dearest. The earlier ranking was a comparison between trees that fed different beds.

Each collapsed form of the plate's heat transfer reaches its limit decades away from air. The exact local Nusselt number of a vertical isothermal plate in its two collapsed forms against the Prandtl number: over (Gr·Pr²)^¼, which tends to Le Fevre's 0.600 as Pr → 0, and over (Gr·Pr)^¼, which tends to 0.503 as Pr → ∞. At Pr = 10⁻⁵ the first reads 0.5994; at 10⁵ the second reads 0.5023. The two limit laws cross at Pr = 0.494, where the exact value is 0.736 of either; air, at 0.72, reads 0.421 and 0.387.

Over a plate in mercury the warm layer flows as if inviscid

The exact solution for a warm vertical plate has been tabulated since 1953 and is usually shot from the wall, which fails in a liquid metal unless the shot starts within a thousandth of the answer. Posed instead as one algebraic system, it solves from a Prandtl number of a hundred-thousandth to a hundred thousand — and shows that in mercury the layer's speed is set without viscosity, that the integral method runs low there rather than high, and that air sits where neither limiting law is within a fifth.

A flat plate's layer is pushed along by the pressure its own warmth leaves, strongest at the wall. Across the layer of a horizontal plate warm side up, in air: the velocity scaled to its peak, the temperature excess, and the along-plate push −∂p′/∂x scaled to its wall value, against the similarity distance η = (y/x)·Gr^⅕. The push is the hydrostatic deficit under the warm fluid, (2/5)(ηθ − P), 0.694 at the wall in units of gβΔT·Gr^−⅕, falling to nothing at the layer's edge; the velocity peaks at η = 1.58 and the warm layer ends near 5.93.

Laid flat, a warm plate is pushed by the pressure of its own layer

A warm vertical plate drives its layer with buoyancy acting along it. Lay the plate flat and buoyancy acts across the layer instead, so the only push along it is the pressure deficit its own warm fluid leaves under the cold — deeper where the layer is thicker. That push gives a heat transfer growing as the fifth root of the Grashof number, not the fourth, works only on the plate's upper face, and survives a tilt of a degree for a third of a metre.

Of the four common plumes, only the laminar one above a line speeds up as it rises. Centreline speed against height for four plumes, each scaled to its own speed at one reference height: laminar above a line source, rising as the fifth root of height; laminar above a point source and turbulent above a line, both constant; and turbulent above a point, falling as height to the minus one third. Each exponent follows from the heat flux being fixed and the width growing by diffusion, √(νx/u), when laminar, or in proportion to height when turbulent.

Above a warm wire the plume speeds up as it rises

A plume above a line of heat has no wall, no imposed speed and one conserved quantity, the heat it carries. That alone fixes a centreline speed growing as the fifth root of height — the only one of the four common plumes that accelerates — a temperature excess falling as the three-fifths power, and a warm core that is narrower than the moving sheet around it in any fluid more viscous than five ninths of its own diffusivity. At that Prandtl number the equations have a closed form, and the solver lands on it to three parts in a million.

Below separation Morison's drag coefficient is a Stokes layer's, and it grows without limit as the motion shrinks. Wang's drag coefficient for a circular cylinder in an attached oscillatory flow against the Keulegan–Carpenter number, for β of 10², 10³, 10⁴ and 10⁵, each drawn up to Hall's onset of the Honji instability. β = 10²: 2.77 at KC = 1, onset at KC 1.83; β = 10³: 0.845 at KC = 1, onset at KC 1.03; β = 10⁴: 0.264 at KC = 1, onset at KC 0.578; β = 10⁵: 0.0831 at KC = 1, onset at KC 0.325. At fixed β the coefficient falls as 1/KC.

Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force

Morison's equation splits the force on a cylinder in an oscillating flow into inertia and drag, and the Keulegan–Carpenter number says which dominates — once the flow separates and leaves a wake. Below that, nothing separates, and the drag term describes the thin oscillating boundary layer on the cylinder's surface. Its coefficient then grows without limit as the motion shrinks, as one over the Keulegan–Carpenter number, and that exactly cancels the number's power to decide anything: the drag's share of the force is fixed by the frequency parameter alone, a few per cent at most, until Honji's instability ends the regime.

Freed by its inertia, a settling particle falls through a lattice of vortices several times faster than through still air. The mean settling speed of a cloud released over one cell, over its speed in still air, against the Stokes number, for still-air speeds of 0.1, 0.3 and 0.6 of the vortices' speed. 0.1: 4.82 at St = 0.02, 4.39 at 1 and 1.86 at 10; 0.3: 2.36 at St = 0.02, 1.22 at 1 and 0.605 at 10; 0.6: 1.43 at St = 0.02, 0.984 at 1 and 0.959 at 10. Without inertia most of the cloud never falls at all.

Inertia empties the vortices that would hold a settling particle, and then speeds its fall

A particle that settles slower than a lattice of vortices turns can be carried round a cell for ever: without inertia, its path is a closed curve, and Stommel showed in 1949 that half a cell can be such a trap. Give the particle the slightest inertia and every trap leaks, in a time that grows as one over its Stokes number; the freed particle is flung to the cells' edges, rides their downdraughts, and falls on average several times faster than it would through still air. More inertia undoes some of that, and a narrow band of it can hold a slow particle up indefinitely on an orbit of its own.

Between plates the flow rate falls with rarefaction to a minimum near one free path across, then climbs without limit. The reduced flow rate G of a gas driven by a pressure gradient between two plates, against the rarefaction parameter δ = √π/(2Kn), from the linearised BGK equation; beside it Navier–Stokes with no slip (δ/6), with Maxwell's slip (δ/6 + 1.016) and the uncollided flux of molecules straight from a wall. The least flow rate is 1.538 at δ = 1.07, Kn = 0.827; at δ = 0.01 the solve gives 3.049, at δ = 0.001 4.449 and at δ = 10 2.769.

A thinning gas flows least at one free path across, and then more without end

Pump the gas out of a narrow channel and, for the same pressure gradient, less of it flows — until the channel is about one mean free path high. Thinner than that, more flows, and between infinite plates the flow rate climbs without limit, because the molecules carrying it are the ones skimming along the walls, whose flights only collisions with other molecules can end. Knudsen measured the minimum in 1909. The linearised kinetic equation puts it at a reduced flow rate of 1.538, at a Knudsen number of 0.83, and shows it is not a dip below the two closed forms but a place far above both.

Shaken at the right frequency, a liquid with one relaxation time moves sixteen hundred times the flow its viscosity allows. The amplitude of the oscillating flow rate over the steady Poiseuille flow the same pressure-gradient amplitude drives through a Newtonian liquid of the same zero-shear viscosity, against the frequency, in a tube 1 cm in radius. A Maxwell liquid with λ = 1 s and η₀ = 30 Pa·s peaks at 1660 times at 6.63 Hz and again at 317 times near 15.2 Hz; a Rouse spectrum with the same longest time and viscosity reaches 20.2 at 19.8 Hz; the Newtonian liquid only falls, to 0.99 at 30 Hz.

A liquid with one relaxation time rings in a tube, and a spectrum of times silences it

Push a viscoelastic liquid back and forth along a rigid tube and, if its relaxation time is long against the cycle, the tube becomes a cavity for shear waves in the liquid itself. At the cavity's resonance a wormlike-micelle solution in a tube one centimetre across moves sixteen hundred times the flow a Newtonian liquid of the same viscosity would. The Deborah number says when the liquid is elastic enough to ring; it cannot say whether it will, because a polymer with a spectrum of relaxation times and the same longest one damps the same resonance to a factor of twenty.

$A flicker of a few per cent empties the suspension band and drains Stommel's traps more slowly. The share of a cloud released over one cell that is still within four cells of its start after the run, against the flicker's standard deviation as a fraction of the cells' mean strength, with a memory of three turnovers: inertialess particles falling at 0.3 of the cells' speed in still air, held in Stommel's traps, and particles of Stokes number 3 falling at 0.1, held in the suspension band. Stommel's traps hold 0.52, 0.41, 0.33, 0.16, 0.03, 0; the suspension band 0.22, 0.055, 0.015, 0.015, 0.005, 0, at flickers of 0, 0.05, 0.1, 0.2, 0.35, 0.5.$

Flickering vortices cannot hold a settling particle up, and still make it fall faster

A steady lattice of vortices can do two things to a heavy particle settling through it: hold it up, on closed paths or on orbits of its own, and speed its fall, by flinging it into the down-draughts. Let the vortices strengthen and fade at random, as real eddies do, and the two come apart. A flicker of five per cent of the cells' strength empties three quarters of the band of particles held up by their inertia; a flicker of a third drains Stommel's traps almost entirely. The speed-up hardly notices a flicker of a tenth, and is halved only when the cells swing by half their strength and remember it for a few turnovers. Trapping needs a steady cell; the speed-up does not.

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