Field

Flows and fields

Streamlines, particle paths and the field that carries them. What is conserved, what a picture of a flow can show, and what it cannot.

Streamlines and pathlines are not the same curve. In an unsteady flow the line tangent to the velocity everywhere at one instant, and the track a single particle actually follows, are different curves. They coincide only when the flow is steady, which is the hypothesis most figures forget to state.

Streamlines are not the paths particles take

Three different curves get drawn through a flow and they are routinely treated as one. In steady flow they coincide, which is why the confusion survives; in unsteady flow they are as different as a photograph and a long exposure.

A streamtube narrows and the flow speeds up. Two neighbouring streamlines bound a tube that no fluid crosses. Where the tube pinches, the same mass has to pass through a smaller gap every second, so it must move faster — which is mass conservation with no equations in sight.

Mass has nowhere to go

Squeeze a stream of fluid and it speeds up, not because anything pushes it but because the same amount has to get through a smaller gap every second. Almost every result in the subject is that observation with more machinery attached.

The velocity field, arrows to scale. The same flow drawn as arrows. Scaled to the local speed the picture is honest and crowded; drawn all the same length it is legible and hides the very variation the figure is about.

What a flow is

A fluid is made of molecules and nobody models it that way. Treating it as a continuous field with a velocity at every point is an approximation, an extremely good one, and knowing why it works is knowing where it stops.

The acceleration field of a steady flow. How hard the fluid is being accelerated at each point of a steady flow past a cylinder. The flow does not change with time anywhere in this picture, and yet almost nowhere in it is a parcel travelling at constant velocity — the pattern stands still while the fluid running through it is thrown about.

Steady does not mean nothing is happening

Photograph the flow past a cylinder twice and the two pictures are identical. Every parcel of air in them is being thrown about — braked to a dead stop, hauled round the shoulder at nearly twice the free-stream speed, braked again. Both statements are exactly true.

Spin is not the same as going round

A whirlpool whose streamlines are perfect circles can have no rotation in it anywhere. A flow whose streamlines are dead straight and parallel can be rotating everywhere. Both statements are true, and getting them the wrong way round is the most expensive confusion in the subject.

The number on a streamline is a flow rate

Streamlines get drawn as decoration — curves the flow follows, spaced however the plotting looked best. Each one carries a number, the difference between two of those numbers is the volume of fluid passing between them every second, and it does not matter what route the measurement takes.

Circulation is vorticity, added up

One of these two quantities is measured by walking round a loop and one by summing over the area inside it, and a theorem says they are the same number. That reconciles the site's most confusable pair — and explains how a flow with circulation can have no spin in it anywhere.

A circle of fluid, 0.90 of a gradient time later. A material circle in a uniform velocity gradient, carried by the exact matrix exponential of that gradient. It becomes an ellipse — always an ellipse, for every gradient — and the axes it stretches along are the eigenvectors of the symmetric half. In an incompressible flow the area is unchanged however extreme the distortion, which is the statement that stretching in one direction is squashing in the other.

What a parcel does in the first instant

Drop a circle of dye into a flow and it becomes an ellipse. The velocity gradient that did it splits into a stretch and a spin in exactly one way, the split is not a convention, and one half of it is the reason a fluid has any stress in it at all.

The spin that feeds itself

Stretch a vortex tube and its spin rises in exact proportion, because the circulation round it cannot change and its area has fallen. Nothing in that argument sets a limit — and the one flow where the limit can be written down exactly puts it at a length of √(4ν/α).

The same patch, 6 periods later, in two flows. A round patch of 848 marked particles, advanced 6 periods by the blinking flow and by a steady flow of the same strength. The steady flow has drawn the patch into a smooth ribbon along a streamline and every particle in it is still on the streamline it started on; the blinking flow has folded the patch through itself repeatedly and its particles are spread across the whole region. Neither flow has any diffusion in it and neither has lost a particle. The difference between them is that one depends on time.

No randomness, and it mixes anyway

A steady two-dimensional flow cannot mix, however fast it is stirred, because its trajectories are its streamlines. Switch two vortices on and off alternately and the same fluid, obeying an exact map with nothing random in it, folds a patch of dye through itself until neighbouring particles separate by a factor of a thousand in six periods.

Two centres, two saddles, and a sum of nothing. A separation pattern: a uniform stream with two counter-rotating cored vortices in it, which reproduces the arrangement of critical points behind a body at a Reynolds number of a few tens. There are exactly four — a saddle where the flow divides, a centre in each recirculating cell, and a saddle where it closes — and their indices sum to 0. The winding number of a loop enclosing all of them is 0, which is what a uniform stream far away requires. A bubble costs nothing in this bookkeeping, which is why one is free to appear.

The count a pattern cannot break

A picture of a flow has stagnation points in it, and they are not free to be arranged as anybody likes. Their kinds and their number obey an integer constraint that has nothing to do with the equations of motion — and an incompressible flow in a plane is allowed only two kinds of them in the first place.

One more condition, and the price of it. The model problem ε u″ + u′ = 0 with a condition at each end, at three values of ε. The outer solution is the flat line at one — that is the whole of the answer when ε is zero, and it is a first-order equation that can meet one condition, so it meets the one at the far end and misses the one at the wall by the whole range. Restoring ε restores the second condition and pays for it with a layer of thickness ε, inside which the gradient is of order 1/ε. The product of those two — which is what a stress is — does not depend on ε at all: it is 1.0000 at every value tried, to nine decimal places. Drag does not vanish as viscosity does. It converges.

How many things a flow must be told

The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.

The wall makes vorticity at a rate with no viscosity in it. At a stationary wall the momentum equation collapses to ν ∂²u/∂y² = (1/ρ) ∂p/∂x, and the left-hand side is the diffusive flux of vorticity out of the surface. So the pressure gradient along the wall is the vorticity source, and the viscosity that made the no-slip condition necessary has cancelled out of what the condition produces. The curve is that flux across the Falkner–Skan family, computed from profiles solved by shooting and differenced at the wall; the straight line is the pressure gradient each of those flows has. They agree to 2.3e-14. At zero pressure gradient the flux is exactly zero: a flat plate creates no vorticity at all after its leading edge, and everything in its layer arrived from there.

Where vorticity comes from

Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.

A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

The mean is not the flow

Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.

Four turns of a wave, and the parcel is not back. Two parcels traced through four periods of a linear deep-water wave of steepness 0.1, by integrating the exact velocity field. Each orbit is very nearly a closed circle and misses closing by a little, every time, in the same direction — that miss is the whole of the Stokes drift. The near-surface parcel advances 0.0357 of a wavelength over the four cycles and the one a tenth of a wavelength down advances 0.0122, a third as far — because the drift falls off twice as fast with depth as the orbit's own size does.

The drift in a wave that has none

The velocity at any fixed point under a passing wave averages to exactly zero, and every parcel of water in it moves steadily forward anyway. The orbits do not close, they miss by the same amount every time, and the missing amount is the square of the steepness times the wave speed.

A wall that is not quite there

A porous surface has structure on every scale below the pore, and no calculation resolves it. The whole of it can be replaced by one length — the square root of the permeability — and the replacement is exact to first order, with what it leaves out identifiable as the flow the wall itself carries.

One flow, two observers, two pictures. The same ideal flow past a circular cylinder, drawn in the frame of the tunnel and in the frame of the undisturbed air. The two are related by subtracting one constant velocity. On the left the flow arrives from infinity, divides at a stagnation point on the nose and closes at another on the tail. On the right the air is at rest far away, the body pushes through it, the streamlines are closed loops, and there is no stagnation point anywhere in the field. Every force, every pressure and every measurement either observer can make is identical.

The picture belongs to whoever is watching

Photograph the flow past a cylinder from the tunnel and it has two stagnation points. Photograph the same flow from a frame moving with the air and it has none at all, and its surface speed is exactly the free stream at every angle. Both pictures are correct and no measurement distinguishes them.

One field, and the two parts the theorem splits it into. A velocity field made of a smooth source, a smooth vortex and a uniform stream, and the two fields the Helmholtz decomposition returns for it. The first carries the whole divergence and has no curl anywhere; the second carries the whole curl and has no divergence. They are computed by solving two Poisson problems on a grid, with the divergence and the vorticity differenced from the field rather than taken from the expressions that built it. Adding the two back together does not recover the field.

Every flow is two flows

Any velocity field splits into a part carrying all of the divergence and a part carrying all of the vorticity. The theorem says so and does not say which split — the two halves can be moved between each other by anything harmonic, and on a bounded region that is an infinite family.

A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

Where a vortex stops

Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies four per cent of a window to one observer and more than half of it to another.

A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Incompressible is not a property of the fluid

A flow whose density varies three to one across it, with a divergence of 10⁻¹⁰ everywhere. And a flow of air at Mach 0.1, whose divergence is three per cent of U/a and which every textbook calls incompressible. The word is about what the flow does to a parcel's volume, and about nothing else.

Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

The chord and the tangent, which are the two speeds. The flux of a conserved quantity against its own density, for a wide river and for traffic. At any point the slope of the chord from the origin is the speed the material moves at, and the slope of the tangent is the speed a disturbance moves at. They are the same number only if the curve is a straight line through the origin. For the river the tangent is five-thirds of the chord at every depth; for traffic the tangent turns negative above half the jam density while the chord never does.

A wave nothing in it travels with

A flood crest moves at five-thirds the speed of the water it is made of, at every depth, whatever the roughness and whatever the slope. A traffic wave moves backwards through cars that are all going forwards. Neither result contains a momentum equation.

One streamline, sectioned, in two steady flows. Every time a single streamline crosses the plane z ≡ 0 going upwards, a point is plotted. On the left the flow is integrable and the points lie on a curve, however long the trajectory is run. On the right one coefficient of the same exact solution has been changed and the same single streamline scatters over a sixth of the plane. Both flows are steady, both are incompressible to machine precision, and both are exact solutions of the Euler equations.

Steady, three-dimensional, and mixing anyway

A steady flow that solves the Euler equations exactly, with its vorticity equal to its velocity to six parts in ten thousand million — and one of its streamlines wanders through a sixth of the box while another, started nearby, lies on a curve for ever.

A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

The knot a flow cannot untie

Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.

$One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.$

The stretching rate that is not one number

A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

A patch of dye, folded. A circle of marker particles carried by the flow, at four times. It is stretched into a filament and folded through itself, and its area does not change at any point of that — which is not visible in the picture, and is the whole difficulty. A scheme that lost eight per cent an orbit would produce a picture indistinguishable from this one.

The area that must not move

A patch of dye in an incompressible two-dimensional flow keeps exactly the area it started with, for ever. Two respectable integrators are put on the same flow: one respects that identically at any step size, the other does not, and the pictures they draw are the same picture.

And the net drift, which is where they disagree. The Stokes drift plus each return flow. All three drift forward at the surface, because the Stokes drift there swamps any return current of the right size. Below that they part company completely: the uniform current has most of the column moving upstream, and the two that satisfy no slip have almost none of it. The reversal depths span 77 per cent of the water column.

The drift a closed box will not allow

A wave in a wave tank carries mass forward, and the tank has nowhere to put it. So a return current appears carrying exactly the opposite transport — exactly, from mass conservation and nothing else. Which fixes a total and leaves the answer anybody wants entirely open.

A uniform scalar in a fluid at rest, under two face rules. Nothing is flowing and the scalar starts at one everywhere. The swept-volume rule leaves it at one to the last bit, at every step of the two time units. The midpoint rule moves it by two parts in ten thousand, on a mesh motion that begins and ends in the same place, and the excursion looks exactly like a physical transient.

The mesh that makes its own mass

The transport theorem holds for a region moving at any velocity, which is what makes a moving-mesh calculation possible. Discretised carelessly it is not an identity but an approximation, and a fluid at rest with a uniform density then gains density from the motion of a grid — smoothly, plausibly, and looking exactly like a physical transient.

What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

One pair of strainings, two orders, two lengths. The stretch of the most-stretched material direction against time, for a simple shear followed by a pure strain and for the same two in the other order. The two curves are identical until the swap and separate afterwards, ending a factor of 2.16 apart.

Two strainings, and the order they came in

A material line is stretched by a shear and then by a pure strain, and then by the same two in the other order. Every instantaneous measure of how hard the fluid was being worked is identical in the two cases. The lengths at the end differ by a factor of 2.16.

Five points, and where each one's fluid came from. Back-trajectories through six units of time in an unsteady double gyre. Each curve ends at the place the fluid now at the marked point started; nothing about that place can be read off the velocity at the marker.

A scalar is a record of where its fluid was

A conserved scalar has no value of its own. Its value at a point is whatever it was at the place that point's fluid started from, which makes a dye field a photograph of the past — and makes the map from now to then the only thing in the flow that carries the past at all.

A drift made of two things that average to zero

Stokes drift is usually explained as a parcel spending longer in the forward half of its orbit. That is true and it is not a formula. The formula is a correlation between a displacement and a gradient, each of which averages to exactly nothing, and it splits into two halves that are equal to twelve figures.

The stretching a window of history did, drawn as a field. The largest finite-time Lyapunov exponent over eight units of time in the double gyre, darkest where two neighbouring parcels were pulled furthest apart. The bright crest is a curve across the domain, and it is a property of the eight units rather than of any instant inside them.

A boundary that only exists over a window

The curve that separates fluid going one way from fluid going another is not in any snapshot of the flow. It is the crest of a field built from a stretch of history, it moves when the stretch is changed, and reversing the direction of time gives a different curve entirely — both of them real.

What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

A duct that does not change, and a parcel that does. A contraction of area ratio four, with the parcel marked at five equal intervals of time. The walls do not move, the field at every point is the same at every instant, and the spacing of the markers grows because the parcel is carrying its own history through the duct.

How long the fluid has been in there

Age is the simplest thing a flow can remember. It obeys the shortest transport equation in the subject — its material derivative is one — and no instrument pointed at a steady flow can read it, because a steady flow's every field is constant and its fluid is getting older all the time.

4 nodes and 2 saddles, and the difference is the shape. The skin-friction pattern on a body, drawn as the streaks a film of oil would leave, looking at the nose. Every critical point of the field is marked: 4 nodes, where the streaks converge or diverge, and 2 saddles, where two streaks cross. Their indices sum to 2, which is the Euler characteristic of a sphere, and nothing about the flow enters that number — the stream is at 0 and the crossflow at 0, and neither may change it. What the flow decides is where the points are and what they look like; what the body decides is how many there are of each.

The count computed on a body

The rule for a closed surface is usually quoted and seldom solved for. This solves for one, and what the computation adds is not confirmation — it is the discovery that the count survives two events in which the number of stagnation points falls, and that both of them have closed forms.

Four kinds of critical point, and the curve that separates them. The invariants of a trace-free velocity gradient, with the discriminant curve 27R²/4 + Q³ = 0 drawn through them. Inside the two upper lobes the cubic has one real root and a complex pair, which is a spiral being stretched along its own axis on the left and squeezed on the right; below the curve all three roots are real and the point is a node with two saddle directions. Of 820 random incompressible gradients, 509 land in the spiral region and 311 in the real one. A plane flow is the vertical line R = 0 and nothing else, which is why a plane has two kinds and space has four. The marked points are the cases the calculation checks that fall inside this window; the two vortex cases it also checks sit at Q = 3.25 and |R| = 4.25, off the top corners, because a window wide enough to hold them would flatten the curve the figure is about.

Two kinds is a plane flow's privilege

A plane incompressible flow has a saddle or a centre and nothing else, and the proof is one line about a trace. The same line in three dimensions constrains three numbers instead of two, which is far less, and what it leaves is four kinds of point separated by a curve — with the one a plane cannot have being the structure the whole of turbulence is made from.

The line the dye actually draws

The streakline is the curve most photographs really show, and of the three curves it is the hardest to compute, because it needs the whole history of the flow. This draws it, in a flow where all four curves have closed forms, and the third curve turns out to be a different kind of object from the other two rather than a third example of the same one.

The curve that measures a gradient

Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.

In clean water a bubble rises nearly three times as fast as the same bubble in tap water. The terminal rise speed of an air bubble in water at 20 °C against its radius, from buoyancy balanced against drag: with a clean, shear-free surface using Moore's law, and with a surface immobilised by contamination using the rigid-sphere correlation. At 0.3 mm the clean bubble rises at 13.0 cm/s against 6.7; at 0.5 mm at 31.0 against 11.2, a factor of 2.76. Beyond a radius of 0.47 mm the clean bubble's Weber number passes one, its shape flattens, and a spherical calculation stops describing it; that region is shaded. Nothing about the bubble's size, gas or liquid changes between the two curves — only whether its surface can move.

The vorticity a clean surface cannot refuse

A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.

Twice as slippery along as across

A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.

A wall puts in exactly its own speed

A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.

The sign a stagnation point carries in space

In three dimensions a stagnation point's index is the sign of one determinant, and that determinant is minus the R of the invariant diagram — so the diagram's left and right halves are the two indices. An exact Euler flow in a periodic box has eight such points, four of each sign, never a spiral among them, and they can only disappear in pairs that meet on the one line where the sign is allowed to change.

A float under swell on a rotating planet goes round instead of away. The track of a float at the surface over 1 inertial periods (16.9 hours) after a 8 s swell of amplitude 1 m arrives at latitude 45°, in kilometres, the waves travelling to the right. Without friction the float runs round a circle of radius Uₛ/f = 0.479 km and comes back to where it started every 16.92 hours. With a drag on the Eulerian current it spirals out into a steady drift veered to the right: 24.3 per cent of the drift at 76.0° for a drag of 0.25 f; 70.7 per cent of the drift at 45.0° for a drag of 1 f; 94.9 per cent of the drift at 18.4° for a drag of 3 f. In a non-rotating ocean the same float would have gone 3.0 km straight on.

The drift a rotating planet takes back

In a wave tank the Stokes drift is cancelled by a return current because the tank has walls. The open ocean has none, and the drift is cancelled anyway: the Coriolis force acts on the water's real motion, drives an Eulerian current that answers it, and leaves the depth-integrated transport exactly zero at every viscosity. A float under steady swell with nothing to stop it goes round a circle instead of away.

The column in height and time: a falling interface, a rising shock, a fan. A batch settling test from a uniform φ₀ = 0.1, height above the bottom against time, both scaled on the column height and the single-particle settling time. The interface with clear water (thick) falls in a straight line at 0.4538; the sediment shock rises from the bottom at 0.1484 until the two meet at t = 1.661, height 0.2464; the thin lines are characteristics of the fan, each carrying one concentration between 0.317 and packing, and the interface bends as it crosses them. Dots are the finite-volume solve on 400 cells: the interface and the sediment front.

The column the chord rule cannot settle

A suspension settling in a closed column is a kinematic wave, and its flux curve bends both ways. At the top the chord rule works: clear water meets the suspension at a single falling front. At the bottom it does not, and the bed grows behind a shock that stops short of packing and a graded layer beneath it — so the interface, instead of arriving, slows for ever.

In the frame of the wave the walls stand still, and a bolus rides between them. Streamlines of a peristaltic channel of amplitude ratio 0.7 over two wavelengths, drawn in the frame moving with the wave, where the flow is steady and the walls are themselves streamlines. The time-mean flow is Θ = 0.5904 of the wave speed times the mean half-width, so the flow rate between centreline and wall in this frame is q = −0.4096 and the pressure rise per wavelength is 0.000 in units of μcλ/a². The centreline velocity changes sign at 0.106π and 0.894π, and the streamline through those points closes round a bolus holding 30.5 per cent of the fluid in each wavelength, which travels with the wave.

A wave on the wall is a pump

A channel whose wall only moves in and out, in a wave travelling along it, delivers a steady net flow with no part of the wall moving along the channel. In the frame of the wave the walls stand still and are streamlines, so continuity alone fixes how the laboratory flow rate follows the wall shape — and the momentum equation is needed only for one number, which also decides whether fluid rides along with the wave or leaks back against it.

Rolls turning side by side, with the fastest downwind water where they sink. The fastest-growing mode at a Langmuir number of 0.13, looking downwind, over two roll spacings of 2.89 decay depths and 4 decay depths down. The closed curves are streamlines of the overturning; the dashed curves are contours of the downwind velocity the rolls carry, positive under the lines where the water sinks. At the surface the cross-wind flow converges onto those lines, which is where floating foam and weed collect as windrows. The amplitude is arbitrary, as in any linear mode.

The drift that turns a current into rolls

A current carrying a Stokes drift feels a force the drift makes out of the current's own vorticity, and under a wind that force is unstable. It turns the surface layer into rolls lined up downwind, with windrows where they sink. The rolls need both the current's shear and the drift's; their growth rate sees only the product; and the split between the two decides which motion gets the energy.

The kept transport spirals into nothing as the sea deepens. The net Lagrangian transport as a vector, scaled on the Stokes transport, traced as the water depth increases from a quarter of an Ekman depth to eight, for an 8-second swell with an eddy viscosity of 0.01 m²/s. Shallow water keeps the whole transport pointing with the waves, at the right-hand end. As the sea deepens the vector shortens and swings to the right, crosses the across-wave axis near two Ekman depths, and winds into the origin, which is the open ocean's exact cancellation.

The floor that gives the drift back

In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.

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

The pump that is better the more it squeezes

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

The line that decides how much a thickener can pass. The batch settling flux against concentration, the same curve the closed column was solved on, with the operating line of a continuous thickener drawn across it. The line runs from the underflow concentration on the axis to the flux the plant can take, and it is placed by one condition: it must touch the flux curve rather than cut it. Where it touches is the concentration at which the thickener is working hardest.

The tangent that sizes a plant

A closed cylinder of settling suspension is a kinematic wave problem with a flux curve. Open the cylinder — feed it at the top and draw thickened slurry from the floor — and the capacity of the whole plant is a straight line laid against that same curve, touching it rather than cutting it.

Three speeds, and the Froude number at which two of them cross. The kinematic wave speed and the two dynamic wave speeds, all divided by the speed of a shallow-water wave on still water, against the Froude number. The kinematic wave is three halves of the water speed and the downstream dynamic wave is the water speed plus one, so they cross at Froude two — and beyond that crossing the news of a change in flux arrives before the news of a change in depth, which uniform flow cannot survive.

When the flux outruns the pressure

A kinematic wave is derived by throwing the momentum equation away, and the derivation never says when that is allowed. It is allowed until the kinematic wave, which travels faster than the water, overtakes the downstream dynamic wave as well — at which point uniform flow ceases to exist and a concrete chute carries a train of surges instead of a sheet.

Every gradient in one plane, and where each one goes. The two invariants of a trace-free velocity gradient, R across and Q up. Under the restricted Euler equation the combination 27R²/4 + Q³ never changes, so every trajectory is one of these curves, and R never decreases, so every curve is travelled from left to right. Vieillefosse's line, where the combination is zero, has two branches: the left one runs into the origin and is the only way to reach it, and the right one is where every other trajectory ends, at infinity, in finite time.

A gradient left to itself

Follow a parcel's velocity gradient with nothing acting on it but its own square and the part of the pressure a single parcel can know about. Every starting gradient but a set of measure zero reaches infinity in a finite time, and it gets there as a sheet with its spin lying in the sheet.

The gas flows in, against the heat. The radial velocity of the gas, in units of α/a, at the same four times — from the motion of the gas shells themselves — with the heat flux, reversed in sign and in the same units, drawn as dots. The two coincide: at constant pressure the gas moves at (γ − 1)/(γp) times the heat flux, against it. Heat leaves the spot outward and the gas comes in, everywhere.

The air flows in against the heat

A gas moving at a ten-thousandth of the speed of sound is as incompressible as anything gets, by the usual arithmetic. Heat one spot of it and let the spot cool: the gas changes its volume threefold, and it flows towards the hot spot while the heat flows out, at a speed fixed by the heat flux alone. At constant pressure, a joule is a volume.

The unburnt gas bends round a wrinkle, and the bending makes it grow. Streamlines of the unburnt gas flowing up towards a flame with a small sinusoidal wrinkle, one wavelength across, for a density ratio of seven, with the wrinkle exaggerated. The expansion behind the flame pushes back on the gas ahead of it where the flame bulges forward, so the streamlines spread there and the gas arrives slower, and they crowd together where the flame lags, so the gas arrives faster. A flame that burns into the gas at a fixed speed therefore advances further where it was already ahead.

A flat flame is unstable at every size

A flame expands the gas it burns, and the expansion pushes back on the fresh gas ahead. Where the flame bulges forward the fresh gas slows, so the bulge burns further forward; where it lags, the gas speeds up and it falls further back. Every wrinkle grows, the shorter ones faster, and what finally gives a real flame a size is how its burning speed responds to its own curvature.

Every particle leaves a flow that is a vortex at every instant. Eight fluid particles starting on a small circle in Haller's rotating-saddle flow, followed for 2.4 time units. At every instant the velocity gradient has complex eigenvalues and a positive Q — the flow is a vortex by every criterion that reads a snapshot — and every particle spirals outward, its distance growing as e to the time. The fluid is not held; it is flung.

Every snapshot says vortex, and every particle leaves

There is a flow in which the velocity gradient has complex eigenvalues at every point and every instant — a vortex by every criterion that reads a snapshot, in the frame the flow is measured in — and in which every fluid particle is flung away exponentially. The snapshot is not wrong about the gradient. It is wrong about the fluid, and it is wrong exactly when the strain's axes turn.

In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

Remembered pressure turns the runaway into a teardrop. The velocity gradient's two invariants, Q against R, both scaled by the mean enstrophy, for an ensemble of parcels whose pressure remembers one Kolmogorov time of deformation, at a memory of a tenth of a large-eddy time. The restricted Euler equation sends every parcel off to infinity along the right-hand branch of Vieillefosse's curve, dashed. With the remembered pressure the parcels stay: they crowd along that branch in the strain-dominated quadrant and above the axis on the left where vortices are being stretched — the teardrop measured in turbulence.

The pressure a parcel remembers keeps it finite

The restricted Euler equation follows a parcel's velocity gradient with the pressure's shape thrown away, and every gradient it follows blows up. Give the parcel back a pressure that remembers how its neighbourhood was deformed over the last Kolmogorov time — nothing more — and no gradient blows up at all. The ensemble settles into the teardrop measured in turbulence, its vorticity lines up with the middle strain axis, and its intermittency grows as the memory shortens. What the memory cannot do is keep the one identity homogeneity demands, and that miss says where the rest of the pressure lives.

A quiet flame has one cusp; a noisy one keeps making more. Two flame fronts twenty neutral wavelengths wide, burning upwards, drawn apart for clarity: above, the quiet front, the exact pole solution with one cusp per period and the most poles it can hold; below, the same flame with a noise of a millionth kicking its growing wavelengths. The noise keeps seeding small wrinkles on the smooth arcs; each grows as it is swept along the arc into the cusp, so the noisy front carries a train of sub-cusps the quiet one never has, and advances at 0.862 against the quiet 0.5.

A flame's speed limit holds only in silence

A quiet wrinkled flame settles into a single cusp and a speed it never exceeds, however wide it grows. Add noise and the limit is gone. A disturbance of one part in a million million already speeds the front up; one in a million makes it a third faster; and a noisy flame, unlike a quiet one, keeps getting faster as it gets wider, because every added width is room for more wrinkles to be born on its arcs and swept into its cusp. The quiet flame's speed is a property of a silence no burner has.

Squeezed past closing, the tube is held open by its own fluid. The channel's half-width along one wavelength of a travelling squeeze that would push an empty tube's wall 1.3 half-widths inward — past the centreline — and the wall the squeeze and the fluid's pressure make together, for three compliances. The empty squeeze would overlap the centreline across a sixth of the wavelength; with fluid in the tube, the pressure that builds ahead of the narrowest point pushes the wall back, and the gap stays open at a hundredth to a tenth of the half-width.

A squeezed tube is held open by its own fluid

A peristaltic pump with a prescribed wall wave gets better the more its wave closes the channel, and at closure it becomes a piston. A real one squeezes an elastic wall and lets the fluid push back. Then the channel never closes: the pressure ahead of the narrowest point holds a gap open that grows as the square root of the wall's give, squeezing harder past closure pumps less, and the pump behaves as a displacement pump until a pressure of about one over its compliance blows the throat open — with an efficiency that stops short of one by an amount that grows with the wall's give.

Every parcel goes straight while the streamlines wave. A uniform stream across the page carrying a pattern of transverse velocity with it, v = 0.6 sin(x − t): the streamlines at one instant, which wave, and the paths of five parcels over the next six time units, which are straight lines. Each parcel keeps the transverse velocity it started with, because the pattern moves with it; the streamlines are a snapshot of a pattern that is sliding past, and no parcel ever follows one.

A parcel goes straight while the streamlines curve

In a steady flow every parcel accelerates while the picture never changes. The opposite also happens: a flow whose picture changes all the time while no parcel accelerates at all, because the local and convective halves of the acceleration cancel exactly. Such a flow has no pressure gradient anywhere, and that is a severe demand. An incompressible flow can meet it only as a pure shear at every point; a cloud that can compress meets it freely and pays later, when its straight paths cross in a caustic.

Moving air needs a quarter of the inertia still air does. The first crossing against Stokes number for a cloud thrown into still air — time in units of the cloud's fastest convergence rate — and for a cloud carried by air converging on a line, in units of the air's strain rate. Still air only brakes: its threshold is one. Converging air keeps pushing: its threshold is a quarter, and heavy particles in it cross in about one strain time. Dots: particles integrated in the converging sine flow; the curve is the linearised oscillator.

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.

$At a coast the swell's water goes back, or goes along. The Lagrangian mean current — the Stokes drift plus the Eulerian current — under an 8-second swell arriving straight at a coast, in seas 8, 25 and 60 metres deep, each as a fraction of its own surface drift against depth as a fraction of the water's. Left, the part across the shore, positive onshore; right, the part along it. The cross-shore parts carry no net water. At 8 metres, inside the Ekman layer, the whole exchange is across the shore; at 25 the return is shared with a current along the coast; at 60 the open shelf's own rotation has already taken the transport back, and the coast has little left to do.$

A coast sends the drift back, or sends it along

A shelf sea under a swell keeps part of the Stokes drift's transport, turned to the right by the rotating Earth, and near a coast some of it points at the land. The sea answers with a slope in its surface. In water shallower than an Ekman depth the slope drives the water straight back as an undertow, as it would without rotation; a few Ekman depths down the coast turns the transport into a current along the shore instead, and there the sea can slope down towards the land it faces.

$Posts slip as one over the root of their fraction. The slip length of a surface of posts, over the period of their square array, against the fraction of the surface that is solid, beside Philip's stripes along and across and the dilute limit in which each post is a lone disc dragged edgewise, (3/16)√(π/φ). At a hundredth solid the posts slip 2.87 periods against the stripes' 1.32; at a tenth 0.597 against 0.591, nearly equal; at a half 0.0663 against 0.11 along the stripes and 0.0552 across them.$

A post holds liquid back by its radius, a stripe by its length

A surface of no-slip posts standing in a gas-filled texture lets liquid slip over it, and the slip length grows as one over the square root of the solid fraction, where a surface of stripes grows only as its logarithm. The reason is the oldest contrast in slow flow: a disc dragged through a viscous liquid has a drag proportional to its radius, which is Stokes' law, and a line has no finite drag at all, which is Stokes' paradox. But sparse posts beat stripes only while they are sparse.

Below a quarter the cloud settles on the cell walls; above it, it crosses them. Paths of particles released with the air's velocity inside one cell of the vortex lattice, at Stokes numbers of 0.15 and 0.4, over 25 time units; the cell's walls, the separatrices joining the saddles, are drawn. At 0.15 the particles are flung outwards by the rotation and approach the walls ever more closely without crossing, so the cloud is compressed onto lines. At 0.4 they reach a wall with enough speed to overshoot it into the next cell, where they meet particles coming the other way: the cloud folds.

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 current along the shore runs fastest three Ekman depths down. The depth-averaged Eulerian current along the shore, positive to the left looking shoreward, against the local depth in Ekman depths, for a swell arriving 20° off the shore-normal at the shelf edge and for one arriving head-on. Both make a jet to the right, fastest at 2.9 Ekman depths for the oblique swell (-1.42 mm/s at 37.6 m) and 2.6 for the head-on one (-0.948 mm/s), and a weak current the other way in water shallower than one Ekman depth. Rotation turns the transport the shelf returns, so even a swell with no alongshore component drives one.

A sloping shelf puts the swell's current three Ekman depths down

A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.

Moving the saddles never lowers the quarter; fast large wobbles raise it. The smallest Stokes number at which a cloud of particles in the oscillating lattice folds within sixty time units, against the oscillation's amplitude ε, at frequencies of 1, 3 and 10 times the saddles' strain rate. With ε = 0 it is the steady lattice's 0.251. Moving the lattice never takes it below a quarter; at ε = 1 it is 0.283, 0.311 and 0.324 at the three frequencies.

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 swept wall makes vorticity along its isobars, not across its flow

A still wall in a pressure gradient puts vorticity into the fluid at a rate set by the gradient, and the source is a vector: it lies in the wall, along the isobars. On an unswept body the isobars run across the flow and so does the vorticity, which is the ordinary boundary layer. On a swept one the isobars run along the span and the flow does not, so part of every new vortex line points along the flow — half of it, over the accelerating front of a cylinder swept 35°. That part is where a swept wing's crossflow comes from.

A stronger strain folds particles below a quarter only if it lasts. The Stokes number above which particles in a lattice with pulsing strain 1 + a sin ωt fold onto its walls, against the pulse's frequency, for amplitudes of 0.15, 0.3 and 0.5, with the steady lattice's quarter dashed. A slow pulse folds them at the peak strain's quarter, 1/(4(1 + a)): 0.194 at a = 0.3 and ω = 0.05, against 0.192. A fast one folds them only at the mean's: 0.249 at ω = 5.

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.

A nose at incidence makes its vorticity along the flow

A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.

Out to a hundred neutral wavelengths the noisy flame is still speeding up. The front's speed in excess of a flat flame's against its width, on logarithmic axes, at noise strengths of a thousandth and a millionth, with the quiet pole front's and a power fitted from 20 to 100. At a thousandth the speed climbs from 1.05 at 12.5 to 2.71 at 100; at a millionth from 0.668 to 1.71. Both follow the 0.42 and 0.43 power of the width to within 4 per cent, and neither bends over; the quiet front stays at a half.

A noisy flame is still speeding up a hundred wavelengths wide

A quiet wrinkled flame gains a fixed, bounded amount of speed over a flat one, however wide it is. A noisy flame keeps getting faster as it widens, and the open question was whether that ever stops. Out to a hundred neutral wavelengths it does not. The speed grows as about the 0.43 power of the width at two noise strengths a thousandfold apart. The noise sets only the prefactor, and the front's slope grows with the speed until it leaves the range the equation was derived for — which is where the growth stops being the equation's to decide.

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.

$Pressed into their grooves, the menisci cost a striped surface at most a quarter of its slip before it floods. The slip length along the stripes as a share of the flat meniscus's, against the liquid's pressure over the gas's as aΔp/γ, for gas fractions of 0.5, 0.8 and 0.95; the groove floods when the arc meets a vertical wall at its advancing angle — 110°, 120° and 150° at pressures 0.342, 0.5 and 0.866. 0.5: 0.93, 0.903 and 0.84 kept at those floods; 0.8: 0.913, 0.879 and 0.804 kept at those floods; 0.95: 0.888, 0.845 and 0.752 kept at those floods.$

A bowed meniscus keeps most of its slip until the groove floods

The slip lengths of a striped superhydrophobic surface are computed with flat gas surfaces. Press the liquid against the texture and every meniscus bows into its groove as an arc pinned at the solid's edges, and the slip falls. It falls by less than the obvious worry says: at the pressure where a groove wall advancing at 120° lets the liquid in, a surface four-fifths gas still keeps 88 per cent of its slip, and even at a 150° wall's flood three-quarters remains. A striped surface does not lose its slip gradually under pressure. It keeps most of it, and then loses all of it when the groove floods.

The two observers disagree twelvefold about where Q says the vortices are, and not at all about the LAVD edge. The Q = 0 contours and the outermost convex LAVD level over one orbit, in the laboratory and in the frame turning with the pair. Q > 0 covers 2.00 of the window in the laboratory and 24.94 in the turning frame; the LAVD edges enclose 6.29 in the laboratory and 6.29 in the turning frame, and the two LAVD fields differ nowhere by more than 8.0e-8 of their peak.

An edge no observer can move still depends on what it is averaged against

The Q-criterion puts a co-rotating vortex pair in four per cent of a window for one observer and in more than half of it for another. The cure is to average, along each fluid path, how far the vorticity departs from its mean — a number no observer's rotation can change — and take the outermost convex level of it round each core. Computed, it works: both observers find two discs of radius 1.00, twice the core radius, and no fluid inside them strays from its core in an orbit. But the edge has a setting of its own. The mean must be taken over some region, and as that region grows from a radius of 4 to 20 the edge moves from 0.90 to 1.13.

A gas's viscosity ignores its pressure until the free path reaches the walls

Maxwell predicted in 1860 that a gas's viscosity does not depend on its density — halve the pressure and half as many molecules carry momentum across the flow, each twice as far — and measured it with swinging discs in 1866: unchanged from an atmosphere to a sixtieth of one. Solved across every pressure with the kinetic equation, the shear a gas transmits between two plates obeys him over five decades and then, once the free path reaches the gap, falls in proportion to the pressure. A millimetre gap reads half the true viscosity at fifteen pascals, and a vacuum gauge reads pressure from exactly that fall.

A stagnation point can leave the flow through the wall, and the wall's point changes kind to pay for it. The heights of the interior stagnation points on the axis of a swirling stagnation flow onto a plate, against the skin friction's spreading rate σ: the lower point, of index +1, and the upper, of index −1; along the wall, the skin-friction focus, a separation of index +1 where σ < 0 and an attachment of index −1 where σ > 0. As σ falls to zero the lower point descends into the wall as z ≈ σ and the wall's focus turns from attachment to separation; as σ rises to one half the two interior points meet at a height of one and vanish. Interior plus half the wall is −½ throughout.

A stagnation point can leave the flow through the wall, and the count is kept by halves

A no-slip wall is a sheet of zeros, so the index count that holds for a flow in open space has nothing to count there. Divide the velocity by the height and the wall's zeros shrink to the skin friction's nodes, foci and saddles; reflect the result through the wall and each of them is counted once while every point in the fluid is counted twice. That is why the points on a wall are halves, and an exact swirling flow onto a plate shows the halves doing their job: a stagnation point in the fluid can disappear without a partner by sinking into the wall, where the skin friction's focus changes from attachment to separation to pay for it. A section through the flow sees none of this, because in the section every one of the points is a saddle.

A stagnation point hangs above the skin friction's saddle while the saddle is a separation, and is gone once it is not. Streamlines in the plane y = 0 of a wall flow with no axis of symmetry, at μ = 0.4: the wall's saddle of skin friction at x = -0.632, its node at 0.632. An interior stagnation point sits at height 0.15 above x = −0.5, and the saddle is a separation: the flow leaves the wall there.

A stagnation point off the axis can land only on a saddle or a node, and turns it inside out

In a swirling flow onto a plate every stagnation point lived on the axis, so one sinking to the wall always met the wall's single critical point. Without the symmetry it might seem free to land anywhere — on an ordinary stretch of skin-friction line, leaving a pair of new points behind. It cannot. Divided by the height, the velocity is continuous down to the wall, where it becomes the skin friction, so a point can only reach the wall where the skin friction already vanishes. It lands on a saddle or a node, and the count does the rest: the point it lands on turns from separation to attachment. An oil-flow picture of the wall, which shows the same pattern before and after, cannot tell which it is looking at.

Surges merge down the chute until about four hundred time units, and then the train stops merging. The number of surges on a periodic chute 200 units long against time, after uniform flow at Froude number 3 is seeded with depth noise, at two resolutions, eddy viscosity 0.05. At the finer the count is 71, 55, 45, 45, 45 at times 100, 200, 400, 700 and 1000 — a spacing of 4.44; at the coarser, 63, 45, 32, 30, 30, a spacing of 6.67.

Roll waves stop merging at a spacing their fronts choose

On a steep chute the uniform flow breaks into a train of surges, and the train coarsens: a taller surge runs faster, catches the one ahead and swallows it, so the spacing grows down the chute. If the merging never stopped, the spacing in a photograph would only record how far the water had come. Marched down a long channel from noise, it stops. After a few hundred units of travel the surges stop catching each other and the train keeps its spacing for as long as it is followed. The spacing it keeps is not set by the chute or the noise but by the dissipation inside the surge fronts — which is the one part of the flow the shallow-water equations describe only roughly.

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