Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Stokes flow — where it appears

Flow in which the inertia terms are negligible against the viscous ones, so the equations of motion become linear and instantaneous. It is exactly solvable, reversible in time, and the regime of almost everything smaller than a millimetre.

Named by 22 essays across 3 fields — each of them below, with the objects they name alongside it.

The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted.

The price of a gradient

Viscosity does not charge for motion. It charges for the rate at which a parcel is being deformed, and a fluid in solid-body rotation at any speed whatever destroys nothing at all. What is charged for is a sum of squares, which is why the bill can be computed twice.

viscous · Dissipation How far downstream the heat is still being made. The dissipation accumulated from a station ahead of a cylinder to a station behind it, as a fraction of the whole of what is made inside the frame, at five Reynolds numbers. At Reynolds number 1 the fluid has finished paying by about a diameter behind the body. At 100 it has not finished at five, and the curve is still climbing at the edge of the picture — the drag is a force on the body, and the heat it stands for is somewhere else.

Where the heat of a drag is made

The power it takes to tow a body through a fluid becomes heat, all of it, eventually. None of the interesting words in that sentence are the first four. It is the "eventually" that decides how a wake behaves, how far a disturbance reaches, and why no box drawn round a body contains its own bill.

viscous · Dissipation The parabola is the cheapest shape the walls allow. The dissipation of the profile u = A(1 − |y/h|ⁿ) carrying a fixed flux between fixed walls, against the exponent n, divided by the parabola's. Every member of the family satisfies the same boundary conditions and carries the same fluid; they differ only in shape. The minimum is at n = 2 exactly, which is not a coincidence — it is Helmholtz's theorem, and the parabola is a solution of the equations because it is the least dissipative shape rather than the other way round.

The cheapest shape the walls allow

The parabola in a pipe is usually presented as what the equations give. It is better understood the other way round — of every profile that could carry that flow between those walls, it is the one that destroys the least energy, and the equations give it for that reason.

viscous · Minimum-dissipation A sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

A force without the flow that makes it

A sphere carried along by a flow does not travel at the speed of the fluid at its centre. It travels at the average of the flow over its own surface — and getting that result needs no solution of the flow around the sphere at all, only the answer to a completely different problem that everybody already knows.

viscous · Reciprocity The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

The exact theory, drawn by viscosity

Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

inviscid · Hele-shaw Where Einstein's line stops being the measurement. The viscosity of a suspension of rigid spheres relative to the liquid's, against the volume fraction. Einstein's 1 + 5φ/2 is exact for one sphere and holds while the spheres cannot feel one another, which is up to about five per cent by volume. Batchelor and Green's two-sphere term takes it a little further; beyond about a fifth nothing derived works and the curve drawn is a fit, which diverges at a maximum packing that is itself a measurement.

A viscosity made of particles

Stir rigid spheres into a liquid and the mixture is thicker, by five halves of the volume fraction. Einstein's coefficient is not an empirical constant — it is a dissipation calculation on one sphere — and doing it as an energy rather than as a stress shows that four-fifths of it comes from somewhere nobody mentions.

viscous · Suspension A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

A swimmer that cannot go backwards

Taylor's waving sheet is the simplest self-propelled object in a viscous fluid, and its arithmetic contains a result that reads like a mistake — the work it does to travel a metre does not depend on how big its waves are. Doubling the amplitude quadruples both the speed and the power, and changes the bill for the journey by nothing at all.

viscous · Swimming The only candidate the far field allows, and the wall it slips past. The general Stokes solution has four constants; the condition at infinity kills two of them and fixes a third, leaving one to satisfy two conditions at the wall. Setting the stream function to zero there uses it up, and the tangential velocity that remains is exactly twice the free stream — for every radius, every speed, and every fluid.

The flow with no solution

Creeping flow past a sphere has a solution and everybody knows it. Creeping flow past a cylinder has none — not a difficult one, not one needing a clever method. The equations, the no-slip condition and the uniform stream at infinity are inconsistent, and the residual is exactly twice the free stream.

viscous · Stokes' paradox A particle released in still fluid, with the history and without. The velocity of a hundred-micron particle released at one metre a second in air, computed with the unsteady Stokes force and with the quasi-steady drag alone. After five particle time constants the one that remembers is still moving nearly four times as fast.

The drag that integrates a whole history

A particle released in still air does not stop exponentially. The unsteady drag on it carries a term that is an integral over everything the particle has already done, weighted by the inverse square root of how long ago — so there is no time constant, and five particle time constants later it is still moving four times faster than the quasi-steady answer allows.

viscous · Diffusion Slip follows the stripes more closely than the shear does. Plan views of a striped surface, with the stripes running across each panel, for a shear at 0°, 30°, 54.7° and 90° to them. The faint arrow is the direction of the shear; the dark one is the slip velocity it produces, whose component along the stripes is the along-stripe slip length times the shear and whose component across them is half that. The slip is turned towards the stripes by 0.0°, 13.9°, 19.5° and 0.0°. It is largest, 19.47°, for a shear at 54.74°, where tan θ = √2. A surface with a tensor for a boundary condition can push a flow sideways, which a scalar slip length never can.

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.

kinematics · Boundary conditions Three strokes, and only the flat one is a theorem. Three cycles drawn in the swimmer's shape space: a square, a circle of the same width, and an out-and-back along the diagonal. The first two enclose area and carry the swimmer forward; the third encloses none and carries it exactly nowhere, which is the scallop theorem with no symmetry argument in it. What the third lacks is not a broken symmetry but an interior.

A stroke is worth the area it encloses

The usual account of swimming without inertia is a symmetry argument about reciprocal strokes, which says what cannot work and nothing about what does. Draw the stroke in the space of the swimmer's own shapes and the displacement is a line integral — so it is an area, it does not depend on how fast the stroke is played, and the scallop theorem is Stokes' theorem.

viscous · Swimming The only term that turns spin into thrust, and what it is made of. The coupling term of the propulsion matrix against the drag anisotropy, with the geometry held fixed. It is exactly proportional to the difference of the two drag coefficients, so it is zero when they are equal — not small, zero — and no helix of any pitch turned at any rate would move. A real filament sits at 1.66, which is a third of the way from useless to the unreachable limit.

Two drags, or nothing swims

A bacterium turns a corkscrew and goes forward, and the reason is not the corkscrew. It is that a thin filament dragged broadside resists more than the same filament dragged end-on. Make the two resistances equal and the thrust is not small but exactly zero, for every pitch and every rate.

viscous · Swimming Every velocity guesses low and every stress guesses high. Two bounds on the flow rate of a duct, in units of G a⁴/μ, against the number of mesh cells per half-side: from above, the best stress field in equilibrium with the pressure gradient (red); from below, the best velocity that vanishes on the walls (green). For the square the series value 0.562308 (grey) lies between them at every mesh. For the L-shaped duct, which has no closed form, they close on 0.21399 to 0.21415.

A flow pinned between two guesses

The minimum-dissipation principle says the true flow is the cheapest one the walls allow, so any guessed velocity carries too little. It has a twin that nobody teaches: any guessed stress in balance with the pressure carries too much, and it needs no wall condition at all. Between the two, the flow through a duct with no formula is pinned down to as many figures as anyone wants.

viscous · Minimum-dissipation The orbits a rod's axis can be on, seen along the vorticity. The tip of the unit vector along a rod of aspect ratio five, tumbling in a simple shear, seen from along the vorticity axis, for five values of the orbit constant. Every orbit is closed and every one takes the same time. A rod near the centre is spinning about the vorticity axis; a rod on the outer circle tumbles end over end in the plane of shear. The flow never moves a rod from one orbit to another.

A viscosity the flow cannot decide

Spheres stirred into a liquid thicken it by a definite amount. Rods do not. A rod in a shear flow tumbles round a closed orbit, the flow never moves it to another, and the extra viscosity depends on which orbit it is on. The equations of slow flow permit a whole range of values and choose none of them. The smallest amount of noise chooses one, and it does not matter how small.

viscous · Suspension 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.

kinematics · Boundary conditions The drag follows the surfactant load, and a surface pressure of μU pays nearly all of it. How far the drag has climbed from the clean bubble's to the rigid sphere's, against the mean surfactant load over the whole bubble as a surface pressure in units of μU. A load of 0.2 μU makes a 60° cap and a third of the climb; 0.56 a 90° cap and 70 per cent of it; 1 a 120° cap and 94 per cent. The dashed line is the cap's share of the surface for the same caps: the drag runs ahead of the area covered.

A thousandth of a monolayer holds a bubble still

A clean bubble rising slowly through water feels two-thirds of a rigid sphere's drag, and real bubbles almost never do, because surfactant swept to the rear holds the surface still over a cap there. Solving the flow with the cap in it shows how little that takes. The drag runs ahead of the area covered — half-way to rigid with a third of the surface held — and the surfactant needed is set by the viscous stress, not by the surface tension. For a bubble a tenth of a millimetre across, a thousandth of a monolayer, spread as a cap, makes it rise within a few per cent of a solid ball.

viscous · Mobile interface A weak push stirs the whole fluid; a strong one makes a jet. Streamlines of the exact solution in a plane through the force, for a weak force (jet Reynolds number 1, left) and a strong one (100, right), the force pointing right from the origin. The dashed line is the cone inside which the fluid moves outwards: 89.4° from the axis for the weak force, nearly a hemisphere, and 22.8° for the strong one. Outside it the fluid is drawn back towards the origin from every direction and turned into the jet.

A point force makes a jet only when it is strong

Push on a fluid at one point and there is an exact solution of the full Navier–Stokes equations for what follows, at any strength. Weak, it is Stokes's point force: fluid pushed forward over a whole hemisphere and drawn in behind. Strong, it is Schlichting's slender jet, with fluid drawn in from every direction outside a narrowing cone. One constant joins them, and it says how strong a push must be before the boundary-layer jet everyone uses is right — at the axis by a jet Reynolds number of a hundred, at the edges only by a thousand.

viscous · Free shear Below 9.8 µm in silicone oil, a gradient of a degree per millimetre outruns gravity. The speed of a clean air bubble in 10 cSt silicone oil against its radius, on logarithmic axes: migrating in a gradient of 1 K/mm, which grows in proportion to the radius, and rising under its weight, which grows with its square. They cross at 9.81 µm and 31 µm/s. The crossing radius, 3|dσ/dT|G/(2ρg), has no viscosity in it: both speeds are set against the same viscous drag.

The warm side pulls a bubble with no force on it

A bubble in a liquid whose temperature varies from place to place moves towards the warm side, with no gravity and nothing pushing it. The surface tension is lower where the liquid is warmer, the surface is pulled towards the cold pole, and the bubble goes the other way. Young, Goldstein and Block's speed comes out of four interface conditions and one more: that the total force on the bubble is zero. That one condition removes the point force a sinking or rising body always carries, so the bubble's disturbance dies a hundred times faster with distance, and the radius at which it balances its own buoyancy has no viscosity in it.

viscous · Mobile interface The same cap costs a rising bubble a quarter of its speed and a migrating one four-fifths. Speed as a fraction of the clean bubble's against the half-angle of a stagnant cap over the rear: for a bubble migrating in a temperature gradient, and for the same bubble rising under its weight. A 90° cap leaves the rising bubble 74 per cent of its speed and the migrating one 21 per cent; a 120° cap, 68 and 4. The rising bubble can only fall to a rigid sphere's two-thirds; the migrating one falls to nothing.

A cap that slows a rising bubble stops a migrating one

Surfactant swept to the back of a bubble holds the surface still there, and for a bubble rising under its weight the worst this can do is turn it into a solid sphere and cost it a third of its speed. A bubble migrating in a temperature gradient has no such floor. Its own surface is the engine, and a cap removes the engine as well as raising the drag. A cap over the cold hemisphere leaves a rising bubble three-quarters of its speed and a migrating one a fifth; a cap of 120° leaves the migrating bubble four per cent. The surfactant it takes is set by the tension difference that drives the migration, and that is tiny.

viscous · Mobile interface 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.

kinematics · Boundary conditions Two spheres settle faster together than apart, faster in line than side by side, and still faster at thirty radii than either alone. The settling speed of a pair of equal spheres over a single sphere's, against the distance between their centres in radii: along the line of centres, exactly (the bispherical solution) and by Batchelor's far-field series; across it, by the series; dashed, each sphere's Stokeslet alone, 1 + 3/(2ρ) and 1 + 3/(4ρ). Touching in line the pair falls 1.55 times as fast; at 5 radii, 1.286 in line and 1.154 side by side; at 30, 1.05 and 1.025.

Two spheres settling at a slant keep their slant for ever, and three do not

Two equal spheres settling through a viscous liquid fall faster together than either alone — half as fast again if they touch in line, and still five per cent faster thirty radii apart. Slanted, the pair drifts sideways as it falls, by up to three and a half degrees, and keeps the slant, the separation and the drift for as long as it falls: the two spheres always move with exactly the same velocity. Release a third and the shape is lost at once, the middle sphere drawing ahead and the outer two closing in behind it. What a pair keeps and a trio does not is a statement about symmetry, and reciprocity is half of it.

viscous · Reciprocity A sphere two per cent larger overtakes its neighbour round a path that is its own mirror image, and leaves at the offset it came in at. The centre of a sphere two per cent larger than its neighbour, relative to the smaller sphere, as it overtakes it in a viscous liquid, from four horizontal offsets; the circle is the distance at which they would touch. Each path bulges outward as it passes and is symmetric about the smaller sphere's level: from 0.85, 1.5, 2.5, 4 radii the larger sphere leaves at 0.8499, 1.5, 2.5, 4, after closest approaches of 2.27, 2.83, 3.74, 5.15 radii.

A larger sphere overtakes its neighbour and leaves on the line it came in on

Two equal spheres settling together never change their shape. Make one of them two per cent larger and the pair becomes an encounter: the larger sphere sinks past the smaller round a path that bulges out and comes back, slowed sixteenfold at contact but never stopped, and it leaves at exactly the offset it arrived with. The smaller sphere swings seven radii sideways while the larger is above it and gives every radius back while the larger is below. In Stokes flow, meeting a neighbour displaces nothing — unless the spheres touch, and a hundredth of a radius of roughness is enough to make every touching path leave on the same line.

viscous · Reciprocity

Named alongside it

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

Creeping flowModel limitBoundary conditionDissipationReciprocityReversibilityReynolds numberSingularityBubbleDragHadamard rybczynskiMarangoni

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