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

Contact angle — where it appears

The angle a liquid surface makes with a solid where the two meet, measured through the liquid. At rest it is set by the three surface energies; when the line moves, the angle seen a millimetre away differs from the one at the solid by an amount growing with speed.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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

The drop a no-slip wall would never let spread

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

misconceptions · The no-slip condition 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.

regimes · Bond One rim, one drop, every angle in the band. Water on a disc the size of a US cent, 19.1 mm across, pinned at its rim. The faint outlines are the drop at apparent angles of 60°, 90°, 120° and 150°; the dark one is the drop at the angle set, 120°, holding 1.13 mL. The rim stays put while the angle climbs, and every shape is an equilibrium: 0.52 mL at 60°, 1.44 mL at 150°. They are flattened by gravity into puddles with rounded edges, not spherical caps.

An edge holds any angle it is given

A liquid's contact angle is taught as a property of the liquid and the solid. On a smooth face it is. On a sharp edge the line stops, the angle is free to take any value in a band as wide as the edge is sharp, and that freedom is how a coin carries a dome of water and a glass stands full above its brim.

misconceptions · Coanda 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.

regimes · Bond 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.

regimes · Bond An edge is worth its sharpness against the capillary length. The extra height a rounded edge holds over a flat plate, as a fraction of what a sharp edge holds, against the rounding radius in capillary lengths, for Young angles of 30°, 60° and 90°. An edge rounded to a thirtieth of a capillary length, about a tenth of a millimetre for water, keeps about ninety-five per cent. One rounded to a third of a millimetre keeps between eight and nine tenths; one rounded to a whole capillary length, about a third; and past that the worth falls as the inverse of the radius.

A rounded edge spills before its line goes round

Gibbs's band says a sharp edge lets a liquid stand at any angle across a range as wide as the edge's turn. No real edge is sharp, and on a rounded one the line does not stop; it slides round the curve, meeting it at the Young angle everywhere. The band survives the sliding. What the rounding costs is height, because the line drops as it goes round, and past a point the drop outweighs the steeper angle and the liquid spills before the line has reached the far face. An edge is worth its sharpness measured against the capillary length, and nothing smaller.

misconceptions · Coanda The pocket escapes at the least pressure its states can stand. The liquid pressure each equilibrium of the pocket needs, against the pocket's volume, for a crevice with a one-micrometre mouth in a wall the water wets at 40° and one it does not, at 110°. As the crown's pressure falls the pocket follows its curve to the right: the rim slides out of the cone, pins at the mouth, and the meniscus bulges into a cap. The curve's minimum is the escape threshold — -106 kPa for the wetting wall and -138 kPa for the non-wetting one — and below it no state exists: the pocket becomes a cavity.

A crevice keeps the nucleus a free bubble loses

A free bubble at a siphon's crown dissolves in milliseconds, so it cannot be what breaks a siphon that has run for an hour. A pocket of gas in a crack of the hose wall can be, because its meniscus is curved by the wall and need not dissolve. Followed through its equilibria, the pocket escapes at a tension set almost entirely by the crack's mouth. What the wall's wettability decides is slower and more consequential: whether the crack fills, or keeps drawing gas in until the siphon breaks.

misconceptions · Siphon 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

Named alongside it

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

Surface tensionModel limitYoung laplaceBond numberMeasurementThresholdCapillary lengthDropBoundary conditionCapillary numberCoandaFree surface

All concepts