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

Surface tension — where it appears

The energy per unit area of a liquid interface, which acts as a force per unit length along it. It scales with a length where every other force in the subject scales with an area or a volume, so its importance is decided by size alone.

Named by 13 essays across 6 fields — each of them below, with the objects they name alongside it.

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

Where a jet stops being a jet

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

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

The drop that is not a tear

A falling raindrop is flattened along the direction it is going, by the pressure of the air passing it rather than by its own weight, and the group that decides is the Weber number. The teardrop of every illustration has the wrong symmetry entirely — there is no up in the problem it is drawn for.

regimes · Drop shape Twice as fast is not twice as thick. The film a plate carries out of water, against the speed it is withdrawn at, both logarithmic. The slope is exactly two-thirds, so doubling the speed thickens the film by 58.7 per cent and never by more. The open marks are outside the range the derivation holds in — above a capillary number of about a hundredth the film is no longer thin against the capillary length, and the measured thickness leaves this line.

What a plate takes with it

Pull a plate out of a bath and it comes out wet. How wet is not set by the plate, the bath or how much liquid there is, but by a competition in a region a fraction of a millimetre long that nobody looking at the plate can see — and the film goes as the two-thirds power of the speed, never as the first.

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

The number that cannot break a drop

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

regimes · Capillary A threshold that is a curve, not a pressure. The tension a rectangular pulse has to reach to make a five-micron bubble run away, against how long the pulse lasts. At a tenth of a microsecond it takes forty bar; at a hundred microseconds it takes 1.05, which is within about one per cent of the static threshold. There is no such thing as the cavitation pressure of this bubble on its own.

A threshold that is also a duration

The cavitation number treats inception as a pressure: below it the liquid tears, above it does not. A five-micron bubble asked to grow in 0.3 microseconds needs 43 bar of tension and the same bubble given a hundred needs 1.05, because it has to make a journey and not merely respond.

applied · Cavitation 100 metres of water, −1.98 MPa absolute at the top. The absolute pressure up a transpiring column 100 metres tall, with the sap rising at 0.25 mm/s through conduits 40 µm across. It starts at 1.3 kPa at the root, falls by 9.79 kPa a metre for gravity and 10.02 for friction, and reaches −1.98 MPa at the top — below zero, which is not a low push but a pull. The pale line is the same column with nothing flowing. The floor is not the vapour pressure but the pore a gas bubble could be drawn through: −2.81 MPa for a 50 nm pore, which this column would reach at 142 metres. A suction pump lifting the same water from a free surface stops at 10.1 metres, because it offers the water somewhere to boil.

Where a liquid does pull

A fluid cannot pull, and the essays that settle what suction is are right about every gas and every liquid with a free surface near it. A liquid with nothing in it to boil on is another matter. Every tree taller than ten metres depends on the difference, and the floor under it is set by the size of a pore rather than by the vapour pressure.

misconceptions · Suction A tube 10 cm across, spun: the lowest pressure is on the axis. The absolute pressure along a water-filled tube spun about its middle, open to the air at both ends, 5 cm from the axis. In the spinning frame the water is at rest under a centrifugal pull, so the pressure falls from atmospheric at each meniscus to its lowest on the axis, as a parabola. 10 thousand rpm puts −1.3 MPa there, 20 thousand rpm puts −5.4 MPa there, 30 thousand rpm puts −12.2 MPa there and 45 thousand rpm puts −27.6 MPa there. The place the water is stretched hardest is the place furthest from both free surfaces, which is the whole merit of the method: a gas cannot reach the liquid where it is weakest.

A breaking strength that is the size of a flaw

Water can be stretched, and how far is a measurement people have made for a century and a half with instruments that agree with one another and not with the theory. Spinning a tube puts the stretch where no gas can reach it, sealing one caps the stretch at water's density maximum, and every measured number turns out to name the size of the worst cavity in the sample.

misconceptions · Suction A litre of water at the crown carries 77 mL of gas it cannot hold. The volume of free gas a litre of air-saturated water can release at a siphon's crown, at the crown's own pressure, against that pressure, on a logarithmic scale, at three temperatures. It is zero at atmospheric and grows without limit towards the vapour pressure. At the reference siphon's starting crown pressure of 51.5 kPa it is 20.6 mL, a supersaturation of 2.02; at the frictionless floor of 23.0 kPa it is 76.7 mL, a supersaturation of 4.79. Cold water carries more: 87.4 mL at 5 °C. This is the equilibrium bound — what would come out if the water stayed long enough, which it does not.

The air that breaks a siphon nothing else can

A running siphon's heights cannot break it, and its friction only postpones the moment it is most exposed. What does break a siphon that has run for a day is the air dissolved in its water, which the crown's low pressure leaves the water carrying far more of than it can hold — and which gathers only once the flow is too slow to carry a bubble away.

misconceptions · Siphon 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 A water sheet two millimetres thick leaves the lip above 0.270 m/s. The effective tension of a sheet of water, 2σ − ρU²h per metre of its width, against the speed it is poured at, for sheets one, two and four millimetres thick. At rest every sheet carries the tension of its two surfaces, 145.4 mN/m, and the momentum it carries along itself subtracts from that. A 1 mm sheet reaches zero at 0.382 m/s, a 2 mm sheet reaches zero at 0.270 m/s and a 4 mm sheet reaches zero at 0.191 m/s. Below its own crossing a sheet bent round a lip is pulled onto it; above, it is flung off. The quantity that decides is a tension, not a pressure, and the speed at which it vanishes is the speed of waves along the sheet.

The teapot effect is a tension, not a pressure

A slow pour runs back under a spout and down its outside, and the name it is usually given is the Coandă effect. The Coandă effect borrows the ambient pressure, and a liquid in air has none to borrow. A liquid sheet is held to a lip by its own surface tension, and it lets go at the one speed that tension cannot carry — the speed of waves along the sheet — whatever the lip's radius.

misconceptions · Coanda The same wavelength drawn as rolls and as hexagons. Plan views of a convecting layer with one critical wavelength, drawn from the amplitude equations' two stable states. On the left, rolls: a single set of parallel bands, rising fluid along one set of lines and sinking along the next. On the right, hexagons: three sets of rolls at 120° to each other with equal amplitudes, whose sum has its maxima on a triangular lattice with spacing 2/√3 of the wavelength, each maximum at the centre of a hexagonal cell. At ε = 0.0250, inside the window, both are stable: rolls with amplitude 0.158 and hexagons with 0.081 in each of their three rolls. A hexagon is not a different kind of cell; it is three roll patterns that the quadratic term lets reinforce one another.

Hexagons remember how the heat was turned up

A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.

turbulence · Convection 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

Named alongside it

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

Model limitYoung laplaceCavitationMisconceptionAbsolute pressureCapillary numberDimensionlessDropLiquid tensionNucleationThresholdVapour pressure

All concepts