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

Young laplace — where it appears

The relation that the pressure jump across an interface equals the surface tension times the sum of the two principal curvatures. It fixes the shape of every static drop, bubble and meniscus, and integrating it along a meridian is how those shapes are computed.

Named by 6 essays across 2 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 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 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 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 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

Named alongside it

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

Surface tensionModel limitDropMisconceptionAbsolute pressureCavitationDimensionlessLiquid tensionThresholdVapour pressureWeber numberBarometric height

All concepts