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

Continuum — where it appears

The assumption that a fluid may be described by fields rather than by molecules, which holds while the mean free path is small compared with everything else. It is what makes velocity at a point meaningful, and it fails in a vacuum chamber and at a shock's interior.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

kinematics · Continuum 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.

kinematics · Deformation 45° of corner, and an infinite number of eddies in it. The creeping flow in a corner of 45 degrees, drawn from Moffatt's similarity solution. Each eddy turns the opposite way to its neighbours and is 3.17 times smaller and 1.6e+3 times weaker than the one outside it. The contour levels are rescaled inside each eddy, because they differ in strength by three orders of magnitude per step and a single set of levels would show the first and nothing else — which is itself the reason nobody has seen the third. The dividing lines between eddies are drawn where the stream function changes sign, and the dots are the centres, both found from the solved field.

The eddies nobody stirs

A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.

viscous · Corner How wrong the ordinary answer already is. The error in a no-slip continuum calculation of the flow through a channel, against the Knudsen number, both logarithmic. The rule at Kn = 1 is where a molecule crosses the whole channel between collisions — the value the number is named for. The error is one per cent at Kn = 1/594, five per cent at 1/114 and ten at 1/54, so the continuum regime of the usual classification, which runs to Kn = 0.01, is a region in which the continuum answer is already six per cent out at its far end.

Where a fluid stops being one

The Knudsen number is the mean free path over the size of the thing, and at one a molecule crosses the whole channel between collisions. The continuum equations with a no-slip wall are already one per cent wrong at one part in five hundred and ninety-four, which is a factor nothing about the definition would suggest.

regimes · Knudsen The velocity through a shock at Mach two, as a function of position. The Becker profile, integrated outwards from its own inflection point. It runs from the upstream velocity to the downstream one — the two states the jump conditions give, which appear here as the equilibria of a first-order differential equation — and its steepest gradient matches the closed form to one part in ten thousand. The horizontal axis is in units of the thickness, which for this shock is 139 nanometres.

The discontinuity that has a thickness

The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.

compressible · Shock The gas that does not stop at the wall. Channel flow profiles with and without slip, at a Knudsen number of a twentieth. The slipping profile does not reach zero at the wall: the gas there is moving, by an amount proportional to the mean free path times the velocity gradient.

Slip is a memory of one mean free path

A molecule arriving at a wall last collided about a mean free path away and carries the velocity from there. Averaged over arrivals and departures, that leaves the gas at the wall moving — by two per cent of the centreline speed at a Knudsen number of a hundredth, and sixty per cent more flow through a microchannel at a tenth.

regimes · Knudsen Sheared between plates, a gas transmits its full viscous stress down to a few free paths across, and then a share proportional to its pressure. The shear stress a gas transmits between two plates, over Navier–Stokes's μU/H, against the rarefaction δ = √π/(2Kn), from the linearised kinetic equation; with Navier–Stokes and first-order slip, 1/(1 + 2·1.016/δ), and the free-molecular δ/√π. The kinetic stress is 0.908 of μU/H at δ = 20, 0.501 at 2, 0.0979 at 0.2; the slip formula is within 2.7 per cent at δ = 1.

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.

kinematics · Continuum Between plates the flow rate falls with rarefaction to a minimum near one free path across, then climbs without limit. The reduced flow rate G of a gas driven by a pressure gradient between two plates, against the rarefaction parameter δ = √π/(2Kn), from the linearised BGK equation; beside it Navier–Stokes with no slip (δ/6), with Maxwell's slip (δ/6 + 1.016) and the uncollided flux of molecules straight from a wall. The least flow rate is 1.538 at δ = 1.07, Kn = 0.827; at δ = 0.01 the solve gives 3.049, at δ = 0.001 4.449 and at δ = 10 2.769.

A thinning gas flows least at one free path across, and then more without end

Pump the gas out of a narrow channel and, for the same pressure gradient, less of it flows — until the channel is about one mean free path high. Thinner than that, more flows, and between infinite plates the flow rate climbs without limit, because the molecules carrying it are the ones skimming along the walls, whose flights only collisions with other molecules can end. Knudsen measured the minimum in 1909. The linearised kinetic equation puts it at a reduced flow rate of 1.538, at a Knudsen number of 0.83, and shows it is not a dip below the two closed forms but a place far above both.

regimes · Knudsen

Named alongside it

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

Mean free pathKnudsen numberSlipViscosityBoundary conditionKinetic theoryModel validityMeasurementModel limitTransportAsymptoticsCorner eddy

All concepts