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

Residence time — where it appears

How long a parcel of fluid spends inside a device between entering and leaving. Its mean is volume over flow rate and is the number usually quoted; the distribution about that mean decides when a bed first breaks through and how much a reaction converts.

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

A million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

Two slow things make a fast one

Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.

regimes · Dispersion The number that settles the argument. How many turns the earth's contribution gets through in the time the vessel takes to drain: the draining time divided by the rotation period at the drain. A bathtub manages a tenth of one and there is nothing to see. The apparatus that settled it manages nearly nineteen hundred, and that is the whole of what its enormous area-to-drain ratio was for.

The bath that was only ever a wait

The Coriolis force is far too weak to steer a draining bath, and it does steer a large enough tank that has been left alone long enough. Both are true, and the number that separates them is not a force ratio — it is how many turns the earth's contribution gets through before the vessel is empty.

misconceptions · Coriolis scale When the tracer leaves, given when it went in. The residence-time distribution of three beds with the same mean residence time and different Peclet numbers. Every one of them has its mean at exactly one, and they are nothing alike: the loosest lets a tenth of the tracer out before a third of the mean time has passed, and the tightest is nearly the spike a plug-flow calculation assumes.

The outlet is the inlet, a while ago

A bed has a mean residence time and everybody quotes it. Six beds with the same mean let their first hundredth through at 0.20 and at 0.85 of it, mix a window of inlet history between 1.53 and 0.18 wide, and convert a first-order reaction by amounts the mean cannot distinguish.

applied · Porous A duct that does not change, and a parcel that does. A contraction of area ratio four, with the parcel marked at five equal intervals of time. The walls do not move, the field at every point is the same at every instant, and the spacing of the markers grows because the parcel is carrying its own history through the duct.

How long the fluid has been in there

Age is the simplest thing a flow can remember. It obeys the shortest transport equation in the subject — its material derivative is one — and no instrument pointed at a steady flow can read it, because a steady flow's every field is constant and its fluid is getting older all the time.

kinematics · Acceleration

Named alongside it

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

MeasurementMemory kernelDiffusionDispersionMixingModel validityPeclet numberRegimeTransportAccelerationAngular momentumAveraging

All concepts