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

Pipe flow — where it appears

Fully developed flow along a straight duct, in which nothing changes with distance and the pressure gradient balances the wall friction exactly. Its laminar form is one Poisson equation with no parameter in it, so its resistance is a pure number belonging to the cross-section's shape.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Seven fluids in one pipe at one pressure gradient. Velocity profiles of power-law fluids in a round pipe, all at the same pressure gradient and the same consistency, scaled to the fastest. A shear-thinning fluid (n below one) is flatter in the middle and steeper at the wall; a shear-thickening one is the reverse. The flattening is often called plug-like, which invites the reading that the fluid is moving more freely — what has actually happened is that all the shear has been pushed into a thin annulus at the wall, which is the expensive place to put it.

A viscosity that depends on the question

Blood, paint, molten polymer and drilling mud have no viscosity. They have a relation between stress and strain rate that is not proportional, so the ratio of the two depends on how hard they are being sheared — and an instrument that reports one number is reporting a property of itself.

viscous · Non-newtonian A factor of 1.8 between the extremes. The same numbers as a bar. The spread is exactly 96 over 160/3, which is 1.8 — and it is a spread in a quantity that a single correlation using the hydraulic diameter treats as a constant. A rectangle twenty times as wide as it is deep is 78 per cent more resistant than a circle of the same hydraulic diameter.

A number that is only the shape of the hole

The friction factor times the Reynolds number for laminar flow in a duct is a pure number that depends on the cross-section's shape and on nothing else — not the fluid, not the size, not the flow rate. It is exactly 64 for a circle and exactly 96 for a slot, and the correlation that treats them as the same is wrong by a third.

applied · Internal flow Five approach profiles a meter might be looking at. The velocity across the pipe upstream of a contraction, for a uniform flow, fully developed laminar flow, two turbulent power laws and an annular jet of the kind a bend or a partly open valve leaves. All five carry the same volume flow. The meter reads a pressure difference and cannot see any of this.

The profile a meter cannot see

A differential-pressure flowmeter measures a force balance and reports a flow rate. The step between them needs two integrals of a velocity profile the instrument has no access to — and two profiles differing by half the mean velocity across the pipe give identical readings, which is why the standards specify straight pipe rather than a correction.

applied · Metering The core that does not move, and where its edge is. Four velocity profiles at the same pressure gradient and four yield stresses. The shear stress in a pipe is G r/2 whatever the fluid is — that is a force balance and not a constitutive law — so the fluid is unyielded exactly inside r = 2 tau_y/G, and the plug radius is known before anything is solved.

The core that does not move

Toothpaste in a tube has a region in the middle that is not being sheared at all, and its edge is at a radius you can write down before solving anything. Four things about that core are exact, and one of them is that the fluid does not flow below a threshold — not slowly, at all.

viscous · Non-newtonian Two lifetimes, crossing. The mean time for a turbulent puff to decay and the mean time for it to split into two, against Reynolds number, on a logarithmic axis spanning thirty orders of magnitude. Below the crossing puffs die faster than they multiply; above it they multiply faster than they die, and the flow stays turbulent.

A puff that does not know how old it is

A patch of turbulence in a pipe below the critical Reynolds number dies at random, and its chance of dying in the next second does not depend on how long it has already lasted. The flow that contains it has a memory anyway, because the patches multiply — and where multiplying overtakes dying is a Reynolds number.

turbulence · Transition Friction lends the crown 28.4 kPa, and the tank takes it back. The absolute pressure at the crown of a siphon with friction in its hose, through a whole drain, for the crown placed at three positions along the hose, against the frictionless constant of 23.01 kPa. With the crown 0.3 of the way it starts at 63.3 kPa, with the crown 0.5 along it starts at 51.5 kPa and with the crown 0.7 of the way it starts at 40.4 kPa. Every curve is above the constant, every curve falls towards it as the level falls, and every curve reaches it at the end — 23.06 kPa with a centimetre of level left. Friction never brings a siphon nearer to breaking; it lends a margin, and the draining tank returns it pascal by pascal, so the worst the crown ever sees is the frictionless value.

The margin friction lends a siphon

Without friction a draining siphon's crown pressure does not depend on the source level at all, and every real hose has friction. It turns out always to raise the crown pressure, by an amount the draining tank hands back pascal by pascal — and how much it lends is decided by where along the hose the crown sits, not by how rough or how narrow the hose is.

misconceptions · Siphon Four fluids under the same stress, and the four profiles that result. Each fluid carries the same linear stress and answers it differently: a Newtonian fluid with a parabola, a shear-thinning one with a blunter profile, a shear-thickening one with a sharper one, and a Bingham plastic with a plug in the middle where the stress is below its yield point.

The stress a pipe knows

A capillary viscometer measures a pressure drop and a flow rate and reports a viscosity. The first half of that inference is a force balance and is exact for any fluid there is; the second needs the slope of a whole flow curve, which is the experiment the instrument was bought to avoid.

viscous · Non-newtonian

Named alongside it

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

MeasurementConstitutive lawModel validityNon-newtonianRegimeControl volumeFriction factorLaminar flowMisconceptionPower lawReynolds numberShear stress

All concepts