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

Non-newtonian — where it appears

A fluid whose stress is not proportional to its rate of strain — shear-thinning, viscoelastic, or possessed of a yield stress below which it does not flow at all. Its apparent viscosity is a property of the measurement as much as of the material.

Named by 6 essays across 3 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 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 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 The flow stops and the stress does not. Shear stress against time for a fluid sheared at a constant rate for two seconds and then left alone. Nothing is moving after the second mark and the fluid is still stressed: 37 per cent of the peak one relaxation time later, and 0.7 per cent after five.

The fluid that has not finished its last deformation

Shear a polymer solution for two seconds and stop. Nothing is moving and the fluid is still stressed — 37 per cent of the peak one relaxation time later, and still measurably stressed after five. Two histories imposing exactly the same total strain leave it in states differing by a factor of 3.7.

viscous · Non-newtonian Three dies, one melt, three flow curves. The apparent flow curve a capillary rheometer reports, for three dies whose radii differ by a factor of four. A material property cannot depend on the instrument, so the fact that these three do not lie on one another is a measurement of something other than a viscosity. The single curve drawn beneath them is what the same melt gives with the wall condition set to no slip.

The wall a melt really does slip on

For a simple liquid the slip length is under two nanometres, which is why no-slip is safe everywhere. For an entangled polymer melt it is a millimetre, which is larger than the die it is being pushed through — so the melt moves as a plug and the viscosity a rheometer reports is a property of the instrument.

misconceptions · The no-slip condition Shaken at the right frequency, a liquid with one relaxation time moves sixteen hundred times the flow its viscosity allows. The amplitude of the oscillating flow rate over the steady Poiseuille flow the same pressure-gradient amplitude drives through a Newtonian liquid of the same zero-shear viscosity, against the frequency, in a tube 1 cm in radius. A Maxwell liquid with λ = 1 s and η₀ = 30 Pa·s peaks at 1660 times at 6.63 Hz and again at 317 times near 15.2 Hz; a Rouse spectrum with the same longest time and viscosity reaches 20.2 at 19.8 Hz; the Newtonian liquid only falls, to 0.99 at 30 Hz.

A liquid with one relaxation time rings in a tube, and a spectrum of times silences it

Push a viscoelastic liquid back and forth along a rigid tube and, if its relaxation time is long against the cycle, the tube becomes a cavity for shear waves in the liquid itself. At the cavity's resonance a wormlike-micelle solution in a tube one centimetre across moves sixteen hundred times the flow a Newtonian liquid of the same viscosity would. The Deborah number says when the liquid is elastic enough to ring; it cannot say whether it will, because a polymer with a spectrum of relaxation times and the same longest one damps the same resonance to a factor of twenty.

regimes · Deborah

Named alongside it

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

MeasurementConstitutive lawModel validityPipe flowViscoelasticityDeborah numberPolymerPower lawRegimeRelaxation timeShear rateShear stress

All concepts