Concept

Deborah number — where it appears

The ratio of a material's own relaxation time to the time the flow gives it. Below one the material has time to relax and behaves as a liquid; above one it has not and behaves as a solid, and nothing about the material itself changed.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present.

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

regimes · Dimensional

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.

Model validity Relaxation time Convolution Measurement Memory kernel Non-newtonian Regime Viscoelasticity Constitutive law Dimensional analysis Frequency response Phase lag

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