Illustrated Physics
physics, drawn
Concept

Viscoelasticity — where it appears

The behaviour of a material that both stores and dissipates the energy put into it, responding partly as a solid and partly as a liquid. Which it looks like depends on how long the observation lasts compared with the material's own relaxation time, so pitch shatters under a hammer and flows in a funnel.

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

Four answers to one push. Shear stress against shear rate for four fluids. The straight line through the origin is the Newtonian definition and is the only one of the four for which the word viscosity names a number. The Bingham fluid does not move at all until the stress passes 0.4, which is why toothpaste holds a shape on a brush and why wet concrete can be stood in a heap.

The fluid that answers back

For water, stress is proportional to how fast it is sheared, and the constant of proportionality is its viscosity. For paint, blood, ketchup and cornflour in water, it is not — and once the proportionality goes, so does the idea that viscosity is a number a substance has.

fluids · Rheology Solid and liquid are answers about a duration. The relaxation time of seven materials, on a logarithmic axis spanning 39 decades, against the length of one observation. A material behaves as a solid when its relaxation time is longer than the observation and as a liquid when it is shorter, so the vertical line is what decides which — and it is a property of the observer. At 1 s, 4 of these are solids. Move the line six decades to the right and pitch joins the liquids; move it far enough left and water is a glass, which is not a figure of speech but what a picosecond pulse measures.

The liquid that remembers

Pitch shatters like glass under a hammer and flows through a funnel over a decade. Neither behaviour is the true one. What decides which a substance shows is not the substance but the length of the observation, and the ratio between the two has a name and a number.

fluids · Rheology The same liquid climbs a thin rod and is thrown off a thick one. The height of the free surface against distance from the axis, for rods of 10.0, 25.0, 60.0 millimetres radius turning at one revolution a second in the same polymer solution, with the Newtonian answer dashed. Two effects compete and they fall off at different rates: a hoop tension along the curved streamlines pulls the fluid inward and lifts it, falling as the fourth power of the distance, while the centrifugal term pushes it outward and lowers it, falling as the second. Near a thin rod the fourth power wins and the liquid climbs — 2.21 millimetres at the thinnest. Beyond a radius of 34.64 millimetres it does not, and the same fluid at the same speed is thrown outward exactly as water would be. The changeover is a property of the fluid and the rod together, so a demonstration that works on a glass stirring rod fails on a spoon handle.

The liquid that climbs the rod

Stir water with a rod and it is flung outward, leaving a dip at the centre. Stir a polymer solution and it climbs the rod instead. Nothing about the viscosity can produce that, however large it is made — what produces it is a second stress that appears in shear and has no Newtonian counterpart at all, growing as the square of the rate where the familiar one grows in proportion.

fluids · Rheology Above a half, the chains stop letting go. The extensional viscosity of a dilute polymer solution, as a multiple of its zero-shear viscosity, against the accumulated stretch (Hencky strain, the stretch rate times the time), on a logarithmic axis, for stretch rates whose product with the relaxation time is 0.1, 0.4, 0.6, 1. The polymer carries 90 per cent of the viscosity. Dashed curves are the Oldroyd-B dumbbell, integrated from its conformation equation and matching its closed form; solid curves are the same chains with a finite length, FENE-P with L² = 400. At 0.1 the Oldroyd-B viscosity reaches 3.37 and the finite chain 3.37 by a strain of 6; at 0.4 the Oldroyd-B viscosity reaches 9.49 and the finite chain 9.25 by a strain of 6; at 0.6 the Oldroyd-B viscosity reaches 58 and the finite chain 49 by a strain of 6; at 1 the Oldroyd-B viscosity reaches 725 and the finite chain 329 by a strain of 6. Below one half the viscosity settles at a few times the shear value. Above it the Oldroyd-B chain stretches without limit and its resistance grows exponentially, while the finite chain grows until it is nearly fully extended and then stops, hundreds of times higher than it began.

The stretch a chain cannot outrun

A Newtonian liquid pulled into a thread resists exactly three times as hard as it resists being sheared, whatever it is made of. A polymer solution resists three times as hard only until the stretch rate passes one over twice its relaxation time. Past that, its chains can no longer recoil as fast as they are pulled apart, and the same liquid that is barely thicker than water in a stirred beaker becomes hundreds of times stiffer in a thread.

fluids · Rheology

Named alongside it

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

Constitutive lawPolymer solutionRelaxation timeRheologyViscosityWeissenberg numberCoil stretch transitionCreepCreeping flowDissipationDumbbell modelExtensional viscosity

All concepts