Illustrated Physics
physics, drawn
Concept

Shear stress — where it appears

Force per unit area applied along a surface rather than across it, which a fluid answers with a velocity gradient rather than a displacement. That is the operational definition of a fluid: a solid reaches a new shape and stops, and a fluid keeps moving for as long as the stress is applied.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The straight line between two plates. A layer of water 10 mm deep with its top plate drawn along at 1 m/s. In the steady state the velocity is a straight line from zero at the fixed surface to the plate's speed at the moving one, and the stress needed to keep it going is μ times that slope: 0.100 Pa. The fluid at each wall is at rest with respect to it, which is an experimental fact rather than a consequence of anything above.

Momentum going sideways

Viscosity is usually described as friction between layers of fluid, which gets the effect right and the mechanism wrong. What is actually happening is that momentum is being conducted across the flow — by the same equation, with the same solutions, as a drop of ink spreading.

fluids · Viscosity The parabola in a pipe, and what it integrates to. Steady flow in a round pipe: the velocity is a parabola, zero at the wall and greatest on the axis, and its average over the cross-section is 0.500 of the peak — exactly a half, by integration. Because the profile scales with r² and the area with r² as well, the flow goes as the fourth power of the radius: widening a pipe from 1 to 2 mm multiplies it by 16.

The fourth power in a pipe

Halve a pipe's radius and the flow through it falls to a sixteenth. The exponent is four rather than two, because narrowing a pipe both removes cross-section and slows what is left — and one law with that exponent in it governs a blood vessel, a hypodermic needle and a water main.

fluids · Viscosity 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 Two ways to weaken a surface. Surface tension is a property of a surface rather than a constant of a liquid, and this shows how far it moves. The steep curve is water with ethanol dissolved in it, against ethanol mole fraction, through nine measured points: two per cent of ethanol takes the tension from 72 to 56.4 mN/m, a fall of 22% for a change of composition small enough to taste and not to see. The initial slope is 837 mN/m per unit of mole fraction, because ethanol collects preferentially at the surface and a little of it covers a great deal of area. The gentle line is pure water against temperature, read on the same horizontal axis as degrees rather than fraction: about 0.15 mN/m per degree, so a difference of ten degrees across a surface is worth about a millinewton per metre. Neither dependence would be interesting if surfaces were uniform. What makes them matter is that a difference in tension across a surface is a force along it, and nothing in the liquid prevents such a difference from existing.

The surface that pulls toward the stronger side

Surface tension is usually treated as a constant of a liquid, and it is not — it depends on temperature and on what is dissolved, and a difference in it along a surface is a force along that surface — which drags the liquid underneath and needs no pressure difference at all.

fluids · Surface tension The thickness a paste can hold on a slope. The greatest thickness a yield-stress fluid can rest at without flowing, against the angle of the surface it is resting on, on a logarithmic vertical axis, for four materials. A layer of thickness h puts a shear stress ρgh sin α on its own base; it stays put while that is below the yield stress and flows when it is not, so the critical thickness is τ_y divided by ρg sin α and it depends on nothing else — not on the viscosity, not on how long it is left, not on how it was put there. On a vertical wall the numbers are 1.4 mm of ketchup, 9.2 mm of mayonnaise, 15.7 mm of toothpaste, 15.1 mm of basaltic lava, which is why toothpaste stays on a brush and ketchup does not stay on a plate held up. Read the other way it is a measurement: a lava flow that came to rest 3 centimetres thick on a 30° slope had a yield stress of about 400 pascals, and that is how the rheology of a flow nobody was standing next to is recovered from its shape a thousand years later. The model stops where the layer is thin enough for surface tension to matter and where the material's yield stress depends on how long it has been left alone, which for most of these it does.

The paste that holds up its own hill

Toothpaste stands on a brush and ketchup does not stand on a plate, and the difference is a single number with the units of a pressure. Below it a material does not flow slowly — it does not flow. That threshold turns a rheological property into a length, and the length is why a lava flow's thickness says what the lava was made of.

fluids · Rheology Force against slip, and the knee between them. The force a tyre delivers against how much faster its tread is going than the road, for a patch 120 mm long under 4000 N with a friction coefficient of 1. The curve is the integral over the bristles, and the dashed line is the cubic the brush model gives in closed form; they agree to 0.00 per cent of the sliding force. The first slope is 80 kN per unit slip, and it belongs entirely to the elasticity of the rubber — at vanishing slip nothing is sliding, so no friction coefficient can appear in it. Full sliding is reached at 15.0 per cent slip and not before. Everything a driver calls grip lives on the rising part of this curve, at a few per cent of slip, where the patch is partly stuck and partly sliding — and the quantity that decides handling in that region is the slope rather than the friction coefficient at the top.

The grip that needs a little slipping

A wheel that transmits any force at all is not rolling. Part of its contact patch is stuck to the road and part is already sliding, and the force it delivers is a measure of how much has given up. The useful part of the curve is a few per cent of slip, the peak is not the end of it, and everything past the peak is unstable.

mechanics · Friction

Named alongside it

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

ViscosityConstitutive lawDiffusionNo slipVelocity gradientYield stressApparent viscosityBingham plasticBoundary conditionContact areaContact mechanicsDissipation

All concepts