Illustrated Physics
physics, drawn
Concept

Viscosity — where it appears

The constant relating the shear stress in a fluid to its velocity gradient, being momentum conducted across the flow rather than layers rubbing. In a gas it is independent of density, because the number of carriers and the distance each carries cancel exactly.

Named by 34 essays across 6 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 A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

thermodynamics · Kinetic theory Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run.

The jiggle that proved atoms

A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

thermodynamics · Diffusion Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do.

The liquid that will not slow down

Cool helium below 2.17 kelvin and it starts flowing through gaps no ordinary liquid could enter, climbs out of its own container, and circulates for as long as anyone has been willing to watch. The viscosity is not small. As far as any measurement can tell, it is zero.

quantum · Superfluidity 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 viscosity of a gas, over 8 decades of pressure. The viscosity of 3 gases at 300 K against pressure, on a logarithmic pressure axis and a linear viscosity one. The lines are flat, and that is the whole figure. Viscosity is the rate at which momentum is carried across a shear, which is the density of carriers times the distance each one carries it: ⅓ρv̄λ. Doubling the pressure doubles the density and halves the free path, and the two cancel exactly, so the same gas at a hundredth of an atmosphere is exactly as viscous as at one — which is not what anybody expects of a thinner gas and is what is measured. nitrogen comes out at 17.9 μPa·s against a measured 17.9, helium comes out at 19.3 μPa·s against a measured 19.9, argon comes out at 21.7 μPa·s against a measured 22.7. The flatness ends when the free path reaches the apparatus rather than the next layer of gas: at a vessel 10 mm across that is around 0.68 Pa for nitrogen, 1.94 Pa for helium, 0.69 Pa for argon, below which there is no gas-to-gas hand-off left to make.

The viscosity that does not care how much gas there is

Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.

thermodynamics · Kinetic theory 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 Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

The whirlpool that comes in one size

Spin a bucket of ordinary liquid and it ends up turning with the bucket. Spin a bucket of superfluid helium slowly and it does not turn at all. Spin it faster and it does not turn either — until a threshold, at which a single line of circulation appears, carrying not some amount but exactly h/m. There is nothing in between, because the velocity is the gradient of a phase and a phase has to come back to itself.

fluids · Superfluidity An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

The distance that takes the treble out

Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles.

waves · Attenuation 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 The height a 1 mK difference lifts helium. The head of liquid that a temperature difference of 1 millikelvin can support across a superleak — a plug fine enough that the normal fluid cannot pass and the superfluid can — against the temperature it is done at. Only the normal component carries entropy, so warming one side makes the superfluid flow toward the warm side until the pressure difference balances, and equilibrium is at ΔP = ρSΔT. The height that supports is SΔT/g, in which the density cancels exactly; computed both ways here the two agree to machine precision. The numbers are the striking part: 2.0 mm at 1.2 K, 4.6 mm at 1.4 K, 9.2 mm at 1.6 K, 19.4 mm at 1.8 K, 46.9 mm at 2 K, from a temperature step a thousand times smaller than anything a hand could feel. Aim a light at the warm side and the liquid does not merely rise but jets out of the tube, which is the fountain effect Allen and Jones found in 1938 and the most direct demonstration that helium II is two fluids rather than one. The entropies used are measured values; everything else on this chart is computed from them.

The fountain a lamp can drive

Below two degrees above absolute zero, liquid helium behaves as though it were two fluids occupying the same space — one carrying all the entropy and all the viscosity, the other carrying neither. It is not a metaphor and not a mixture. Shine a light on one side of a fine plug and the liquid jets out of the tube, because a temperature difference of a thousandth of a degree is a pressure of a hundred pascals.

fluids · Superfluidity One word, two mechanisms, opposite signs. Viscosity on a logarithmic axis against temperature over the range 280 to 360 kelvin, where gases and liquids can both be measured. The gases rise and the liquids fall, and the two families are separated by three decades of magnitude as well as by sign. The logarithmic slopes at the middle of the range are 0.76 for air, 0.69 for helium, -6.13 for water, -5.41 for ethanol, -23.00 for glycerol, so the steepest liquid responds 30 times more strongly than the gas and in the other direction. Nothing about the word viscosity requires this: what is being measured in both cases is the ratio of a shear stress to a shear rate, and that definition says nothing about what carries the momentum. In a gas it is molecules in free flight, so heating speeds up the carriers; in a liquid the molecules are permanently in contact and what has to happen is one of them getting past its neighbours, so heating removes an obstacle rather than adding a carrier. The obvious question this raises is what a dense gas near its critical point does, where neither picture holds, and the honest answer is that neither formula on this chart applies there at all.

The thickness that goes both ways

Heat a liquid and it thins; heat a gas and it thickens. The two are not a strong effect and a weak one but opposite signs, differing by a factor of thirty in size as well — and the word viscosity names one measurement made on two mechanisms that have almost nothing in common.

fluids · Viscosity The motion that arrives late and does not go far. A flat plate sliding back and forth in its own plane, with the fluid above it drawn at 0°, 60°, 120°, 180° of the cycle. The speed at depth y is exp(−y/δ) times a cosine whose phase lags by y/δ, and both halves are checked: the largest speed reached at any depth follows exp(−y/δ) to a part in a thousand million, and the fluid one skin depth out is fastest a full radian after the plate is. Two skin depths out the motion is an eighth of the plate's and a quarter of a cycle behind; four skin depths out there is essentially nothing. Alternating shear does not diffuse away without limit — it fills a fixed depth and stops.

The shear that only reaches so far

Viscosity carries momentum sideways without limit when the driving is steady. Reverse the driving and it stops: the motion fills a depth set by the viscosity and the frequency, arrives there late, and beyond that depth the fluid does not know the wall exists.

fluids · Viscosity The wedge that proves it. A wedge of water 4 mm on its vertical side, at a depth of 3 m, with the pressure on each of its three faces as an unknown. The two force balances decide them. Horizontally, the sloping face's push has a component that must exactly cancel the vertical face's, and since the sloping face is longer by exactly the factor its slope reduces the component by, the two pressures are equal — the geometry cancels, at every angle, for every size. Vertically the same cancellation happens except for the wedge's own weight, which needs the bottom face to carry 0.0667 per cent more. That excess falls in proportion to the size of the wedge, so at a point it is nothing and the three pressures are one number. Pressure being the same in every direction is the conclusion of that argument, not an assumption in it.

The push that has no direction

That the pressure at a point in a still fluid is the same whichever way the surface faces is not a definition. It is a theorem, and its proof is an argument about how two kinds of force scale with size — which is also the exact statement of when it stops being true.

fluids · Hydrostatics A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation The four vortices a standing wave leaves behind. Streamlines of the steady flow that a standing sound wave sets up in a channel, over half an acoustic wavelength, with the horizontal axis in units of the wave's own phase and the vertical axis scaled to the channel. The sound itself is a back-and-forth motion that averages to nothing; this is what does not average to nothing. Four closed cells fill each wavelength, two above the centreline and two below, turning in opposite senses, with the fluid moving along the walls toward the velocity nodes and back along the centre. The boundary layer that generates all of it is 69 micrometres thick, which is 0.7 per cent of the channel and is thinner than the width of a line in this drawing. The cells are not in the layer; they fill the channel.

The drift a sound leaves behind

A sound wave moves fluid back and forth and puts it back where it started. Over many cycles it does not: a steady circulation appears, four cells to a wavelength, driven entirely from inside a boundary layer seventy micrometres thick. Its speed contains the sound speed and the amplitude, and it contains no viscosity at all — so making the fluid thinner does not make the drift weaker.

fluids · Viscosity The pressure at which an oil becomes a glass. Viscosity against pressure for 4 liquids on a logarithmic axis, from Barus's rule with the pressure coefficient each one actually has. The rule is an exponential, so a gigapascal multiplies an ordinary oil's viscosity by a hundred million or more, and the curves cross the line conventionally taken to mark a glass — a million million pascal-seconds — at 1.37 GPa for a mineral oil, 0.98 GPa for a traction fluid. The vertical line is the peak pressure inside the loaded contact this figure is about, 1.32 gigapascals, computed from the Hertz solution for that geometry and load. Water is drawn for contrast: its pressure coefficient is thirty times smaller, it never approaches a glass over this range, and that is why it is useless as a lubricant in a rolling contact however clean it is. Each curve is drawn solid up to the glass line and dashed above it, because past that point the material is not a liquid and its viscosity is not what decides how it shears — the exponential continues, and the substance it was written for does not.

The oil that is a glass for a quarter of a millisecond

Every other account of viscosity here varies the temperature. Pressure does something larger and in the same direction for every liquid: viscosity rises exponentially, by a factor of ten to the eight or more at the gigapascal inside a loaded gear tooth. That is not a curiosity — it is the only reason there is a film there at all. Remove the pressure dependence from the calculation and the predicted film is five nanometres, under the roughness, and the surfaces touch.

fluids · Viscosity 14 decades of fluid, and a floor none of them reaches. The ratio of viscosity to entropy density for 8 substances, in units of the proposed lower bound, on a logarithmic axis. The quantity is a viscosity divided by how much entropy a cubic metre of the substance holds, and it has the dimensions of Planck's constant over Boltzmann's — so a bound on it is a statement with no material properties in it at all. The span here is a factor of 1.6e+14, from pitch at 20 °C at the top to the quark–gluon plasma at the bottom, which sits a factor of 1.6 above the floor. The values marked as computed are worked out from a viscosity and a tabulated entropy; the rest are quoted from the compilations, because a minimum along an isobar is an inference from many measurements rather than a single one.

Whether a fluid can be made arbitrarily thin

Viscosity has no units anybody would call fundamental, and nothing obviously stops it being as small as you like. Divide it by the entropy in a cubic metre and the units become Planck's constant over Boltzmann's — and a conjecture from 2005 says that ratio has a floor. Across fourteen decades of ordinary fluid nothing has been measured below it, the closest thing to it is the hottest matter ever made, and a kinetic-theory argument reaches the same number from the opposite direction.

fluids · Viscosity The column that overshoots its own height. The height of water in a vertical glass tube, as a fraction of Jurin's equilibrium height 2γ/ρgr, against time for tubes of radius 0.2, 0.4, 1, 2 mm, integrated from the capillary pull, the weight and the Poiseuille drag with the inertia of the rising column. The 0.2 mm tube (Jurin height 74.4 mm) creeps up without overshooting; the 0.4 mm tube (Jurin height 37.2 mm) creeps up without overshooting; the 1 mm tube (Jurin height 14.9 mm) overshoots to 1.33 of it; the 2 mm tube (Jurin height 7.4 mm) overshoots to 1.46 of it. The linearised motion about the equilibrium is underdamped when the radius exceeds (32η²γ/ρ³g²)^(1/5) = 0.48 mm for water: wider tubes ring, narrower ones do not.

The pore that lifts highest fills slowest

Jurin's law says how high water climbs in a tube and nothing about how long it takes. The rise has three laws in succession — the column's inertia, then its viscous drag, then its weight — and the same narrowness that lifts water highest throttles it hardest, so for any deadline there is a best pore. A wide enough tube does something Jurin's law cannot: it overshoots and rings. And whatever the route, exactly half the energy the surface releases is lost.

fluids · Capillarity The flow past a thread takes thousands of widths to recover. The speed of the fluid beside a body, at right angles to the stream, as a fraction of the free stream, against distance from the body's axis in body radii, in slow viscous flow. Past a sphere, from Stokes's solution, the flow is back to 90 per cent of the stream by 7.5 radii whatever the Reynolds number. Past a cylinder, from Lamb's inner solution, the speed grows only as the logarithm of distance, with a slope set by the Reynolds number: it reaches 90 per cent of the stream at 29, 232, 1,843 and 1.5 × 10⁴ radii for Re = 10⁻¹, 10⁻², 10⁻³ and 10⁻⁴ in turn. Each cylinder curve is drawn out to where the inner solution reaches the stream speed, beyond which Oseen's outer flow takes over. Each tenfold drop in the Reynolds number pushes the recovery ten times further out, because what finally stops the disturbance is inertia, at a distance of order ν/U.

The drag a distant wall decides

A sphere moving slowly through a liquid has a drag that belongs to the sphere: six pi times the viscosity, the radius and the speed, and the liquid a few radii away hardly knows it is there. An infinitely long cylinder has no such drag. The equations of slow viscous flow have no solution past it at all — Stokes found this in 1851 — because a force spread over circles instead of spheres never finishes spreading. What rescues the cylinder is whatever distant thing supplies a largest length, and its drag carries the logarithm of that length for ever after.

fluids · Viscosity Flow curves that bend back, and one that stops. Shear stress against shear rate, both on logarithmic axes, for volume fractions 0.5, 0.55, 0.565, 0.575, 0.6, from the same model: stress in units of σ, rate in units of σ over the liquid's viscosity. Below φ = 0.569 every curve rises steadily — the suspension thickens but continuously. Above it and below the frictional jamming point 0.58 the curve bends back into an S: over a range of stress, more stress gives less flow, and a suspension driven at a fixed rate must jump between branches. At φ = 0.6 the curve returns to zero rate at 2.47 σ*: pushed that hard, the suspension stops flowing altogether and holds the stress like a solid.

The suspension that seizes when pushed hard

Cornflour stirred into half its volume of water runs off a spoon like cream, and a fist driven into it bounces off as if it had hit a floor. The particles have not changed and neither has the water. What changes, above a threshold stress, is the kind of contact between neighbouring grains: lubricated, they slide past one another and the suspension can pack to 64 per cent before it locks; pressed into frictional contact, they lock at 58. A suspension sitting between those two numbers is a liquid when handled gently and a solid when handled hard.

fluids · Rheology The triangle of collisions that survive. Every collision of a particle ε above a filled Fermi sea with a partner from the sea, at zero temperature: the partner's energy across, one product's energy up, both from the Fermi level, the other product's fixed by energy conservation. Without exclusion every point in the plotted square would be an allowed collision. With it, the partner must come from below the Fermi level, and both products must land above it: the partner no deeper than ε below, the first product above zero, the second product above zero too. What survives is the shaded triangle, whose area is half of ε squared, and the collision rate is proportional to it. Halve the particle's excess energy and the triangle shrinks to a quarter; at the Fermi level itself it vanishes, and a particle there cannot scatter at all.

The collisions the exclusion principle forbids

A copper wire holds one free electron per atom, packed a quarter of a nanometre apart and repelling one another with the full Coulomb force. It seems impossible that an electron could travel any distance through that crowd without colliding. It travels micrometres — thousands of atoms — before another electron deflects it. The collisions are not weak. They are forbidden: to collide, both electrons need empty states to move into, and in a sea filled to the brim almost none exist. The same rule makes cold helium-3 thicken as the square of how cold it gets.

quantum · Exclusion The same rate for every size of molecule. The diffusion-limited rate constant for two equal spherical molecules meeting in water at 25 °C, against their radius from 0.1 to 100 nm: 8kT/3η = 7.43·10⁹ per molar per second, flat. The dashed line is what a larger target would give if its partner kept the mobility of a 0.3 nm molecule — 7.4·10¹¹ at 30 nm — the intuition that a bigger molecule is easier to hit. Stokes and Einstein's law takes it away again: a molecule twice as big is twice as easy to hit and diffuses half as fast, and the two cancel exactly. Every reaction whose partners react at the first touch runs at close to this one number, whether the partners are ions, small organic molecules or proteins.

The speed limit that does not care how big the molecules are

Two molecules in water cannot react faster than they can find each other, and how fast they find each other is set by diffusion. A bigger molecule is an easier target, so it seems a reaction between big molecules should be faster. It is not: a molecule twice as big is twice as easy to hit and wanders half as fast, and the two cancel exactly. Every reaction that happens at the first touch runs at the same rate, about seven billion per molar per second in water, whether the partners are ions or proteins — a speed limit written by the viscosity of water and the temperature, and nothing else.

thermodynamics · Diffusion The same paste stopping or running, according to the push. A thixotropic paste, sheared hard and then held under a constant stress: its viscosity against time, on a logarithmic axis, for stresses of 1.5, 1.8, 2.0, 2.4 Pa. Its structure builds at rest and is broken by flow, and the model's critical stress is 1.89 Pa. Below it the paste's viscosity climbs without limit: it slows, its structure builds faster than the slowing flow can break it, and it creeps to a stop, its viscosity growing in the end as the 1.5 power of time. Above it the viscosity falls: it speeds up, breaks its structure faster, and runs at a steady rate. After a thousand seconds the viscosities stand at 514.1, 238.3, 2.0, 1.5 Pa s: stresses a few per cent apart have gone opposite ways — a bifurcation, not a smooth change — and the paste hesitates longest when the stress is closest to the critical value.

The paste that forgets it was stirred

A jar of clay slurry left on a shelf sets like a jelly; shaken, it pours. That is thixotropy, and the tempting description — its viscosity falls when it is stirred — misses what makes it strange. A thixotropic paste has no viscosity at a given rate of stirring until it has been told how long it has been stirring, and under a steady push it cannot settle on a slow flow at all. Push slightly less than a critical stress and it creeps to a halt; push slightly more and it runs. Between the two there is nothing, and a paste made to flow slowly splits into a layer that flows and a layer that does not.

fluids · Rheology Blood's viscosity, measured in tubes of different sizes. The viscosity of blood at a haematocrit of 45 per cent, as a multiple of the plasma's, measured in glass tubes of each diameter (the fit of Pries and colleagues to many experiments), against the diameter on a logarithmic axis, with the two-layer model of a core inside a cell-free sleeve 2.5 μm thick, the core 4 times as viscous as plasma (dashed). In wide tubes blood is 3.20 times as viscous as plasma. From about half a millimetre down it gets steadily thinner, to 1.25 times at 6.8 μm — about the size of one red cell — and then climbs steeply, to 5.5 at 3 μm, where cells must fold to get through at all. Which viscosity blood has depends on the tube it is in.

The viscosity that belongs to the tube

Measure the viscosity of blood in a wide glass tube and it is about three times that of the plasma it is made from. Measure it in a tube a tenth of a millimetre across and it is lower; in one a hundredth of a millimetre across, lower still, barely more than plasma's. Then, below about six micrometres, it shoots up. The blood has not changed. The red cells push themselves away from the wall and leave a sleeve of plain plasma a few micrometres thick for the rest to slide in, and in a narrow tube that sleeve is a large part of the flow. A viscosity that depends on the pipe is no longer a property of the fluid alone.

fluids · Rheology The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not.

The wobble that radiating makes larger

Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

astrophysics · Gravitational waves The highest note the air can carry, with height. The frequency at which a sound wave in the air loses its amplitude by a factor e in a single wavelength, in hertz on a logarithmic scale, against height, with the band of human hearing shaded. Because viscosity does not depend on pressure, the collision time grows as the pressure falls, and the ceiling falls with it. It is about 212 MHz at sea level, 0.17 MHz at 50 km and 3 kHz at 80 km. It passes the top of human hearing, 20 kHz, at about 68 km, and the bottom, 20 Hz, at about 110 km; at 100 km it is 92 Hz. Above about a hundred kilometres the air cannot carry anything a person could hear, at any loudness: the molecules are too far apart to pass a compression on before it has dispersed.

The note too high for thin air

Sound in a gas is a compression handed from molecule to molecule by collisions, and it can only be handed on if the molecules collide many times in each period of the wave. At sea level they collide seven thousand million times a second, and the limit is far above anything audible. But a gas's viscosity does not depend on its pressure, so as the air thins with height the time between collisions grows in proportion, and the highest note the air can carry falls with it: two hundred megahertz at the ground, three kilohertz at eighty kilometres, below twenty hertz above a hundred and ten. The upper atmosphere is not silent for want of loudness. It is silent because its molecules are too far apart to pass a sound along.

thermodynamics · Kinetic theory A rod in a shear flow turns in fits and starts. The angle between a rigid rod's axis and the flow direction, unwrapped, against the strain of a steady shear flow (shear rate times time), for spheroids of aspect ratio 1, 5, 20 lying in the plane of the shear, each starting along the flow. A sphere (r = 1) turns steadily at half the shear rate. A rod spends most of its time nearly aligned with the flow, turning slowly, and then flips through half a turn quickly; at r = 5 it is within 10° of the flow direction 46 per cent of the time, at r = 20, 82 per cent. Each flip is half of Jeffery's period, π(r + 1/r) units of strain.

The rod that tumbles for ever in a stream

Drop a short rod into honey being sheared between two plates and it does not settle into line with the flow. It lies nearly along the stream for a long time, then flips end over end in a moment, then lies along it again, and it keeps doing that for as long as the shear lasts. George Jeffery worked out the motion in 1922: every orientation belongs to a closed orbit, every orbit takes the same time, and the flow has no way to choose between them — so a suspension of rods never reaches a steady state, and its thickness rises and falls in time with the tumbling.

fluids · Viscosity A tilt that reaches zero at a definite time. The tilt of the disk against time from 10°, when the losses are set to end the motion after 100 s, for two ways of losing energy: viscous air squeezed out of the thin wedge under the rim (red), which takes energy faster the smaller the tilt, so that α³ falls in a straight line; and a steady rolling resistance at the contact (blue), which takes energy in proportion to the contact rate, so that α to the power 3/2 does. Both fall slowly at first and steeply at the end, and both reach zero tilt at a definite time rather than approaching it for ever: at 99 s, one second before the end, the tilt is 2.15° under the air model and 0.46° under rolling resistance. The disk's energy, (3/2)Mgaα, runs out in a finite time, and with it the tilt.

The disk that spins faster as it stops

A coin spun on a table settles with a whirr that rises in pitch and then stops dead. Everything about it is losing energy, yet the rattle gets faster to the very end. The point where the rim touches the table races round faster and faster as the coin tips flatter, while the coin's face, and the head on it, turns more and more slowly. Two plausible mechanisms of loss both predict that the tilt reaches zero, and the rattle infinity, at a definite instant — a singularity a coin on a kitchen table runs into every time.

mechanics · Rotation The suction that holds two wet plates together. The force needed to pull apart two discs 10 cm across, starting 10 micrometres apart with water between them, at constant opening speeds of 1, 10 and 100 micrometres a second, against time; the dashed line is the largest suction the water can transmit, ½πR²pₐₜₘ = 398 N, reached when the pressure at the centre falls to zero and the water cavitates. At 1 µm/s the force starts at 29.5 N and falls as the cube of the widening gap, to 10.7 N after four seconds, when the gap is 14 micrometres. At 100 µm/s the viscous law would demand 2.9 kN; the water cannot pull that hard, and the force sits on the cavitation limit until the gap has opened enough for the viscous suction to fall below it.

The wet plates that hold on when pulled

Lift a wet glass plate straight up off another and it resists with a force far beyond its weight; slide it sideways and it comes away at a touch. The grip is not surface tension. It is the water that must rush into the widening gap, and the suction it takes to draw it in grows as the inverse cube of the gap — until the water tears open and the atmosphere sets the limit. The same cube makes a flat base float on a wet table where a ball would settle, and gives a micro-machined accelerometer exactly the damping it needs from a few micrometres of air.

fluids · Viscosity Nine drops in eighty-four years. The time between successive drops of the pitch in Thomas Parnell's funnel, cut open in 1930, against the year each drop fell: 8.2, 7.2, 8.1, 8.3, 8.7, 9.3, 12.4, 13.4 years. The first seven intervals average 8.3 years and scatter by about a year either way; after the building was air-conditioned in the late 1980s the next two took 12.4 and 13.4, averaging 12.9. Read as a viscometer through the funnel's geometry, the early rate gives a viscosity of about 2.3 × 10⁸ Pa·s — some two hundred billion times water's.

The drop that takes a decade

In 1927 a physics professor in Brisbane poured heated pitch into a sealed funnel, waited three years for it to settle, and cut the stem. Nine drops have fallen since. The record is a viscometer read about once a decade, and it gives the pitch a viscosity two hundred billion times water's. It also records the building's air-conditioning: after 1988 the drops slowed from eight years apart to thirteen, and a fall of a degree and a half in the average temperature is enough to explain it. The same reasoning, applied to medieval window panes, settles an old story without guessing what glass does at room temperature.

fluids · Viscosity Stopped for half a second, a raw egg starts again by itself. Two eggs spinning at 3 revolutions a second, each touched with a finger at t = 4 s so that its shell is held still for 0.5 s and then released. The cooked egg (solid) stops and stays stopped. The raw egg's shell (dashed) stops too, but its contents (dotted) are still turning — slowed to 2.15 revolutions a second during the hold, by viscosity working against the held shell — and once the finger lifts they drag the shell back into rotation, reaching 1.81 revolutions a second, 60 per cent of its speed before the touch. The table's friction is left out here, so that nothing else slows either egg. It is the oldest kitchen test of whether an egg is cooked, and the whole of the physics is a liquid's reluctance to stop turning.

The egg that keeps turning after it is stopped

Spin an egg on a table, touch it for a moment to stop it, and let go. A cooked egg stays still. A raw one starts turning again by itself, as if something inside had kept going — which it has. A raw egg is two bodies, a thin shell and a liquid that the shell can turn only through the slow grip of viscosity, and the kitchen test that tells them apart is a measurement of how long that grip takes. The same two-body arithmetic says that a raw egg wastes most of the energy of a flick, starts out spinning six times faster than a cooked one given the same twist, and keeps a record of its past rotation in its interior for a second or two after its shell has forgotten.

mechanics · Rotation

Named alongside it

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

DiffusionDissipationMean free pathRheologyShear stressDimensional analysisKinetic theoryPoiseuille flowTransportAngular momentumGlass transitionPressure

All concepts