Field

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

The pressure that only knows depth

A litre of water and a swimming pool press equally hard on a floor at the same depth. Pressure in a still fluid is a scalar with no direction of its own, it depends on how far down and on nothing else, and the shape of the container falls out of the arithmetic entirely.

Force multiplied, distance paid. Two pistons on one body of fluid, of areas in the ratio 16 to 1. A force of 200 N on the small one holds 3.20 kN on the large one, and pushing the small piston 16 cm raises the large one by 10.0 mm. The two products are the same number: nothing is gained except the shape of the bargain.

Force multiplied, and nothing gained

A push on a small piston becomes a much larger push on a large one, in the ratio of their areas, with no machinery in between except the liquid. What the liquid will not do is give anything away — the distances shrink by the same factor the forces grow by, and the product is untouched.

Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

The weight of the water that is not there

A submerged object is pushed up by the weight of the fluid it has displaced — not by something like it, not approximately, but exactly. The reason is that the pressures on its faces do not cancel, and the sum that survives has forgotten everything about the object except its shape.

A heeled hull, and the couple it makes. A rectangular hull of beam 3 m heeled 18°, with the waterline solved so that it displaces the same volume it did upright. The centre of buoyancy has moved 0.244 m to the low side, and weight and buoyancy now act along two lines 0.059 m apart — a couple that turns the hull back upright.

Why a ship comes back upright

Whether a floating body rights itself or rolls over is not decided by its weight, its density or how deep it sits. It is decided by the shape of the slice the water cuts through it, and the number that settles it can be worked out before the vessel is built.

Surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.

The skin that is not a skin

A drop of water behaves as though it were wrapped in a stretched membrane, and there is no membrane. What there is instead is an energy cost per unit of surface, and almost everything the apparent skin does follows from a liquid trying to have less of one.

The small bubble empties into the large one. Two soap bubbles of radius 4 mm and 12 mm joined by an open tube. The excess pressure inside each is 4γ/R — 72.8 Pa and 24.3 Pa — so the 4 mm bubble is at the higher pressure and blows itself into the other. The smaller a bubble gets the harder it pushes, so the process runs away rather than settling: there is no equilibrium anywhere except both bubbles equal.

The small bubble blows up the big one

Connect two soap bubbles of different size and the small one empties into the large one. Everybody expects the opposite, and the reason it happens is one equation with a radius in the denominator — which also means the process runs away rather than settling.

Which wavelength wins. The growth rate of a disturbance on a liquid thread against kR, the circumference divided by the wavelength. Everything to the right of one decays; the maximum sits at kR = 0.697, which is a wavelength of 9.01 radii or 4.51 diameters. That number, and not a property of any particular liquid, is what sets the spacing of the drops a tap breaks into.

The thread that cannot stay a thread

A stream of water from a tap breaks into drops, and it does so at a spacing that is always about four and a half diameters. Nothing chooses that number — it is the wavelength that grows fastest out of a competition between all of them, and it can be computed before any water is poured.

Four tubes, four heights. Water in tubes of radius 0.2, 0.4, 0.8, 1.6 mm, with each meniscus drawn as the spherical cap a 20° contact angle forces and each height computed from it. The narrowest rises 70 mm and the widest 9 mm — in inverse proportion to the radius, with nothing about the glass or the volume of water entering it.

How high water will climb

Water rises up a narrow tube against gravity, and the narrower the tube the higher it goes. The height is set by a curved surface pulling on a circumference while gravity pulls on an area, and the two scale differently — which is the whole of 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.

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.

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.

Speed against wavelength. Phase and group speed for waves on water 4 m deep. Short waves are dispersive — the speed rises as the square root of the wavelength and the group travels at half the phase speed — and long ones all travel at √(gh) = 6.26 m/s together, which is why a tsunami keeps its shape across an ocean while a wind sea spreads out into swell.

The speed that depends on the length

Long waves on water travel faster than short ones, which is why a distant storm arrives as a slow swell and a tsunami crosses an ocean without spreading. One relation covers both, and the two familiar rules taught separately are its two limits.

One volume of liquid, several solids. 4 drops of the same 5 µL of liquid, on 4 solids it meets at 20°, 60°, 90°, 140°. Each is the spherical cap that volume and that angle force, so the footprint radius is computed rather than chosen: 20° gives 2.61 mm, 60° gives 1.71 mm, 90° gives 1.34 mm, 140° gives 0.69 mm. At every contact line the three interfacial tensions are drawn to scale, and only their horizontal components balance — γsv = γsl + γlv cos θ. The vertical pull of the liquid surface is taken up by the solid, which is why the angle belongs to three interfaces at once and to no single liquid: change the solid and nothing about the water has changed.

The angle a liquid makes with what it sits on

Everything capillarity does — climbing, beading, wicking, waterproofing — is the sign and size of one cosine, and that cosine belongs to three interfaces at once rather than to the water. Change the solid and nothing about the water has changed, yet the same five microlitres goes from a footprint 2.61 mm across to one of 0.69 mm.

The surface a spin decides

Spin a dish of liquid and its surface settles into a paraboloid — exactly, with nothing about the liquid in the shape. A parabola of that form has a focal length of g over twice the spin rate squared, so a bucket of mercury turning at twenty revolutions a minute is a telescope mirror figured by a clock instead of by grinding.

$Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting.$

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

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

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.

Where a drying drop loses its liquid. The rate at which liquid leaves the surface of a drying drop, against distance from the centre in units of the drop's radius, for 5 contact angles. The flux is not uniform, and it is not a property of the liquid: it is set by how vapour diffuses away from a lens-shaped object, which is the same boundary-value problem as the field around a charged lens and has the same answer — a power law in the distance from the rim, with an exponent that depends only on the contact angle. at 10° the exponent is 0.471, and the loss has doubled by 87.8 per cent of the way out, at 40° the exponent is 0.357, and the loss has doubled by 92.5 per cent of the way out, at 70° the exponent is 0.182, and the loss has doubled by 98.9 per cent of the way out, at 90° the exponent is 0.000 and the drop dries evenly everywhere, at 120° the exponent is -0.500 and the flux falls toward the rim. Below a right angle the flux diverges at the contact line; at exactly a right angle it is uniform; above it the edge is the slowest-drying part of the drop. Since a pinned edge must be resupplied from the interior, that sign decides which way the liquid inside the drop flows — and therefore whether everything suspended in it ends up in a ring at the rim or in a spot at the centre.

The ring the drop leaves behind

A drop of coffee dries into a ring rather than a disc, and nothing about coffee is responsible. The pattern is produced by a boundary condition — an edge that cannot move — and it survives replacing the coffee with anything else that will stay suspended.

The height a siphon cannot pass

A siphon will not lift water more than about ten metres, and the usual explanation for the limit is also given as the explanation for the mechanism. It cannot be both. A siphon runs in a vacuum, with degassed water, over a crown no atmosphere could support.

The same block, one of them with no upthrust at all. Two identical blocks 0.8 m tall with their tops 1.2 m under the surface, drawn with the pressure on every wetted face at its true relative size. On the right the block is clear of the floor and the pressure on its underside exceeds that on its top by 7.8 kPa, which is ρgh and is exactly Archimedes' 7.8 kPa. On the left the bedding is perfect and there is no water under it, so nothing pushes up: the resultant is 11.8 kPa downward and the block presses on the floor with more than its own weight. Buoyancy is not something the fluid has. It is what the bottom face is doing, and a face the fluid cannot reach does nothing.

The block the water does not lift

A block bedded flat on the bottom of a tank, with no water underneath it, feels no upthrust at all. It is fully submerged, Archimedes' principle is not suspended, and it presses on the floor with more than its own weight — because buoyancy is not something a fluid has, it is what the bottom face is doing, and a face the water cannot reach does nothing.

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.

The stress that stops growing with depth. Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.

The silo that does not weigh what it holds

Pour water into a tall vessel and the pressure at the bottom is the depth times the density times g, whatever the shape above it. Pour grain in and the floor stops learning anything new after the first couple of metres, because the walls have quietly taken the rest — and the length over which they take it contains no property of the grain at all.

The column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

The pressure that comes from counting

Dissolve a teaspoon of anything in a litre of water, put a membrane between it and pure water, and the solution will hold up a column of water two and a half metres tall. Nothing is pulling. The pressure does not depend on what was dissolved, only on how many particles it made — which is the ideal gas law, with the solute in place of the gas.

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.

A pore that lifts 100 m has to be 0.1 µm or finer. Capillary rise against pore radius, both logarithmic, for a liquid of surface tension 72.8 mN/m at a contact angle of 20°. The relation is a straight line of slope −1 — halve the pore and double the rise — and the two horizontal marks are the height in question, 100 m, and the 10.3 m that one atmosphere supports. 0.01 µm lifts 1394.7 m; 0.1 µm lifts 139.5 m; 1 µm lifts 13.9 m; 5 µm lifts 2.8 m; 20 µm lifts 69.7 cm; 50 µm lifts 27.9 cm. The conducting vessels of a tree are tens of microns across and lift under a metre; the pores in the membranes between them are tens of nanometres and would lift kilometres. Those are the same expression at two scales, and only one of them is a pipe.

The column that is pulled, not pushed

A capillary fine enough to lift a hundred metres is far too fine to carry any flow, and one wide enough to carry the flow lifts under a metre. Neither is how the water gets up a tree. The column is under tension — an absolute pressure of −0.88 MPa at the top, which a gas cannot have — held together by cohesion and prevented from tearing by pores a few tens of nanometres across.

The layer a parcel cannot leave

Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.

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.

The cross a shaken cylinder leaves in a stratified fluid. The beams radiated by a small body oscillating in a fluid of buoyancy frequency 1.053e-2 per second — a period of 9.9 minutes — at 0.3, 0.6, 0.9 times that frequency. The disturbance does not spread in circles. It leaves along four rays, and the angle of those rays to the horizontal is fixed entirely by the ratio of the driving frequency to the buoyancy frequency: 17.5° at 0.3N, 36.9° at 0.6N, 64.2° at 0.9N. Nothing about the size of the body, the amplitude of the shaking or the wavelength enters. Drive it faster and the beams stand up; drive it slower and they lie down; drive it above N and there are no beams at all, because the dispersion relation ω = N cos φ has no solution. The short arrows across each beam are the wavevector, which is perpendicular to the beam — the dot products drawn here are 6e-17 — so the crests travel sideways across the ray while the energy travels along it, and a fluid doing this looks, in a photograph, as though its waves are moving at right angles to where they are going.

The wave that picks an angle

Shake a rod slowly in a tank of salty water layered by density and the disturbance leaves along four straight beams, at an angle fixed entirely by how fast the shaking is. Change the wavelength and the angle does not move. Change the frequency and it does. Above the buoyancy frequency there is no wave at all.

The slope a heap of grains settles at. The steepest slope a cohesionless heap can hold, against the friction between its grains. A slab of thickness h on a slope is pushed down it by the weight's along-slope component and held by friction acting on the weight's across-slope component, and both are proportional to the same ρgh — so the thickness cancels and the criterion is tan θ = μ. Checked here at thicknesses of 2 mm, 50 mm, 2000 mm, the ratio of driving to holding stress differs by 2.2e-16, which is zero. That is why the quantity is an angle: nothing about the size of the pile, the size of the grains, the density or the strength of gravity survives into it, and a heap of sand on the Moon stands at the same slope as one on Earth. The marked materials are glass beads at 24°, dry sand at 33°, crushed gravel at 40°. The band between the two curves is the hysteresis: a slope steeper than 31.0° will keep flowing once started, and one shallower than 35.0° will not start — so a pile has a range of stable angles rather than one, and which it is found at depends on how it was built.

The angle that does not know the size of the heap

Pour sand and it makes a cone with a definite slope, and pouring more makes a larger cone with the same slope. The reason the answer is an angle rather than a length is a cancellation — the force pulling a surface layer downhill and the friction holding it back are both proportional to its weight, so everything about the size of the pile divides out — and everything that puts a length back in is a story about cohesion.

What it costs to take the salt out

Salt dissolves in water because mixing is overwhelmingly the more probable arrangement, and separating the two again means paying back what the mixing gave away. The bill can be computed before any apparatus is chosen — 0.79 kilowatt-hours for the first cubic metre from seawater — and it rises with every further cubic metre taken, because what is left behind is saltier than what was started with.

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.

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.

Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains.

The heap that becomes a solid

Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else.

Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn.

The angles a film has no choice about

A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.

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.

$The speed of a wave that carries no pressure. Second-sound speed against temperature, computed from the two-fluid equations with the normal component treated as a phonon gas — which it is below about six-tenths of a kelvin. The upper line is ordinary sound at 238 m/s, which moves the two components together. The lower curve is the other mode, and at low temperature it sits at 137.4 m/s, which is 238/√3 to three figures: a result with no adjustable constant in it, and the reason to believe the two-fluid model rather than merely to use it. What oscillates in this wave is not the density — the two components move in opposite directions and their sum stays put — but the fraction that is normal, which is a temperature. So a temperature disturbance in helium II propagates, with a speed, a reflection and a resonance, where in every ordinary liquid it diffuses and has none of those. Above a kelvin the rotons take over from the phonons and the measured curve falls to about 20 m/s; the model here is the low-temperature one and it is drawn only where it holds.$

The heat that arrives as a wave

Two fluids with two velocities give two wave equations, not one. In the first the components move together and the density oscillates, which is ordinary sound. In the second they move oppositely, the density stays put, and what oscillates is the temperature — so a heat pulse in liquid helium has a speed, a front and a reflection.

The two holes a grain can fall through, and their exact sizes. Three equal spheres in contact, and four, drawn with the largest sphere that passes between them. The numbers are geometry and nothing else. Three mutually touching spheres put their centres on an equilateral triangle of side 2R, whose circumradius is 2R/√3, so the gap admits a sphere of radius 0.154701R — about a seventh. Four in a square admit 0.414214R, nearly half. A real packing contains both arrangements and everything between, so a grain smaller than the first threshold gets through everywhere, one larger than the second gets through nowhere, and one in between percolates slowly through the loosest routes. That is the whole size-dependence of segregation by percolation, and it is why the effect is reliable below about a seventh and erratic between a seventh and a half.

The big one comes to the top

Shake a jar of mixed grains and it sorts itself, which is the opposite of what shaking a mixture of gases does. There is no thermodynamic paradox in it because there is no temperature to speak of — and the mechanism is a piece of geometry with an exact number in it: a sphere fits through the gap between three touching spheres only below a radius ratio of 0.1547.

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.

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.

$The hourglass that keeps time. Discharge against how much is left above the opening, for grain and for liquid through the same 40 mm hole, each as a fraction of its own rate at a full hopper. The grain rate is a horizontal line: it does not appear in Beverloo's law at all, because the pressure at the outlet does not depend on the head — the walls carry the weight, which is what Janssen's argument establishes, so the grains at the opening are pushed by their immediate neighbours and by nothing else. The liquid falls as the square root of the head and is down to 32 per cent by the time a tenth is left. That is why an hourglass keeps time and a water clock does not, and why the water clocks that worked were built with a float and an overflow to hold the head constant.$

The hourglass that keeps time

Grain leaves a hopper at a rate that does not depend on how much is above it, and that goes as the orifice to the five-halves power rather than the one half a liquid gives. Both facts follow from the same thing: the weight is carried by the walls, not by the grains at the opening.

$The fraction of the pressure a membrane can hold. The osmotic pressure actually developed across a membrane against a 300 mol/m³ solution at 298 K, as the reflection coefficient runs from a membrane the solute passes freely to one it cannot pass at all. The line is straight with the van 't Hoff pressure 743.69 kPa as its slope — checked against the drawn line — because the coefficient enters as a simple factor. At σ = 1 the pressure is 743.69 kPa; At σ = 0.6 the pressure is 446.21 kPa; At σ = 0.2 the pressure is 148.74 kPa. Van 't Hoff's law is the ceiling rather than the answer, and a membrane's coefficient against a given solute is as much a property of the pair as the concentration is of the solution.$

The membrane that almost holds

Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper.

One line decides whether it climbs for ever. The wetting condition for a corner, drawn as a map. The horizontal axis is the corner's half-angle and the vertical axis the liquid's contact angle with the walls; the diagonal is θ + α = 90°. Below it the meniscus in the corner curves into the liquid, the capillary pressure grows without bound as the corner narrows, and the liquid wicks along it indefinitely. Above it the curvature has the other sign and the liquid stays put. The boundary is tested by evaluating the meniscus a millionth of a degree either side of it at seventeen half-angles, and it wicks on one side and not the other every time. water on clean glass, right-angled corner: α = 45°, θ = 5° — wicks; water on glass, a 20° groove: α = 10°, θ = 40° — wicks; water on plastic, right-angled corner: α = 45°, θ = 75° — does not. A right-angled corner needs a contact angle under 45°, which water on clean glass has and water on most plastics does not.

The corner a liquid never stops climbing

A narrow tube lifts a liquid to a definite height because it has a smallest width. A corner has none, so the capillary suction it can develop is unbounded — and whether the liquid takes advantage is decided by a single inequality between the contact angle and the corner's own angle.

The pressure a charged gel develops. The swelling pressure against the gel's fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts with different slopes, and both limits are checked against the solution rather than read off the plot. Where the fixed charge is small compared with the salt the pressure goes as the square of the charge divided by four times the salt: the bath's ions screen the fixed ones, and doubling the salt halves the pressure. Where the fixed charge dominates, every counterion it demands is an extra particle in the gel and the pressure goes as the charge itself. At 15 mM salt, a gel with 200 mM of fixed charge develops 427 kPa; At 50 mM salt, a gel with 200 mM of fixed charge develops 306 kPa; At 150 mM salt, a gel with 200 mM of fixed charge develops 150 kPa; At 500 mM salt, a gel with 200 mM of fixed charge develops 49 kPa. Cartilage carries a fixed charge of a couple of hundred millimolar from the sulphated sugars on its proteoglycans, sits in a bath of about 150 mM, and develops a swelling pressure of an atmosphere and a half — which is what holds a joint apart and carries the load across it.

The swelling a membrane cannot stop

Give the thing a membrane holds back an electric charge and the small ions that can cross are no longer free to distribute themselves. Two conditions — neutrality on each side, and equal chemical potential for the salt — fix where every ion goes, and they leave the charged side with more particles than the other. That excess is what holds a joint apart, and it is why a cell has to spend a third of its energy pumping.

$The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly.$

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

$A hundred thousand taps, and still not finished. The packing fraction of a column of grains against the number of taps it has been given, on a logarithmic horizontal scale, for several tap intensities. The grains start where pouring leaves them, around 0.55, and climb towards something near 0.64. At an intensity of 1.2 the packing reaches 0.6318 after a hundred thousand taps; At an intensity of 2 the packing reaches 0.6327 after a hundred thousand taps; At an intensity of 3 the packing reaches 0.6338 after a hundred thousand taps, which is still 0.0097 short of the asymptote. The shape is what matters. On a logarithmic axis the curve is close to a straight line over four decades, which means the packing improves by about the same amount for each factor of ten in the number of taps — not for each additional thousand. Going from a hundred taps to a thousand buys as much as going from a thousand to ten thousand. An exponential relaxation is over after a few time constants and this is not one. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure.$

The pile that is never finished settling

Tap a jar of grains and it settles. Keep tapping and it goes on settling — logarithmically, so that each factor of ten in the number of taps buys the same small improvement as the last. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure, and the asymptote everyone quotes is a fitted number rather than a measured one.

The pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream.

The wave that is required to stand still

A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.

A reflection that keeps the angle to gravity and not to the wall. An internal wave beam of frequency 0.5N reflecting from a slope of 12°, with the wavelength ratio for slopes of 12°, 20°, 28° computed beside it. The frequency fixes the angle the energy makes with the horizontal — 30.0° here — because the restoring force is gravity and gravity is vertical, so the reflected beam must leave at that same angle whatever the wall is doing. Incident and reflected rays are therefore not mirror images, and the wavelength changes on reflection by sin(θ+α)/sin(θ−α). A flat floor gives one, checked exactly. A slope approaching the ray's own angle gives infinity, checked as the limit, and that is where a basin's internal tide is compressed until it breaks.

The reflection that changes the wavelength

An internal wave's frequency fixes the angle its energy makes with gravity, so a sloping wall cannot send it back the way a mirror would. The reflected beam leaves at the same angle to the vertical rather than the same angle to the wall, its wavelength changes by a factor that diverges when the slope matches the ray, and in a closed basin the changes accumulate until every ray in the fluid lies on one line.

The pore size the air decides. The largest pore radius that holds condensed water in equilibrium with air at a given relative humidity, for water at 25 °C, on a logarithmic radius axis. A concave meniscus lowers the vapour pressure over it by exp(−2γVₘ/rRT), so a pore whose meniscus would be tighter than a radius set by the humidity is in equilibrium only when full. The upper curve is the emptying condition, through a hemispherical meniscus with two curvatures; the lower is the filling condition, through the cylindrical film that lines a pore before it closes, with one — so the same pore fills at a higher humidity than it empties at. At 50% humidity a pore empties below 1.51 nm and fills below 0.76 nm; at 90% humidity a pore empties below 9.96 nm and fills below 4.98 nm; at 99% humidity a pore empties below 104 nm and fills below 52 nm. The radii run from molecular at low humidity to a tenth of a micrometre at 99 per cent, and every one was checked by putting it back into the vapour-pressure relation.

The pore that fills from dry air

Water condenses when the air is saturated — on a flat surface. Over a curved one the vapour pressure is different, higher over a drop and lower over a meniscus, and in a pore a few nanometres across it is low enough that the pore fills with liquid from air at half humidity. The water it holds is under a tension of a hundred megapascals, and the pore empties at a lower humidity than it filled at, for a reason that needs no roughness at all.

An angle set by a voltage. The contact angle of water on a fluoropolymer coating with a rest angle of 115°, against the voltage across the coating, for coatings 1 and 3 µm thick with a relative permittivity of 1.93. The Lippmann–Young law cos θ = cos θ₀ + ε₀εᵣV²/2dγ makes the cosine rise as the square of the voltage, so the curve is symmetric about zero volts and steepest where the angle is near 90°. For 1 µm the law reaches complete wetting at 110 V; for 3 µm the law reaches complete wetting at 191 V. It does not get there. In most reported experiments the angle stops falling somewhere in the shaded band and then stays put however high the voltage is raised — contact-angle saturation — and why is still argued: charge trapped in the insulator, ionisation of the air at the sharp edge of the drop, and the thermodynamic stability of the contact line have all been proposed and none accounts for every case.

The angle a voltage can set

A contact angle is treated as a fact about three materials — a solid, a liquid and the air — fixed the moment they are chosen. Put a voltage across a micrometre of insulator under a drop and the angle falls as the square of the voltage, with nothing about the materials changed. Drops can be steered across a chip with no pump, and a lens can focus with no moving part. The law that describes it is exact in its model, and real surfaces stop obeying it at an angle nobody has fully explained.

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.

$Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it.$

The counterions that never leave the chain

Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

The latitude past which a tide cannot split

The ocean's internal tide carries about a terawatt, and somewhere it has to be broken into waves small enough to mix the water. One of the ways it breaks is by pumping waves at half its own frequency, the way a child on a swing pumps at twice the swing's. Those half-frequency waves cannot exist where the planet's rotation forbids oscillations that slow — poleward of 28.8° for the semidiurnal tide — so the route has an edge on the map, fixed by the Moon's period and the Earth's spin.

A body at an interface, and the difference that holds it. The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary — each curve scaled by its own largest value so that three very different cases fit on one axis. The displaced weight has to be counted twice, once with each density, so the force is linear in the displacement with a stiffness set by gravity, the cube's cross-section and the difference of the two densities — the difference, and neither of them alone. air over water holds the cube with its underside 60 per cent of the way through, at a stiffness of 97.8 newtons per metre; oil over water holds the cube with its underside 47 per cent of the way through, at a stiffness of 14.5 newtons per metre; water over mercury holds the cube with its underside 24 per cent of the way through, at a stiffness of 1229.4 newtons per metre. Both numbers are read off the drawn curve rather than substituted. The consequence is the one worth the figure: a body at an oil–water interface is held six times less stiffly than the same body at an air–water one, because the density difference is six times smaller, and the ordinary intuition that a denser fluid holds a body more firmly is exactly wrong — what matters is the contrast across the surface the body is sitting in.

The body that displaces two things

Every earlier argument has a body wetted by one fluid, so the displaced weight is a volume times a density. A body at an interface displaces two, and what holds it is the *difference* between them — so the same block is held six times less stiffly at an oil–water boundary than at an air–water one. In a continuously stratified column the neutral depth becomes stable, which is the exact opposite of the compressible case.

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.

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.

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.

$A thrust that cannot push past the hump. The resistance a hull meets against its speed in knots, in two layers whose fastest interfacial wave travels at 1.02 knots: ordinary friction rising as the square of the speed, plus the drag of the interfacial waves, whose hump sits just below the wave speed. The horizontal lines are 2 steady engine thrusts. A ship settles where its thrust meets the resistance curve. Thrust 0.6 meets it at 0.81 knots; Thrust 1.3 meets it at 0.94, 1.00, 1.82 knots. A ship accelerating from rest reaches the first crossing and stops gaining speed there, below the hump, even when a faster crossing exists beyond it — which is the dead water sailors reported, a ship held to a fraction of its usual speed by a wave it cannot see.$

The wave that holds a ship back

In 1893 the polar ship Fram, which could make four or five knots, was held to about one in a calm Arctic sea with nothing visible in the water. The sea was layered — a metre or two of fresh meltwater over salt — and the ship was making a wave on the boundary between the layers, a wave that travels at about a knot and carries away almost all of a slow ship's power. Below that speed the drag is a hump no steady thrust can climb; above it the wave cannot keep up and the drag falls away.

Mean field always pushes; correlations pull. The pressure between two planes of equal charge with only their own counterions between them, against their separation, in units of the Gouy–Chapman length μ for the separation and 2πℓ_Bσ²kT for the pressure. The upper curve is the Poisson–Boltzmann result, solved from k·tan(kd/2) = 1: it is the density of counterions at the midplane and is positive at every separation, falling from the ideal-gas 2/d at contact to π²/d² far apart — 1.71 at 1μ, 0.290 at 4μ. The lower curve is the strong-coupling limit, 2/d − 1, which the same ions reach when their valence and the surface charge are high. It crosses zero at d = 2μ and is negative beyond, tending to −1: the two like-charged planes attract, and the separation 2μ is where they come to rest.

The like charges that pull together

Two surfaces carrying the same charge, with nothing between them but the ions that neutralise them, ought to repel, and the standard mean-field theory proves that they always do. With calcium or spermine as the counterions they attract, and come to rest a fraction of a nanometre apart. The mean field misses it because it averages the ions into a smooth cloud, and multivalent ions are too strongly repelled by each other to form one. Each keeps a patch of surface to itself, and the pressure between the plates becomes a single ion's business.

A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it.

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

$A gas of grains that lose 51 per cent of their energy per collision, cooling into clumps. 2000 discs in a square box with periodic walls, covering 25 per cent of its area, started with random velocities and left alone. Each collision keeps a fraction e = 0.7 of the relative velocity along the line of centres. The panels are the same gas at three temperatures, where temperature means the mean kinetic energy of a grain: 0.85 T₀, 0.019 T₀, 0.00049 T₀. It starts uniform — the grain counts in a 10 × 10 grid have a variance 0.3 times their mean, more even than scattered points because discs cannot overlap — and ends in dense bands and clumps with empty space between, the variance 17 times the mean. Nothing attracts anything. The clumps form because a dense patch collides more often, cools faster, loses pressure and is squeezed denser by the hotter gas around it.$

The gas that cools itself into clumps

Shake a box of grains hard enough and they fly about like the molecules of a gas. Stop shaking and the gas cools, because every collision destroys a little of the motion — and it does not cool evenly. A denser patch collides more often, cools faster, loses pressure, and is squeezed denser by the hotter gas around it. With nothing attracting anything to anything, the gas gathers itself into clumps and bands, and in the limit three grains can collide infinitely many times in a finite time.

Water gets denser the deeper it is. The density of seawater at 0 °C and a salinity of 35 against depth, from the surface to the 10.99 km of the Challenger Deep, with the pressure integrated down the column from the equation of state rather than assumed. At the surface it is 1028.1 kg/m³. At the ocean's mean depth of 3.68 km it is 1045.3, 1.67 per cent more; at the bottom of the deepest trench it is 1076.5, 4.71 per cent more. An incompressible ocean would be the vertical line. The curve is almost straight; what curvature it has, too little to see here, is the water stiffening as it is squeezed: its secant bulk modulus rises by 3.24 pascals for every pascal of pressure.

The sea that stands lower because water gives

Water is the textbook incompressible fluid, and at the bottom of the ocean it is 4.7 per cent denser than at the top. That squeezing lowers the whole sea by about thirty metres, adds the weight of 266 metres of water to the pressure in the deepest trench, and speeds sound up so steadily with depth that it digs a channel in which a low note can cross an ocean.

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

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.

The thin parcel that keeps sinking. A parcel pushed one centimetre down a column in which heat stabilises and salt destabilises, with a density ratio of 2, integrated for three hours, for parcels whose pattern of sinking and rising water repeats every 2 m, 20 cm, 6 cm. The 2 m parcel keeps its heat and its salt and swings back and forth with the buoyancy period of 23.6 minutes. The 20 cm parcel crosses its starting level 8 times while its swing dies away, and is left sinking slowly, at the 0.48 e-folds an hour its width allows; the 6 cm parcel never comes back and passes five centimetres below it after 60 minutes. The thinnest loses its heat to its surroundings in 11 minutes and its salt in 18 hours, so it arrives salty and cooled, heavier than the water around it, and runs away at 2.33 e-folds an hour — the rate the linear dispersion relation gives for that width.

The salt that sinks through a stable sea

Warm salty water lying on cold fresh water is heavier at the bottom at every depth, and by every test of static stability it should stay where it is. It does not. Heat leaks sideways out of a thin tongue of sinking water a hundred times faster than salt does, so the tongue arrives cooled and still salty — heavier than its new surroundings — and keeps going. A column can be stable by density and unstable by diffusion, and most of the subtropical ocean is.

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

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.

The sand that chooses a side

Divide a box of grains into two compartments with a slot in the wall between them and shake it from below. Shaken hard, the grains share themselves equally, as molecules of a gas would. Shaken a little more gently, they gather in one compartment and leave the other almost empty, and the crowded side is cold and the empty side hot. Nothing sorted them. Every collision between grains loses energy, and a compartment that happens to hold a few more grains loses more of it and stops throwing grains over the wall.

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

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.

The deepest water reads warmer than the water above it. A column of seawater 10.9 km deep whose potential temperature — the temperature each parcel would have if brought to the surface without gaining or losing heat — falls from 20 °C at the top to 1 °C in the deep, and the temperature a thermometer lowered into it would read. The two agree near the surface and part company with depth: the thermometer reads a minimum of 1.38 °C near 3.9 km and then rises to 2.35 °C at the bottom, although the water there is the same water, with the same potential temperature, 1.00 °C, as the water two kilometres higher. The warming is compression alone; no heat reaches the deep trench from anywhere.

The water that warms on the way down

Lower a thermometer into the Mariana Trench and it reads colder and colder for four kilometres and then starts to warm, reaching about two and a half degrees at the bottom, eleven kilometres down. No heat is coming up from the sea floor to do it. The water at the bottom is the same water as the water two kilometres higher, carried down and squeezed, and squeezing warms it. In fresh water near four degrees the same squeezing does nothing at all, and below four degrees it cools — so a parcel of cold lake water pushed deep enough keeps sinking on its own.

Chains counted, and then blobs. The osmotic pressure of a polymer solution in a good solvent against concentration, on logarithmic axes, for chains of 1,000, 10,000, 100,000 segments. Dilute, each chain is one particle and the pressure is van 't Hoff's, proportional to the number of chains: a line of slope one, lower for longer chains by the ratio of their lengths. Past the overlap concentration of each, marked, the coils interpenetrate and the curves bend up onto a single line of slope 2.31 that does not depend on chain length at all. The overlap concentration falls as the chains lengthen, by a factor of 5.8 for each tenfold increase in length, which is why the longest chains leave the counting of molecules first.

The pressure that stops counting molecules

Van 't Hoff's law counts dissolved particles, and for a dilute polymer each chain is one particle, however many thousand segments it has. That is how osmometers weigh polymers. Concentrate the solution until the coils overlap and the count changes. The pressure stops depending on how long the chains are and grows as the 2.3 power of the concentration, the same for chains a thousand segments long and a hundred thousand. The unit being counted is no longer a molecule. It is a stretch of chain as long as the mesh of its neighbours allows, and it has no chemical identity at all.

$The pull of a meniscus against the stiffness of a wall. For two neighbouring lines of a wet pattern leaning towards each other, the net Laplace pressure pulling them together (heavy curve) and the bending stiffness pushing them back (straight lines), against the lean as a fraction of the half-gap. The pull grows faster than linearly, because narrowing a gap raises its suction, and it diverges as the gap closes. The stiffness is a straight line whose slope falls as the fourth power of the line's height; the lines drawn are for 0.8 times, 1 times, 1.25 times the critical height. At 0.8 of the critical height the straight state is stable and an unstable balance lies at a lean of 0.77 of the half-gap, a barrier the lines must be pushed over; at the critical height the two start with the same slope and the straight state is marginal; at 1.25 times the straight state is unstable and the pair collapses.$

The walls a rinse pulls together

Every result about capillary rise assumes the walls hold still. Make them thin enough and they do not. The water between two walls is below atmospheric pressure, so it pulls them together, and as the gap narrows the suction grows. Beyond a critical height or length the walls cannot push back and they snap shut. It is why wet hair clumps, why a paintbrush comes out of water as a point, and why a chip factory cannot rinse lines of resist taller than about three times their width without laying them flat.

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.

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.

Why the mixture of two equal waters is heavier than either. A temperature–salinity diagram for seawater near the surface, with lines of equal density (curves) 0.3 kg/m³ apart. Two waters lie on the same line: one at 4 °C and 33.93 g/kg, the other at 16 °C and 36.53 g/kg, both 1026.00 kg/m³. Mixing them in any proportion moves along the straight line between them, because temperature and salt add linearly; the lines of equal density curve, because cold water's density responds less to temperature than warm water's. The straight line lies on the dense side of the curve. An even mixture, at 10 °C and 35.23 g/kg, is 0.182 kg/m³ denser than either parent, and sinks between them.

The mixture heavier than either water

Two bodies of seawater can have exactly the same density — one cold and fresh, one warm and salty — and sit side by side without either wanting to sink. Mix them, and the mixture is denser than both, and sinks. Nothing has been added and nothing removed; the extra weight comes from the fact that water's density is a curved function of temperature while mixing averages temperature in a straight line. The same curvature, at its most extreme, is why a lake overturns twice a year and freezes from the top, and why the open ocean, too salty to have a density maximum above freezing, does neither.

Five square logs of different densities, as they float. Cross-sections of five long square logs in their stable floating positions, with relative densities 0.1, 0.25, 0.5, 0.75 and 0.9, each drawn with its water line and its submerged part shaded, and a dot at its centre. The lightest floats flat and high; at 0.25 it has tilted to about 27°; at 0.5 it floats on its corner with a diagonal vertical and the water line across the middle; at 0.75 it tilts to the same angle as at 0.25, and at 0.9 it floats flat again, low in the water. The heavy logs are the light ones turned upside down, with air and water exchanged.

The log that floats on its corner

A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.

Like menisci attract, unlike menisci repel. Cross-sections of a water surface with two small floating bodies, drawn with menisci of the shape K₀(r/ℓc) that a small floater makes, exaggerated vertically. Left: two light bodies, each pulling the surface up round itself as buoyancy pushes it up against the surface; each sits on the other's slope and is drawn towards it, as a bubble rises — the two climb together. Right: a light body that raises the surface beside a heavy one that depresses it; each sits on the other's slope the wrong way and they drift apart. The deformations reach a few capillary lengths, 2.71 mm for water, and the rule is the whole of the Cheerios effect: cereal clumps, and cereal gathers at the wall of a bowl the milk wets.

The cereal that gathers at the edge of the bowl

The last few pieces of cereal in a bowl of milk clump together and drift to the side of the bowl. Nothing is pulling on them but the milk's surface. Each floating piece bends the surface round itself, and a second piece sitting on that bend has somewhere to slide: like menisci attract, unlike menisci repel, and the reach of the force is a few capillary lengths, 2.7 millimetres for water. The arithmetic is the arithmetic of charged lines in two dimensions, screened by gravity, and it says why leaves gather on a pond, why bubbles on coffee drift to the cup's rim, and why particles smaller than about ten micrometres do not clump this way at all.

The lift of a zero-pressure balloon as it climbs. The gross lift, in tonnes, of a zero-pressure balloon whose envelope holds at most 1.1 million cubic metres, against height, for launches with 3,500 and 5,000 cubic metres of helium; the dashed line is its 3 tonnes of envelope, payload and ballast. At launch the envelope is mostly empty. As the balloon climbs the gas expands exactly as fast as the air thins, so the mass of air it displaces stays the same and the lift stays at 3.69 or 5.28 tonnes all the way up. The envelope fills at 37.5 km with the larger fill and 39.9 km with the smaller; above that, gas vents through the open base, the volume is fixed and the lift falls with the air's density. Both reach the same float, 41.3 km, where the lift has fallen to the weight. More helium only makes the climb faster; the ceiling is set by the envelope's size and the mass it carries.

The balloon that floats at a density

A helium balloon let go rises until something stops it, and what stops it depends entirely on how it is built. A party balloon or a weather balloon stretches until it bursts. An open-bottomed science balloon climbs at constant lift — its gas expanding exactly as fast as the air thins — until its envelope is full, then floats where the air is thin enough. A sealed, unstretchable balloon floats where its whole density equals the air's and bobs back if pushed away. None of them floats at a height; each floats at a density. The difference between the three is how compressible the balloon is compared with the air around it, the same comparison that decides whether a Cartesian diver sinks.

The rise a watt-year buys at each depth. The rise in sea level, in millimetres, from one watt-year per square metre of heat (3.16 × 10⁷ joules) stored in a thin layer at each depth, for two water columns: a subtropical one, 24 °C at the surface cooling to 1.5 °C in the deep, and a polar one, 1.5 °C at the surface and 0.5 °C below. In the warm column the yield is 2.36 mm at the surface, falls to 1.02 mm near 2.0 km where the water is cold but the pressure still modest, and recovers to 1.49 mm at 6 km as pressure takes over. In the polar column the same heat buys 0.63 mm at the surface and 1.13 mm at 4 km. A joule stored in cold, deep water raises the sea about twice as much as one stored in the cold water above it.

The rise a joule of heat buys

Warm a column of seawater and it stands taller without weighing any more, so the sea rises while the pressure on the sea floor stays exactly the same. How much it rises depends on where the heat goes. Seawater's expansion coefficient grows with temperature and with pressure, so a joule stored in warm tropical water lifts the sea more than six times as far as the same joule in Arctic surface water. Cold water in the deep, squeezed by four kilometres above it, expands about twice as much per joule as the same cold water at the surface. That is why the budget of sea-level rise turns on thousandths of a degree in the least-measured part of the ocean.

A cell swells until its membrane runs out. The volume of a red cell of 90 fL against the salt concentration of the solution round it, in per cent sodium chloride by weight, with 0.9 per cent isotonic. By the Boyle–van 't Hoff law the water inside follows the outside's tonicity: the cell's volume is an inactive part, 40 per cent of the normal volume in haemoglobin and other solids, plus a part inversely proportional to the concentration. Its membrane's 135 μm² can enclose at most 147.5 fL, as a sphere (dashed). The swelling curve reaches that at 0.44 per cent salt: below it the cell has no more shape to give up and bursts. In a 1.5 per cent solution the same cell shrinks to 68 fL and crenates, with membrane to spare.

The cell that bursts by geometry

Put a red blood cell in salt water weaker than blood and water flows in. The cell does not stretch to take it, because its membrane will not stretch: it can change shape almost for nothing and change its area by about two per cent before it tears. So the cell swells by changing shape, from a dimpled disc towards a sphere, and when it reaches the sphere its membrane can enclose, it has no more shape to spend. That happens at about 0.44 per cent salt, half the strength of blood, and the clinical test that measures it — called a test of fragility — turns out to measure a ratio of area to volume, not the strength of anything.

A parcel that leaves the air behind. A parcel of air lifted from the ground at 30 °C with a dew point of 23 °C, against height, beside the air it rises through, which cools at 6.5 K per kilometre to the tropopause at 15 km. Unsaturated, the parcel cools at the dry rate and is soon colder than its surroundings; at 890 m it reaches saturation, a cloud base, and from there cools along the moist adiabat. From 2080 m it is warmer than the air round it and accelerates upward, until at 13.4 km it is cold again. The shaded area, the buoyancy integrated over that climb, is the convective available potential energy: 2543 J/kg. If all of it went into motion the updraft would reach 71 m/s, the square root of twice the CAPE. The 99 J/kg below the level of free convection is the push it needs to get started.

The cloud that cools more slowly than the air

Dry air carried upward cools at 9.8 degrees per kilometre, and the atmosphere, which cools at about 6.5, is stable against it: a dry parcel nudged upward ends up colder than its surroundings and sinks back. Saturated air is different. As it rises its vapour condenses, the latent heat warms it, and it cools at only four to six degrees per kilometre in the lower atmosphere. The same column of air is then unstable to it. Most of the troposphere, most of the time, is stable until something lifts air to saturation, and then it is not — and that one asymmetry runs from a fair-weather cumulus to a thunderstorm, to the warm wind that comes down the far side of a mountain.

A floating needle and the water it displaces. A cross-section of a steel darning needle 1.36 mm thick (density 7.85) floating on water, with a contact angle of 100°, its meniscus computed from the balance of surface tension and pressure; one capillary length is 2.73 mm. The needle's centre sits 1.43 mm below the undisturbed surface, and the water meets it 117° from the top, sloping at 37°. Two regions of water have been displaced. Below the level, under the needle, a region whose weight — 21 per cent of the support — is what pressure on the needle's underside supplies. Beside it, the two dents in the surface, whose weight — the other 79 per cent — is exactly the vertical pull of surface tension along the two contact lines. Together they weigh what the needle weighs: Archimedes, counting the dent.

The water the dent displaces

A steel needle laid carefully on water floats, and the usual explanation is that surface tension holds it up instead of buoyancy, as though the two were rival mechanisms. They are not rivals. The needle presses a dent into the surface, and the upward pull of surface tension along its edges is exactly the weight of the water missing from that dent. Archimedes' principle holds without amendment: a floating needle displaces its own weight of water. Most of that water is simply not underneath it.

The speed of sound in dry sand rises with the load on it. The speeds of compressional and shear waves in a dry random packing of quartz grains, porosity 0.36, against the pressure on it, from the Hertz–Mindlin effective-medium theory, for coordination numbers from 6 to 9 (the bands). The pressure axis is logarithmic. Both speeds rise as the sixth root of the pressure. At 1 kPa — the load about six centimetres down in dry sand — compressional waves travel at 316–362 m/s; at 100 kPa, about six metres down, 681–780 m/s; at 10 MPa, 1468–1680. The dashed lines are the speed of sound in air, 343 m/s, and in water, 1,480 m/s. Near the surface dry sand carries sound no faster than air; measured shallow speeds are lower still, by up to a factor of two, because real contacts are fewer and weaker than the theory assumes.

The sound sand carries faster the deeper it goes

A solid carries sound at a speed set by what it is made of. A heap of quartz grains is made of quartz, and near its surface it carries sound more slowly than the air above it. The grains are not the solid; their contacts are, and a contact between two spheres grows stiffer the harder it is pressed. So the speed of sound in dry sand rises as the sixth root of the load on it — from about air's speed a few centimetres down to twice that at six metres — the paths of sound in a beach curve back up to the surface, and a water table is a mirror to one kind of wave and almost invisible to the other.

Six cells of a two-dimensional foam, by number of sides. Cells of a two-dimensional soap froth with three to eight sides, each drawn with its films meeting three at a time at 120°, as surface tension requires. A hexagon can do that with straight sides. A cell with fewer than six sides cannot: to make its corners 120° its sides must bulge outward, so its gas is at higher pressure than its neighbours' and leaks out. A cell with more than six must have sides curving inward, lower pressure, and gains gas. The rate of change of area, in units of πκγ/3 where κ is the films' permeability, is n − 6: −3 for a triangle, +2 for an octagon.

The bubble that grows by counting its sides

A froth of soap bubbles squeezed between two sheets of glass coarsens over a few days: small cells vanish and large ones swell. The obvious guess is that size decides which cells win, as it does for two bubbles joined by a tube, where the small one always empties into the large one. In a froth it is wrong. John von Neumann showed in 1952 that a cell's area changes at a rate fixed entirely by how many sides it has — shrinking if fewer than six, growing if more — and not at all by how big it is. The proof is a single sum of angles.

Water's surface tension, all the way to the critical point. The surface tension of water against its liquid vapour, from the international standard correlation, against temperature from 0 °C to the critical point at 373.946 °C. 20 °C: 72.7 mN/m; 100 °C: 58.9 mN/m; 200 °C: 37.7 mN/m; 300 °C: 14.4 mN/m; 370 °C: 0.4 mN/m. It falls almost in a straight line for most of the range and then bends down to reach zero exactly at the critical point, where liquid and vapour stop being two things. The bend is a power law, the gap to the critical temperature raised to 1.256.

The tension that vanishes where the phases meet

Surface tension is usually treated as a property of a liquid, like its density. It is a property of a boundary between two things, and it can only exist while there are two things. Heat water in a sealed tube and its surface tension falls steadily, then faster, and reaches exactly zero at 374 °C, where liquid and vapour stop being different. On the way the surface stops being a surface: the step in density spreads from a molecule's width to thousands, and the rate at which the tension dies turns out to be a number the whole family of critical points shares and the simplest theory gets wrong.

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.

A drop in a tapered tube. A section through a glass tube tapering from 1.2 mm to 0.2 mm in radius over 60 mm, holding a 2 μL plug of water (contact angle 20°), not to scale across. Both menisci curve into the empty tube, so the water just inside each is below atmospheric pressure — by 269 Pa at the narrow end, radius 0.50 mm, and by 252 Pa at the wide end, radius 0.54 mm. The pressure is lower at the narrow end, so the water flows towards it: a net push of 17 Pa, with no gravity, no pump and no motion of the tube.

The drop that runs to the narrow end

Put a drop of water into a glass tube that narrows towards one end and it moves, on its own, towards the narrow end — horizontally, with nothing pushing it. Francis Hauksbee reported it to the Royal Society in 1712, and the explanation is two curved surfaces pulling with different strengths: each meniscus lowers the pressure of the water behind it by an amount that grows as the tube narrows, so the end in the narrower part sucks harder. A drop of mercury, which does not wet glass, goes the other way. Which way a drop goes depends on the cosine of one angle, and, for a short drop, on its length.

A pond that is hottest at the bottom and does not turn over. Temperature, salt and density against depth in a salt-gradient solar pond: a fresh upper layer 0.3 m deep at 25 °C, a gradient layer 1.2 m thick in which the salt rises from 2 to 20 per cent and the temperature from 25 to 80 °C, and a storage layer of hot brine 2 m deep (shaded bands are the two mixed layers). The density rises steadily downward, from 1011 to 1122 kg/m³, so the hottest water is also the heaviest and stays at the bottom. The dashed line is the same temperatures with no added salt: the bottom water would then be 25 kg/m³ lighter than the top, and the pond would overturn.

The pond that is hottest at the bottom

Water heated from below rises, which is why a pond in the sun is never much warmer than the air: the heat is carried to the surface and lost there. Put a metre of salt gradient in the way and the hot water at the bottom is also the heaviest, cannot rise, and keeps its heat, behind an insulating layer that sunlight passes straight through. Ponds built this way reach eighty degrees under an ordinary sky. The salt does not collect any extra light. What it buys is temperature.

Where the level goes when a floating ice cube melts. The change in level when a 30 g ice cube floating in a glass 7 cm across has melted completely, against the density of the liquid it floats in (the liquid must be denser than ice, 917 kg/m³, or the cube sinks). In fresh water the level does not move: the meltwater exactly fills the volume the ice displaced. In seawater it rises 0.21 mm and in saturated brine 1.3 mm, because the ice was displacing a denser liquid than it melts into; in castor oil, which is lighter than water and does not mix with it, the level falls 0.32 mm as the meltwater sinks beneath the oil. The change is the meltwater's volume minus the displaced volume, m/ρw − m/ρ, spread over the glass.

The floating ice that still raises the sea

An ice cube melting in a glass of water leaves the level exactly where it was, and the reason is exact: a floating body displaces its own weight, and the meltwater has that weight. The same argument is often used to say that melting sea ice and ice shelves cannot raise the sea. It nearly works. The ice was floating in salt water and melts into fresh, which is lighter, so the meltwater overfills the hole by almost three per cent — and the ice that really fills the sea is the part of a grounded glacier above the thickness at which it would float.

The waterfall hidden in a river mouth. The height of waterfall that would release as much energy per cubic metre of fresh water as mixing it into a large body of salt solution, against the solution's salt content by mass, for sodium chloride at 25 °C: the osmotic pressure divided by the weight of a cubic metre of water. Seawater is worth a 290 m fall; the brine rejected by a desalination plant, about twice as salty, 617 m; a saturated salt solution 3864 m. The curve bends upward because concentrated brine is less ideal than dilute and its osmotic coefficient rises above one. Every river mouth on Earth is a waterfall of this height that nobody sees, since all of its energy goes into the stirring of the estuary.

The waterfall at every river mouth

Where a river meets the sea, fresh water mixes into salt and the free energy of mixing is released as nothing more useful than a little warmth and stirring. It is not a little energy: a cubic metre of river water entering the sea gives up as much as if it fell 290 metres. A desalination membrane run backwards can catch some of it, and the arithmetic of doing so is the arithmetic of every source with an internal resistance — the most power comes at half the pressure, and half the work is the price of getting it quickly.

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.

$The level at which a hinged gate swings open. The height of the centre of pressure on a vertical gate, as a fraction of its height H, against the depth of water above the gate's bottom (solid), approaching the centroid at half the height (dotted) as the water deepens; the dashed lines are hinges at 0.3H, 0.4H, 0.45H. A gate hinged on a horizontal axis and resting against a sill on the downstream side is held shut while the centre of pressure is below its hinge and swung open the moment it passes above. A hinge at 0.3H opens when the water is 0.90H deep, before it reaches the gate's top; at 0.4H, at 1.33H; at 0.45H, at 2.17H. A hinge at or above the centroid never opens at any depth, and the opening level grows steeply as the hinge approaches it — the setting is a choice of hinge position, made once, with no mechanism, sensor or power.$

The floodgate that opens at a chosen depth

Hang a flat gate on a horizontal hinge across a channel and it can be made to stay shut while the water rises, press harder against its seat for a while, and then swing open on its own at one particular level — with no float, no motor and no sensor, only the position of the hinge. The trick is that the water's push on a submerged surface acts below the surface's middle by an amount that shrinks as the water deepens, so the point the push acts at climbs as the flood rises. Put the hinge on that climb and the gate opens where the climb crosses it.

$The force to pull a foot out of packed sand. The suction that resists pulling a foot (a sole of 250 cm², 10 cm long) straight up out of sand, as water is drawn through the sand into the space the foot leaves — Darcy's law with a Kozeny–Carman permeability — against the pulling speed, on logarithmic axes, for compacted sand (fraction 0.6) of three grain sizes and for loose sand (0.41) of 0.1 mm; the dotted line is a 70 kg person's weight. The force is proportional to the speed and inversely to the square of the grain size. Pulled at a centimetre a second from compacted 0.1 mm sand, a foot resists with 2.5 kN, 4 times the person's weight; the speed at which the suction equals that weight is 2.71 mm/s. In the loose sand before it compacted, the same pull would need 15 per cent of the force; with clay filling the pores of the compacted sand, as in natural quicksand, a thousand times more, and the speed a body's weight can pull at falls to microns a second.$

The sand that lets a body in and will not let it out

Quicksand cannot swallow anyone: liquefied, it is nearly twice as dense as water, and a person in it floats with their chest clear. What makes it dangerous is the opposite asymmetry. Going in is easy, because a step breaks the gel holding the grains apart and the sand turns liquid. Coming out is hard, because the sand settles and packs around whatever sank, and a trapped foot can only rise as fast as water can seep through the packed grains to fill the space it leaves. Pulling harder barely helps; letting water in does.

The sand whose friction depends on how fast it flows

A heap of sand stands at its angle of repose and starts to slide a few degrees steeper; between those angles there are steady avalanches, and their speed is decided by one number. The inertial number compares how long a grain takes to fall into a gap opened under it with how long the flow takes to open the gap. Friction rises with it, from the static value to a ceiling, and that single dependence explains why grains flow steadily on only a twelve-degree window of slopes, why a flowing layer has the velocity profile Bagnold measured, and why a deeper avalanche is faster — but not as a liquid would be.

One height inside the bottle where the pressure is the air's. Gauge pressure against height inside a sealed bottle with an open tube reaching down to 10 cm above its outlet, while it is bubbling, at three water levels: 30.0, 22.0, 13.0 cm. Each line rises at the same hydrostatic slope, ρg, from the air pocket at the top, which is below atmospheric, to the outlet at the foot. All three cross atmospheric pressure at the tube's tip — 7.4 mm below it, the extra a bubble needs to leave a 2 mm tube — and so all three reach the outlet at the same 906 Pa above atmospheric, whatever the level. Dashed: an open bottle at the same levels, whose pressure at the outlet is 2937, 2154, 1273 Pa.

The bottle that pours at one rate however full it is

Water running from a hole in the foot of an open bottle slows as the bottle empties, because the pressure at the hole is the depth of water above it. Seal the bottle and push an open tube through the stopper, down into the water, and it stops slowing: the flow holds steady from full until the level reaches the bottom of the tube. Nothing regulates it. The tube's tip is the one point inside where the pressure is the atmosphere's, the water above it hangs from an air pocket below atmospheric, and the outlet sees only the height of the tip.

The grounding line with nowhere to stop

Where an ice sheet flowing into the sea lifts off its bed and begins to float, buoyancy fixes how thick it is: just thick enough to touch the rock, about 1.1 times the depth of the water. The ice crossing that line goes as the 19/4 power of the thickness, so twice the depth sends out twenty-seven times the ice. On a bed that deepens inland, a grounding line that retreats a little moves into deeper water, exports more, and retreats further. No steady state can sit there, and an ice sheet whose edge reaches such a slope crosses the whole of it.

A hopper that runs in bursts. Grains discharged against time, on a logarithmic scale, from a silo of 1 mm glass beads through outlets 3.5, 4.0, 4.4 grain diameters wide; each time the outlet jams it is cleared after 2 s (a tap on the silo) and the flow restarts. Avalanche sizes are drawn at random from an exponential distribution with each outlet's mean. At 3.5 diameters the mean avalanche is 53 grains and the outlet jammed 57 times in two minutes, passing 2,537 grains; at 4.0 diameters the mean avalanche is 1,000 grains and the outlet jammed 37 times in two minutes, passing 33,568 grains; at 4.4 diameters the mean avalanche is 45,821 grains and the outlet jammed 3 times in two minutes, passing 116,983 grains. While it flows, each outlet runs at its full Beverloo rate; a narrower outlet does not flow more slowly, it stops more often, and the flat stretches are the waits.

The outlet that jams by chance

A hopper whose outlet is a few grains wide does not flow slowly. It flows at the full rate and then stops dead, when a handful of grains wedge into an arch across the opening, and it flows again only when something knocks the arch down. The number of grains that pass between one jam and the next is random and exponentially distributed, as if every grain leaving had the same small chance of being the last. Its average rises so steeply with the opening's size that it looks like a threshold, and whether there is one is still argued: two laws agree wherever an outlet can be watched, and they part only after millions of grains.

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.

Three vessels emptying through the same hole. The water level against time in three vessels that each hold 25 litres to a depth of 35 cm and drain through the same 0.8 mm hole in the floor, the outflow following Torricelli's law. The cylinder empties in 16.0 hours, but its level falls fast at first and slowly at the end — a parabola in time — and it is half empty in depth after only 32% of the run. The bowl whose radius grows as the fourth root of the height lowers its level at one rate from top to bottom, a straight line, and empties in 12.9 hours. A straight-sided cone sloping outwards, the shape a potter can actually throw, comes between: half its depth goes in 49% of its run.

The vessel whose level falls evenly

Water leaves a hole in the floor of a vessel faster when the vessel is full, because the pressure at the hole is the depth of water above it. In a cylinder that makes the level fall quickly at first and slowly at the end, so hour marks on its wall crowd towards the bottom. A level that falls at one rate needs a vessel that widens as the outflow quickens: its radius must grow as the fourth root of the height. The Egyptians who made the oldest surviving water clock, about 1400 BC, gave it sloping sides, which halves the error of a cylinder. They also gave it a scale for every month, because the hours they wanted to count were not all the same length.

A surface whose thickness depends on how much of it is looked at. The rms roughness of a water surface at 20 °C, in nanometres, against the size of the patch over which it is measured, from the thermal ripples that fit inside the patch — with gravity (solid) and without it (dashed) — for ripples down to a molecular wavelength of 0.6 nm. Over 10 nm the surface is 0.16 nm rough; over a micrometre, 0.26; over a millimetre, 0.36; and beyond a few millimetres gravity flattens the longest ripples and the roughness stops at 0.390 nm. In free fall it would keep growing as the square root of the logarithm: 0.43 nm over a metre. An X-ray reflectivity measurement sees the ripples smaller than its own coherence length, typically some micrometres, and reports a thickness of about a third of a nanometre — a number that belongs to the instrument as much as to the water.

The surface whose thickness depends on the view

The surface of a glass of water looks like a perfect plane, and over most distances it nearly is. But it is at room temperature, and every ripple it could carry — from a few molecules long to a few millimetres — is shaken by heat with its own share of thermal energy. The ripples are tiny, a fraction of a nanometre high, and they add up in an odd way: every factor of ten in wavelength contributes the same amount of roughness. So the thickness of a water surface has no single value. It depends logarithmically on how large a patch is looked at, until gravity stops it at a few millimetres, and the X-ray measurements that report a third of a nanometre are reporting a property of their own resolution as much as of the water.

Two columns of air side by side, falling at different rates. The air pressure against height inside a 100 m building kept at 21 °C (solid) and outside it at −10 °C (dashed), relative to the pressure outside at the ground, for a building whose openings are equal at top and bottom. Both fall by about 12 Pa a metre, the weight of a metre of air on each square metre, but the cold outside air is 12 per cent denser and its pressure falls faster. The two lines cross at the neutral plane, half-way up (dot). Below it the outside pressure is higher and pushes air in at every crack; above it the inside pressure is higher and pushes air out — 69 Pa at the roof, 69 Pa at the ground. The building is a chimney.

The building that breathes through its doors

In winter the lobby doors of a tall building are hard to push open and, once open, let in a gale, though there is no wind outside. The building is a chimney. Its warm air and the cold air outside are two columns of fluid standing side by side, and the lighter one loses pressure more slowly with height, so the two pressures can be equal at only one level. Below it the outside pushes in through every crack; above it the inside pushes out. The difference is a few tens of pascals, the weight of a few millimetres of water, and over the area of a door it is the push of a strong hand — set entirely by hydrostatics, and running backwards in summer.

$A plume rising into a stratified fluid, until it cannot. A turbulent plume from a small steady source of buoyancy, rising through a fluid whose density increases steadily downward, integrated from the Morton–Taylor–Turner equations with an entrainment coefficient of 0.1; height across, in units of F₀^(1/4)N^(−3/4), the only length the source's buoyancy flux and the stratification can make. Up: the buoyancy flux, as a fraction of the source's (solid); the momentum flux, as a fraction of its peak (dashed); and the plume's radius, as a fraction of its radius at the neutral level (dotted). The buoyancy flux falls from the start, because the fluid the plume entrains came from lower down and is denser than the fluid it is carried up into; it reaches zero at 4.37 — the level at which the plume will spread sideways — and goes negative. The momentum flux, built up by the buoyancy, peaks at 4.37 and is then spent carrying the now-heavy plume on to 5.75, where it stops. The radius grows almost in proportion to height until near the top, where the model's top-hat radius runs away as the momentum vanishes — the point at which the plume stops being a plume and becomes a spreading intrusion the equations do not describe.$

The plume that stops below the surface

Hot smoke rises, oil and gas from a seabed leak rise, and in a fluid of uniform density they would go on rising for ever. Real oceans and atmospheres are stratified, denser lower down, and a rising plume carries its own downfall with it: it rises by drawing in the fluid around it, and the fluid it draws in near the bottom is denser than the fluid it is carried up into. Its buoyancy runs out, it overshoots, and it spreads sideways at a height set only by how buoyant its source is and how strongly the surroundings are layered — a quarter power of one and a minus three-quarters power of the other.

The height at which hot air stops being light enough. Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load.

The balloon a cold morning lifts

A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.

The film that thins too slowly. The thickness of the air film under a silicone-oil drop of radius 0.38 mm pressed onto its pool, against time, on logarithmic axes, by Reynolds's law: dh/dt ∝ −h³, so the film thins quickly at first and then ever more slowly, its thickness falling only as the inverse square root of time once it is thin. The shaded band is one bounce, 4.1 ms, in which the film falls from 10 µm to 1240 nm; the dashed line is the 100 nm at which van der Waals forces pull the surfaces together, reached after 0.65 s. The air does not have to be held up; it has only to be slow to leave, and the cube of the thickness makes it slower and slower the thinner it gets. A vibrating bath that lifts the drop off every few milliseconds refills the film each time.

The drop that will not merge

A drop of oil let fall onto a pool of the same oil ought to vanish into it at once: merging releases surface energy, and nothing holds the two apart. Instead it bounces. A film of air is trapped between drop and pool, and air, though it offers no strength at all, is slow to leave a gap a few micrometres thick: the film takes the better part of a second to drain to the hundred nanometres at which the surfaces can touch, while a bounce lasts four milliseconds. Shake the pool and the drop can bounce for ever, in rhythms that double as the shaking grows — and, near the pool's own instability, it starts to walk across the surface on the waves of its own making.

The tube that is stiff because it is full. The bending moment a thin-walled tube 1 mm in radius can carry when inflated, in newton-millimetres, against its internal pressure in megapascals: the moment at which the side in compression goes slack and wrinkles, πPr³/2, and the moment at which the whole section folds, πPr³. Both are proportional to the pressure and contain nothing about the wall's own stiffness: a membrane that cannot resist compression at all carries bending only by being pre-stretched, and the pressure is the pre-stretch. At 0.74 MPa the tube wrinkles at 1.17 N·mm; with no pressure, at nothing. That is the whole of how a lettuce leaf, a petal or a young shoot keeps its shape. It is also how an inflatable boat, a bouncy castle or the boom of an inflatable antenna holds up a load, and how they fail: by wrinkling on the compressed side first.

The stem that stands by pressure

A lettuce leaf has no bones, and a young sunflower stem has hardly any wood, yet both hold their shape against gravity and wind. They do it with pressure. Each cell holds more dissolved solute than the water around it, water flows in until the cell wall pushes back, and the wall ends up stretched taut by a pressure of seven or eight atmospheres — three times a car tyre's. A tube of membrane under tension resists bending in exact proportion to its pressure, whatever the membrane is made of, so a soft plant is stiff exactly as long as its cells are full. Lose a few per cent of their water and the pressure is gone, and the plant wilts.

The sandcastle a spoonful of water holds

A dry heap of sand stands at thirty degrees and no steeper. Add a spoonful of water to a bucketful, and the same grains will stand in a vertical wall and carry a person's weight. Add ten times as much water and the wall is no stronger, because a bridge of water between two grains pulls with almost the same force whatever its size. Its strength belongs to the grain and the surface tension, and the amount of water hardly enters.

The lake that holds its gas until it does not

On the night of 21 August 1986, Lake Nyos in Cameroon released about a cubic kilometre of carbon dioxide in a few hours, and some 1,700 people died in their sleep in the valleys below. The lake was not boiling and it was not erupting. Its deep water had been holding the gas safely for decades, in a layering that was stable as long as nothing moved. Lift that water far enough and it cannot stop rising. The same instability, in a pipe, now drains the danger away.

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