Illustrated Physics
physics, drawn
Concept

Hydrostatic pressure — where it appears

The pressure in a fluid at rest, equal to the density times g times the depth and independent of the vessel's shape. It follows from a fluid having no shear strength, so a vertical wall can carry none of the weight, and it fails for grains, which have one.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

fluids · Buoyancy 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.

fluids · Granular matter 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.

fluids · Capillarity 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.

fluids · Buoyancy 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.

fluids · Hydrostatics 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.

fluids · Hydrostatics 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.

fluids · Buoyancy 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.

fluids · Hydrostatics 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.

fluids · Hydrostatics The boiling point in a column of water, and the depth heating reaches it first. The temperature at which water boils against depth in a geyser's conduit 2240 m above sea level, where the air's pressure is 77.6 kPa (solid): 92.7 °C at the surface, 103.6 °C at 4 m, 115.5 °C at 10 m and 131.8 °C at 22 m — rising steeply near the top and more gently below, because each extra metre adds the same pressure to water whose boiling point responds less as it gets hotter. Dashed: the water as it heats from beneath, cooled to 80 °C at the top by the pool and below that warming by only 0.8 °C per metre, mixed by convection; 6 °C and 3 °C short of the curve at 4 m, and then touching it. It touches at 4 m, not at the bottom: there the water is still 13.7 °C below boiling.

The column that boils from the middle

Water at the bottom of a geyser's tube is hotter than water anywhere else in it, and it is not the water that boils first. The weight of the column raises the boiling point with depth faster than the heat from below raises the temperature, so the column reaches its boiling curve partway down, under a cooler surface layer. A little boiling there lifts the water above. Everything beneath is then at a lower pressure than the one it was heated under, so it is superheated and flashes, and the eruption feeds itself: past about fourteen centimetres of water thrown out, each slug of steam makes more room than it was given.

thermodynamics · Phase change 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.

fluids · Hydrostatics 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.

fluids · Hydrostatics

Named alongside it

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

BuoyancyCapillary lengthArchimedes principleCompressibilityContact angleEquation of stateEquilibriumLaplace pressureNucleationSea levelSurface tensionTorricellis law

All concepts