Module 11 · fluids
A fluid is anything that cannot hold its shape — liquid or gas, and for most of what follows it makes no difference to the physics. Three ideas do most of the work: pressure grows with depth, a floating thing displaces its own weight, and fast-moving fluid has low pressure. From those come submarines, ships, and the reason your shower curtain attacks you. Two more are needed for the rest — fluids are sticky, and their surfaces behave like skin — and blood pressure lives on the first of those. Aeroplane lift, we will see, is not really a fast-fluid story at all.
Pressure is force spread over area: P = F/A, where P is the pressure and F the force, measured in pascals (one newton per square metre). It is why a drawing pin goes into a wall when your thumb does not — same force, area a thousand times smaller, pressure a thousand times higher.
Now go inside a liquid. Picture a flat one-square-metre patch at some depth, and the column of liquid standing on it up to the surface. The pressure at that depth is the weight of that column, plus whatever is pressing down on the surface — usually the atmosphere:
Three vessels stand side by side, filled with water to exactly the same depth: one a narrow cylinder, one a wide cone flaring outward, one a cone narrowing to a tiny neck at the top. Which has the greatest pressure on its base?
This bothered people for two centuries, and it is called the hydrostatic paradox. The resolution is that the walls take up the difference. Do it as a force balance on the water. The vessel that flares outward holds more water than a plain cylinder on the same base would, so the total weight of that water is more than the base could be feeling — the sloping walls must be holding the extra up. The vessel that narrows towards the top holds less water than the base pressure alone would account for, so there the sloping walls must be pressing down on it. Either way the base feels only the column standing directly above it. The base only ever feels Patmosphere + ρgh — and the atmosphere is the same for all three vessels, so only the ρgh part could tell them apart, and it does not. Pascal is said to have demonstrated it by attaching a long thin tube to a sealed barrel and pouring in a few cups of water from an upstairs window — the barrel burst, because a three-metre column of water in a pipe the width of a finger produces exactly the same pressure as a three-metre lake.
Take a block sitting in water. Its bottom face is deeper than its top face, so the pressure pushing up on the bottom is greater than the pressure pushing down on the top. That difference is a net upward force — and that is the whole of buoyancy. It is not a new law of nature; it is ρgh applied twice and subtracted.
Here is that arithmetic, in four lines. Take a block of base area A, with its top face at depth h₁ and its bottom face at depth h₂. The push up on the bottom face is (Patmosphere + ρgh₂)A; the push down on the top face is (Patmosphere + ρgh₁)A. Subtract them: the atmospheric parts cancel, and the net upward force is ρg(h₂ − h₁)A. But (h₂ − h₁)A is the volume of the block, so this is the weight of that much fluid. That is the whole of it:
One line of algebra turns this into the number the bench shows. A floating object is in balance, so its weight equals the upthrust: ρobject Vobject g = ρliquid Vsubmerged g. Cancel g, divide through, and Vsubmerged / Vobject = ρobject / ρliquid. The fraction of an object sitting below the waterline is just the ratio of the two densities — a 600 kg/m³ block in water floats with six-tenths of itself under.
A steel ship floats, though steel is eight times denser than water. What is the honest explanation?
| Case | The numbers |
|---|---|
| An iceberg | ice is 917 kg/m³, sea water 1025, so 917/1025 = 89% sits below the surface. The visible tip really is about a tenth, and the phrase is, unusually, accurate. |
| You, in a pool | the human body averages roughly 985 kg/m³ with lungs full — just under water's, so with your face down, arms loose, and your lungs held full, you float with the crown of your head just out of the water. That is the dead-man's float. Breathe right out and your average density climbs above water's, and you sink. |
| The Dead Sea | brine at 1240 kg/m³, about a quarter of it dissolved salt. Your fraction submerged drops to about 0.79, so you ride noticeably higher — the famous newspaper-reading photograph is buoyancy, not a trick. |
| A submarine | the only vehicle that deliberately sits at neutral buoyancy, adjusting its average density by pumping water in and out of ballast tanks until upthrust exactly equals weight and it neither rises nor sinks. |
| A hot-air balloon | the same law in air. Heat the air inside and it expands (Module 4), fewer molecules occupy the envelope, average density falls below the surrounding air's, and the balloon floats up in air exactly as a cork floats up in water. |
Float an ice cube in a full glass of water, marking the level. When it melts, does the water overflow? Work it out first from Archimedes, then check. (The floating ice displaces exactly its own weight of water — and when it melts it becomes precisely that weight of water. The level does not move by a millimetre.) Then the harder version: if all the Arctic sea ice melted, would sea level rise? (Arctic ice is floating sea ice, so the glass-of-water answer applies almost exactly — melting it changes sea level hardly at all. Almost, not exactly, because sea ice is fresh and the sea is salty.) Antarctica is a completely different answer for one reason: most of its ice sits on rock, well above sea level, and is displacing no sea water at all. Melting that is like pouring water into the glass from outside, and sea level rises.
Now let the fluid move. Two rules govern almost everything.
The first is bookkeeping: fluid does not pile up or vanish, so whatever flows into a pipe per second must flow out per second. Narrow the pipe and the fluid must speed up to keep the account balanced.
The second is energy, which is Module 5 again. Follow a small parcel of fluid as it moves; the path it traces out is called a streamline, and what follows holds along one such path, in flow that is smooth and steady. That parcel carries three kinds of energy — pressure energy, kinetic energy, gravitational — and their total does not change as it goes:
One more thing to have in hand before the gate: a moving fluid drags its neighbours along with it. Falling water pulls the air beside it downward, the way a passing lorry tugs at you on the footpath. That dragging is viscosity, which we come back to at the end of this section.
You are in the shower and the curtain billows inward against your legs. Why?
| Where you meet it | What is going on |
|---|---|
| A spray gun or perfume atomiser | air blown fast across the top of a tube drops the pressure there; atmospheric pressure on the liquid's surface pushes it up the tube and into the stream. |
| A curveball or a topspin tennis shot | the spinning ball drags air round with it, making the flow faster on one side than the other. Lower pressure on the fast side, and the ball is pushed sideways in flight — the Magnus effect. |
| A swinging cricket ball | a different mechanism, and worth knowing: the raised seam, held at an angle, trips the air on one side into turbulence while the other side stays smooth. The turbulent side clings to the ball for longer before separating, so the wake is thrown to one side and the ball swings the other way. The bowler's spin only keeps the seam pointing where he wants it. |
| A carburettor, a Venturi flow meter | both are Bench 3 built as a device: measure how far the pressure drops at the narrowest point — the throat — and you can work back to the flow speed. |
| Blowing across the top of a sheet of paper | it lifts — but not for the reason usually given. The jet from your mouth is at ordinary atmospheric pressure, so "fast air above, still air below" compares two streams that have nothing to do with each other, and Bernoulli says nothing about the pair. What happens is that the jet follows the paper's droop and is thrown downward, and the paper is pushed up in reaction — the aerofoil argument below, in miniature. Blow across a sheet held flat and taut and it barely moves. |
| Wind lifting a roof off | fast air over the top, still air in the loft. In a storm, roofs are more often pushed off from below than pulled off from above. |
The explanation in most textbooks: the wing's curved top is longer, so air going over it must travel faster to "meet up" with the air underneath, and Bernoulli then gives lower pressure above. Take the two objections one at a time. First, no law says the two parcels of air must arrive at the back of the wing together; when you actually measure it, the air that went over the top gets there earlier, not simultaneously. Second, if the curved top were the reason for lift, no aeroplane could fly upside down — and aerobatic pilots do it at every airshow.
The honest account is Newton's third law, and it is simpler. A wing is tilted slightly nose-up, and as it goes it throws a large mass of air downward every second — for an airliner, roughly a tonne of air a second, sent down at a few metres per second. Giving that air downward momentum takes a downward force, and by Newton's third law the air pushes back up on the wing with a force of exactly the same size. That upward push is the lift. None of this makes Bernoulli wrong. The air really does move faster over the top of a wing, and the pressure really is lower up there. What is wrong is the direction of the explanation: the faster flow is a result of the way the wing deflects the air, not the cause of the lift. Both descriptions are views of the same event, and if you want to know how much lift there is, count the downward momentum given to the air per second. Module 3, doing aerodynamics.
Viscosity is internal friction — the resistance of one layer of fluid sliding over the next. Honey has it in abundance, water much less, air very little. It is why real pipes need a pressure difference just to keep fluid moving at all: Bernoulli's ideal fluid would coast forever, but a real one is always losing energy to its own internal rubbing. It is why blood pressure drops steadily along your arteries. And it is why a narrow tube is much worse than you would guess. Careful here, because this is a different question from Bench 3. There, a fixed amount of fluid per second was forced through a constriction and had to speed up. Here we ask how much gets through at all when the pressure pushing it is fixed. Two things now count against a narrow tube: there is less area to flow through, and a larger share of the fluid is close to a wall that is holding it back. Put together, the flow through a tube goes as the fourth power of its radius, so halving the radius cuts the flow to a sixteenth. A modest narrowing of an artery is a serious event for exactly this reason.
Surface tension is what you get when the pulls on a molecule stop cancelling. Deep in a liquid, every direction pulls equally and the pulls cancel; at the surface a molecule has neighbours below and beside it but none above, so the net pull is inwards. Bringing a molecule up to the surface costs energy, so a liquid keeps its surface as small as it can. The stretched-skin picture is worth keeping — but the "skin" is no separate material, and never stronger than the molecular pull that makes it. Set a paperclip down edge-first and it floats; nudge it, or drop a speck of soap on the water, and it gives way at once. Small droplets are spherical for the same reason (big raindrops are flattened by the air rushing past them), a water strider walks on ponds, and water climbs a narrow tube by itself — capillary action, the same effect that pulls water up the fibres of a towel and, in part, up the trunk of a tree.
1. A dam is built across a valley. To hold the same depth of water, a dam holding back a lake 10 km long must be built:
2. You are floating in a boat on a small pond, holding a heavy stone. You throw the stone overboard. The water level in the pond:
3. Water flows through a pipe that narrows to half its cross-sectional area. In the narrow section, the speed and the pressure are: