From still to flowing: the ideal fluid
So far the fluid just sat there. Now let it run: rivers, blood in your arteries, air streaming over a wing. Moving fluids are genuinely harder, so we begin with a deliberate idealisation — the ideal fluid: incompressible (constant density), non-viscous (no internal friction), in steady flow (the pattern does not change with time). Real fluids break these rules, and Guide 5 is all about how — but the ideal fluid captures the core ideas cleanly first.
A useful picture is the streamline: a curve that a tiny speck of fluid follows. In steady flow, streamlines are fixed paths, and fluid never crosses from one to another. A bundle of streamlines forms a “tube” of flow — and thinking about what enters and leaves such a tube gives us our first law.
Continuity: what goes in must come out
Put your thumb over the end of a garden hose and the water leaps out faster and farther. You added no energy — you narrowed the exit. Because the fluid is incompressible and none leaks away, the same volume must pass every cross-section each second. Where the pipe is wide the fluid ambles; where it narrows it must speed up to keep the volume flowing. That bookkeeping is the continuity equation.
Continuity for an incompressible fluid: cross-sectional area times speed is constant along the flow. That product is the volume flow rate Q (in m³/s).
The consequence is sharp: halve the radius of a pipe and you quarter its area (A = \pi r^2), so the fluid must flow four times faster. This one relation, the volume flow rate Q = Av, is why a nozzle speeds a jet and why a river races through a narrow gorge. It also has a surprising biological twist: blood crawls through the capillaries even though each is microscopic, because there are billions of them and their combined area dwarfs the aorta's.
Bernoulli's principle
Continuity tells us fast and slow spots must exist; Bernoulli's principle tells us their pressure. Its headline is counter-intuitive: where an ideal fluid moves fast, its pressure is low; where it moves slowly, its pressure is high. The principle is really just energy conservation written per unit volume of fluid — pressure energy, kinetic energy and gravitational energy trading off so their sum stays fixed along a streamline.
Bernoulli's equation: static pressure + kinetic energy per volume + potential energy per volume is conserved along a streamline of an ideal fluid.
Why should speeding up cost pressure? Follow a parcel of fluid into the narrow section. To speed up, it had to be pushed — and the only thing that can push it is a pressure difference, higher behind than ahead. So the region where the fluid ends up moving fast is precisely the region of lower pressure. Bernoulli is Newton's work–energy idea wearing a fluid's clothing.
Bernoulli everywhere — and Torricelli
Once you see the speed–pressure trade, you spot it everywhere. Blow fast air across the top of a straw and the low pressure sucks liquid up into the stream — that is a perfume atomiser and a paint sprayer. Wind racing over a roof lowers the pressure above it, and the still, higher-pressure air below can lift the roof clean off in a storm. A spinning ball drags air faster on one side, so the pressure imbalance curves its flight. The same idea, run backwards, lets a Venturi tube or pitot probe measure flow speed from a pressure reading.
For a clean quantitative case, poke a small hole in the side of an open tank at depth h below the water surface. Apply Bernoulli between the calm top and the jet at the hole: both are at atmospheric pressure, so the pressure terms cancel and the drop in height turns entirely into speed. The result, Torricelli's theorem, is startling — the water leaves exactly as fast as if it had simply fallen the distance h.
Torricelli's theorem: fluid escaping a hole at depth h below the surface leaves at the free-fall speed for that height.