Fluids don't move on their own — something has to push them. That something is a difference in pressure between two locations. Fluid flows from high pressure toward low pressure, the same way any other object accelerates in the direction of a net force.
For a tube that's open at both ends and completely filled with an incompressible fluid, matter can't pile up or disappear anywhere inside it. That means the rate at which matter enters must exactly equal the rate at which it exits — this is just conservation of mass.
The rate at which matter flows past any cross-section is proportional to that section's area and the fluid's speed there. Setting the rate in equal to the rate out gives the continuity equation:
A narrower cross-section must have a faster flow speed to keep the same amount of matter moving through per second. This is exactly why water speeds up when a hose nozzle narrows.
Water flows through a pipe that narrows from a cross-sectional area of 0.020 m² to 0.005 m². If the fluid's speed in the wider section is 1.5 m/s, find its speed in the narrower section.
A difference in a fluid's height between two locations produces a difference in gravitational potential energy — and conservation of energy says that difference has to show up somewhere else, as a difference in kinetic energy and pressure. Bernoulli's equation captures exactly this trade-off along a streamline:
Pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume — the three terms trade off against each other, but their sum along any streamline never changes.
Shrink the narrow section and watch particles speed up exactly as much as continuity demands — and watch pressure drop right along with it.
Notice: shrinking A₂ increases v₂ (continuity) and that increase in speed drops P₂ (Bernoulli) — the two laws are working on the same pipe at once.
Water flows through a horizontal pipe (constant height) that narrows partway through. At the wide section, pressure is 1.50×10⁵ Pa and speed is 2.0 m/s. At the narrow section, the speed is 8.0 m/s. Find the pressure at the narrow section. (ρ_water = 1000 kg/m³)
Torricelli's theorem isn't a new law — it's Bernoulli's equation applied to one very specific, very common situation: fluid draining out of a small opening in a tank.
A large open tank has a small hole in its side, 2.0 m below the water's surface. Both the surface and the exiting stream are open to the atmosphere. Find the speed of the water as it exits the hole. (g = 9.8 m/s²)