Fluid Flow

Squeeze the end of a garden hose and the water shoots out faster. Blow across the top of a sheet of paper and it lifts. Watch a river narrow into a gorge and the current picks up. Two laws are behind all three: continuity (a narrower pipe carries the same volume per second by speeding up) and Bernoulli's equation (pressure, $\tfrac{1}{2}\rho v^2$, and $\rho g y$ sum to a constant along a streamline). The trap most students fall into is sitting right between them: faster flow does not mean higher pressure. It means lower.

Continuity is the rule that narrowing a pipe makes the flow faster: the same volume per second has less area to pass through, so it has to speed up. Bernoulli is the rule that faster flow means lower pressure at constant height, because the kinetic term has to grow at the expense of the pressure term. Tilt the pipe downhill and a third term, $\rho g y$, joins the trade; release the fluid through a hole in the side of a tank and Bernoulli reduces to Torricelli's law, $v = \sqrt{2 g h}$. One equation, three terms, four scenes.