Put your thumb over a garden hose and the water leaves faster. Yet for a pipe that narrows, the confident answer often flips: less room, so it must slow down.

Continuity: the pipe cannot hide any water#

If the volume crossing every station per second is the same number $Q$, then at each station that number is the area times the speed:

$$A_1 \vel{v_1} = A_2 \vel{v_2} = Q$$

The dot rows leave the inlet at a fixed interval, so the gap between two rows is $v\,\Delta t$: a speedometer you read with a ruler.

---

Now look up. The vertical tubes are manometers: the liquid in each stands as high as the pressure under it can push it, $h = p/\rho g$. Over the wide run they stand at 0.82 m; over the throat, at 0.51 m. The pressure is lower where the fluid is faster.

Bernoulli: the trade, written down#

Follow one parcel of fluid along a streamline. The pressure behind does work on it, it does work against the pressure ahead, and gravity does work on it if it changes height. Add that up and you get a quantity that stays the same along the streamline:

$$\frc{p} + \tfrac{1}{2}\rho \vel{v}^2 + \rho g\, \pos{y} = \text{constant}$$

---

Uphill#

This case separates the two equations cleanly. Continuity says something about the speed that intuition does not expect. Bernoulli says something about the pressure that it does.

Where Bernoulli stops#

Every equation above assumed an ideal fluid: no viscosity, nothing rubbing. A real fluid sticks to walls and shears against itself. Both cost energy, and the energy comes out of the pressure.

---

What comes next#

You sped water up through a throat, watched its pressure fall, and found where friction takes over. Set $v = 0$ and Bernoulli collapses to $p + \rho g y = \text{constant}$, the hydrostatic law of the previous lesson: its third term was hydrostatics, riding along. Next the unit turns from fluids to heat, and to the question of what temperature two objects settle at when they touch.