Fluid flow

Define the volume flow rate as the volume of fluid crossing a section of the pipe per second:

$$Q = A\,v$$

Here $A$ is the cross-sectional area at the section and $v$ is the average flow speed there. If the fluid is incompressible (density does not change) and the pipe is full, mass conservation forces the same volume to pass every section per second. So $Q$ is the same everywhere along the pipe:

$$A_1 v_1 = A_2 v_2$$

Read that as a balance: shrink the area, the speed grows by the same factor. Halve the area, double the speed. Quarter the area, quadruple the speed. It is what your finger over the hose tells you: squeeze the end (smaller $A$), the water shoots out faster (bigger $v$).

Energy bookkeeping per unit volume of fluid. Every bit of fluid along a streamline carries three terms,

• pressure energy density: $P$
• kinetic energy density: $\tfrac{1}{2}\rho v^2$
• gravitational energy density: $\rho g y$

For an ideal fluid (incompressible, inviscid, steady), their sum is constant along a streamline:

$$P_1 + \tfrac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \tfrac{1}{2}\rho v_2^2 + \rho g y_2$$

For a horizontal pipe, the $\rho g y$ terms cancel and Bernoulli reduces to a trade between pressure and kinetic energy:

$$P_2 - P_1 = \tfrac{1}{2}\rho\bigl(v_1^2 - v_2^2\bigr)$$

Where $v_2 > v_1$, the right side is negative, so $P_2 < P_1$. The extra kinetic energy at section 2 has to come from somewhere; pressure pays.

Tilt the pipe so $y_2 < y_1$ and the lost height contributes to the kinetic or pressure term, depending on whether the area also changes. Keep the pipe horizontal and the height term goes away. Keep the area constant and the kinetic term goes away. Bernoulli is one equation; the question is always which terms are doing work in the given setup.

Four working examples of the same two laws:

Airplane wing. If air moves faster over the top surface than under the bottom, Bernoulli predicts lower pressure on top than underneath, and the net upward force is the wing area times this pressure difference. (AP-1 treats this qualitatively only; what makes the air speed up over the wing, and full lift theory, is beyond the course.)

Atomizer. A perfume bottle has a vertical tube reaching into the reservoir and a squeeze bulb. Squeezing the bulb sends a fast jet of air across the top of the vertical tube. The fast air has lower pressure than the still air inside the bottle; the pressure difference drives liquid up the tube and into the airstream, where it breaks into a fine spray.

Hurricane lifts the roof. When wind rushes over a roof, the air on top can move much faster than the still air in the attic below. By Bernoulli, faster outside airflow lowers the pressure on top, so the pressure difference pushes the roof upward. This is an idealized picture: real roof failure also depends on openings, turbulence, uplift paths, and how the structure is built.

Torricelli's law. A tank of water with a small hole punched in the side at depth $h$ below the surface drains horizontally. Apply Bernoulli between the open surface (pressure $P_0$, speed $\approx 0$, height $h$) and the exit (pressure $P_0$, speed $v$, height $0$): the pressure terms cancel and $\rho g h = \tfrac{1}{2}\rho v^2$, giving

$$v = \sqrt{2 g h}$$

That is the speed of a falling object dropped through height $h$. Gravitational potential converts cleanly into kinetic energy. Pressure drops out because both ends of the streamline are open to the atmosphere.

Ten scenarios across the two laws and the four traps. Pick the option, hit Check, read the trap explanation when you miss. Progress saves locally as you go.