Points to Ponder

Pressure is a scalar quantity. The common description "force per unit area" can mislead you into thinking pressure is a vector. The force in that definition is only the component of the force that acts normal (perpendicular) to the area. When studying fluids, you must shift your thinking from particle or rigid-body mechanics. Instead of tracking a single object, you now care about properties — like pressure — that can change from one point to another within the fluid.

Do not imagine that pressure exists only where a fluid touches a solid surface (like a container wall or an immersed object). Pressure is present at every point inside the fluid. A small fluid element stays in equilibrium because the pressures acting on all its faces are equal — not because it is being pushed only from the outside.

The formula $P = P_a + \rho g h$ is valid only for an incompressible fluid. In practice, this works well for liquids, which hardly change volume under pressure. For a liquid, $\rho$ is constant with height, so the equation holds.

Gauge pressure is defined as the difference between the actual pressure and the atmospheric pressure: $P_g = P - P_a$. Many common pressure-measuring devices — such as a tyre pressure gauge or a blood pressure gauge (sphygmomanometer) — actually read gauge pressure, not absolute pressure.

A streamline is a map of the flow pattern. In steady (non-turbulent) flow, two streamlines never cross. If they did, a fluid particle at the intersection would have two different velocities at the same instant — which is impossible.

Bernoulli's principle fails when viscous drag is present. Viscous forces do work on the fluid, dissipating energy. In that case, the pressure $P_2$ downstream will be lower than what Bernoulli's equation predicts, because the lost energy must be accounted for. …