Bernoulli's principle — pressure in a flow

What just happened?

You narrowed the pipe, the fluid stream sped up and the static pressure dropped. It is not magic — it is a direct consequence of energy conservation for a flowing fluid.

Continuity equation (incompressible fluid):

— the same amount of fluid passes through every pipe cross-section in the same time. A smaller cross-section $S$ ⇒ a higher speed $v$. That is why the streamlines bunch up in the constriction.

Bernoulli's equation (horizontal pipe):

— the sum of the static pressure $p$ and the dynamic pressure $\tfrac12\rho v^2$ is constant. As the speed rises, the "dynamic" term takes more and more, so the static pressure $p$ must drop. $\rho$ — the fluid density.

What does this mean in practice?

Fast flow = low pressure. This is how a carburettor and an atomiser work (the Venturi effect), how an aircraft gains lift, and why two boats sailing close together are "sucked" toward each other.

Why you need this for the exam:

In pipe-constriction problems, first use $S\cdot v=\text{const}$ to find the speed in the constriction, then $p+\tfrac12\rho v^2=\text{const}$ to compute the pressure. Trick: where it is tighter and faster, the pressure is lower.

Remember: faster-flowing fluid exerts lower pressure — p + ½ρv² = const. In the constriction the speed rises and the pressure drops.