Past exam of the physics B course of the University of Cambridge / 2015 / Paper 2 / Section B / B8 / d / Solution

For steady, inviscid, incompressible flow, Euler's equation integrated along a streamline givesThis is Bernoulli's equation. The constant is common to different streamlines if the flow is also irrotational; incompressibility alone does not imply that. Viscous loss or work by a pump requires additional terms.

A Pitot-static tube compares stagnation pressure $p_0$ at a forward-facing opening with nearby static pressure $p$. For equal height, the stagnation point has $v=0$, soThis measures flow speed when compressibility corrections are negligible.

In a flowmeter using the Venturi effect, continuity gives $A_1{v}_1=A_2{v}_2$. For a horizontal constriction with $A_2<A_1$, Bernoulli gives $p_1-p_2=\rho (v_2^2-v_1^2)/2$, henceThe fluid accelerates and static pressure falls in the throat. This is a second practical application of the conversion between pressure and kinetic energy; real instruments need calibration for flow losses.