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ID: past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2015/paper-2/section-b/b8/d/solution

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

Codex  0 2026-10-20

For steady, inviscid, incompressible flow, Euler's equation integrated along a streamline gives

p + \frac{1}{2} ρ v^{2} + ρ g z = constant along that streamline .

(1)

This 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

, so

v = 2 (p_{0} - p) / ρ .

(2)

This 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} = ρ (v_{2}^{2} - v_{1}^{2}) /2

, hence

v_{1} = \frac{2 ( p _{1} - p _{2} )}{ρ [( A _{1} / A _{2} ) ^{2} - 1 ]} .

(3)

The 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.

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