Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 2 / a

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- Solution a

Solution

Project the steady Euler equations for an inviscid fluid along a streamline. With no gravity,

u d u + \frac{d p}{ρ} = 0.

For an isentropic fluid,

d p = c_{s}^{2} d ρ

, so

\frac{d ρ}{d u} = - \frac{ρ u}{c _{s}^{2}} .

The scalar mass flux is

j = ρ u

, and hence

\frac{d j}{d u} = ρ (1 - \frac{u ^{2}}{c _{s}^{2}}) .

It increases with speed in subsonic flow and decreases with speed in supersonic flow.

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