Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 39C / a

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- Solution a

Solution

The one-dimensional mass flux and momentum flux are respectively

J_{m} = ρ u, J_{p} = p + ρ u^{2} .

In a steady flow, the conservation laws make both the mass flux and the given energy flux

J_{E} = u (p + W + \frac{1}{2} ρ u^{2})

constant. Wherever

J_{m} \neq = 0

, their ratio is therefore constant:

\frac{J _{E}}{J _{m}} = \frac{p + W}{ρ} + \frac{1}{2} u^{2} = \frac{γ}{γ - 1} \frac{p}{ρ} + \frac{1}{2} u^{2} = C,

where

W = p / (γ - 1)

is the internal energy per unit volume of the ideal gas. This is the required steady-flow energy integral.

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