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

Table of contents
- Solution c

Solution

The Bernoulli function for steady unmagnetized flow without gravity is

C_{B} = \frac{u ^{2}}{2} + h .

For a polytropic equation of state,

h = c_{s}^{2} / (γ - 1)

. Evaluation at the critical point found in part b gives

C_{B} = \frac{c _{cr}^{2}}{2} + \frac{c _{cr}^{2}}{γ - 1} = \frac{γ + 1}{2 ( γ - 1 )} c_{cr}^{2} .

Consequently

c_{cr}^{2} = \frac{2 ( γ - 1 )}{γ + 1} C_{B} .

Since

C_{B}

is conserved along a streamline, that streamline has a unique critical speed.

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