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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 314

The <Bernoulli function> for steady unmagnetized flow without gravity is
$$
C_B=\frac{u^2}{2}+h.
$$
For a <polytropic equation of state>, $h=c_s^2/(\gamma-1)$. Evaluation at the critical point found in part b gives
$$
\boxed{
C_B=\frac{c_{\rm cr}^2}{2}
+\frac{c_{\rm cr}^2}{\gamma-1}
=\frac{\gamma+1}{2(\gamma-1)}c_{\rm cr}^2}.
$$
Consequently
$$
c_{\rm cr}^2=\frac{2(\gamma-1)}{\gamma+1}C_B.
$$
Since $C_B$ is conserved along a streamline, that streamline has a unique critical speed.

Solution

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

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

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

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