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

Project the steady Euler equations for an inviscid fluid along a streamline. With no gravity,For an isentropic fluid, $dp=c_s^{2}d\rho$, soThe scalar mass flux is $j=\rho u$, and henceIt increases with speed in subsonic flow and decreases with speed in supersonic flow.

The continuous transonic branch passes from $dj/du>0$ to $dj/du<0$, so $j(u)$ has a maximum where its derivative vanishes. At this critical speed of a polytropic flow,The maximum mass flux is therefore attained at the sonic point.

The Bernoulli function for steady unmagnetized flow without gravity isFor a polytropic equation of state, $h=c_s^2/(\gamma -1)$. Evaluation at the critical point found in part b givesConsequentlySince $C_B$ is conserved along a streamline, that streamline has a unique critical speed.

Across a stationary normal shock wave, the Rankine-Hugoniot conditions for a perfect gas conserveMomentum conservation and $1/\rho =u/j$ giveDirect elimination of the two pressures and densities from these three jump conditions givesPart c identifies the right side with the square of the critical speed, so the Prandtl shock relation isIt maps the unique upstream supersonic state on a given Bernoulli streamline to its downstream subsonic state.