Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 1 / c

The functionhas its unique minimum at $\mathcal{M}=1$, where $f(1)=4$. At large $r$, mass conservation and the reservoir boundary conditions give $u\propto r^{-2}$ and hence $\mathcal{M}\to 0$. The physical solution therefore begins on the subsonic branch $0<\mathcal{M}<1$. It cannot pass smoothly to $\mathcal{M}>1$ without reaching the minimum, where the two algebraic branches meet. For $\gamma =5/3$ that meeting can occur only in the limiting central behavior of the critical solution, so the flow remains subsonic for every $r>0$.