 Critical speed of a polytropic flow

For a steady isentropic flow with $p=K{\rho }^{\gamma }$, $K>0$ and $\gamma >1$, write the Bernoulli function asAt a fixed position, hold $C_B$, $K$ and the external potential $\Phi$ fixed and assume $H=C_B-\Phi >0$. The positive scalar mass flux is $j=\rho u$ for $u>0$. Differentiating the Bernoulli constraint gives $d\rho /du=-\rho u/c_s^2$, henceThe maximum occurs at $u=c_s=c_{\mathrm{cr}}$. Substituting this equality back into the energy constraint givesBoth the small-speed endpoint and the zero-enthalpy endpoint have vanishing $j$, so this interior stationary point is the maximum. The familiar formula with $C_B$ alone assumes a zero potential at the chosen section, or defines $C_B$ to exclude that fixed contribution.

For a perfect gas with no potential change, the result is equivalent to $T_{\mathrm{cr}}=2T_{\mathrm{stag}}/(\gamma +1)$. The local maximum explains choked flow; a complete nozzle solution must also connect the prescribed upstream state to the controlling section.^[ref]