Project the steady <Euler equations for an inviscid fluid> along a streamline. With no gravity,
$$
u\,du+\frac{dp}{\rho}=0.
$$
For an isentropic fluid, $dp=c_s^2d\rho$, so
$$
\frac{d\rho}{du}=-\frac{\rho u}{c_s^2}.
$$
The scalar mass flux is $j=\rho u$, and hence
$$
\boxed{\frac{dj}{du}=\rho\left(1-\frac{u^2}{c_s^2}\right)}.
$$
It increases with speed in <subsonic flow> and decreases with speed in <supersonic flow>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-314/2/a/solution