Consider a body of perfect gas in smooth, steady, isentropic motion with body force $\boldsymbol{b}=\mathbf{0}$, heat supply $r=0$, density $\rho(\boldsymbol{x})$, velocity $\boldsymbol{v}(\boldsymbol{x})$, and entropy $\eta_{0}$ (constant). Let $\boldsymbol{y}(s), s \in \mathbb{R}$, denote an arbitrary streamline defined by the equation $\boldsymbol{y}(s)^{\prime}=$ $\boldsymbol{v}(\boldsymbol{y}(s))$, and let $\vartheta(s)=|\boldsymbol{v}(\boldsymbol{y}(s))|$ denote the local flow speed and $m(s)=\rho(\boldsymbol{y}(s)) \vartheta(s)$ the local mass flux along the streamline.
(a) For any streamline show
$$
\frac{d \rho}{d s}=\nabla^{x} \rho \cdot \boldsymbol{v}, \quad \vartheta \frac{d \vartheta}{d s}=\boldsymbol{v} \cdot\left(\nabla^{x} \boldsymbol{v}\right) \boldsymbol{v}
$$
(b) Use balance of linear momentum to show
$$
\frac{d \rho}{d s}=-\frac{\rho \vartheta}{c^{2}} \frac{d \vartheta}{d s}, \quad \frac{d m}{d s}=\rho\left[1-M^{2}\right] \frac{d \vartheta}{d s}
$$
where $c(s)$ is the local sound speed and $M(s)$ is the local Mach number along the streamline.
Remark: The first result in (b) shows that mass density always decreases with increasing flow speed. The second result shows that mass flux may increase or decrease depending on the local Mach number. In subsonic regions of a flow $(M<1)$, mass flux increases with increasing flow speed. However, in supersonic regions of flow $(M>1)$, mass flux decreases with increasing flow speed. Many phenomena in the dynamics of perfect gases can be attributed to this change in behavior of the mass flux between subsonic and supersonic speeds.