Isentropic Flow

In the limit of vanishing viscosity and heat conduction, the equations of compressible ideal gas flow, introduced in Section 1.15, can be written

$$\frac{D\rho}{Dt} = -\rho\,\nabla\cdot{\bf v},$$ (14.30)

$$\frac{D{\bf v}}{Dt} = - \frac{\nabla p}{\rho} - \nabla{\mit\Psi},$$ (14.31)

$$\frac{D}{Dt}\left(\frac{p}{\rho^{\,\gamma}}\right) = 0,$$ (14.32)

where ${\bf v}$ is the flow velocity, and ${\mit\Psi}$ the potential energy per unit mass. It is clear from Equations (14.25) and (14.32) that the specific entropy is constant along a streamline, but not necessarily the same constant on different streamlines. Such flow is said to be isentropic. From Equation (14.29), isentropic flow is characterized by

$$d{\cal H} = \frac{dp}{\rho}$$ (14.33)

along a streamline. More generally, isentropic flow is characterized by $p/\rho^{\,\gamma}$ , $\rho/T^{\,\gamma-1}$ , and $T^{\,\gamma}/p^{\,\gamma-1}$ , constant along streamlines.