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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

$\displaystyle \frac{D\rho}{Dt}$ $\displaystyle = -\rho\,\nabla\cdot{\bf v},$ (14.30)
$\displaystyle \frac{D{\bf v}}{Dt}$ $\displaystyle = - \frac{\nabla p}{\rho} - \nabla{\mit\Psi},$ (14.31)
$\displaystyle \frac{D}{Dt}\left(\frac{p}{\rho^{\,\gamma}}\right)$ $\displaystyle = 0,$ (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

$\displaystyle d{\cal H} = \frac{dp}{\rho}$ (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.

Richard Fitzpatrick 2016-03-31