Isentropic flow

Remarks

For an ideal gas with constant ratio of specific heats $\gamma$, isentropic flow means that $p / \rho^\gamma$ is constant along particle paths. In steady isentropic flow with no heat exchange, along a streamline, $$\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2} M^2, \qquad \frac{p_0}{p} = \left(\frac{T_0}{T}\right)^{\gamma/(\gamma - 1)}, \qquad \frac{\rho_0}{\rho} = \left(\frac{T_0}{T}\right)^{1/(\gamma - 1)},$$ where $M$ is the Mach number and the index $0$ marks the values at rest. Example: smooth inviscid flow of air through a nozzle; with $T_0 = 300\ \mathrm{K}$ and $\gamma = 1.4$, the temperature at $M = 1$ is $300 / 1.2 = 250\ \mathrm{K}$. Non-example: flow through a shock wave, where the entropy of each particle jumps up, so $Ds/Dt = 0$ fails. Viscous friction and heat conduction also raise the entropy, so flow inside a boundary layer is not isentropic.