Crocco's theorem

In aerodynamics, Crocco's theorem relates the flow velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. This theorem gives the relation between the thermodynamics and fluid kinematics. The theorem was first enunciated by Alexander Friedmann for the particular case of a perfect gas and published in 1922:

$\frac{D𝐮}{Dt}=T\mathrm{\nabla }s-\mathrm{\nabla }h$

However, usually this theorem is connected with the name of Italian scientist Luigi Crocco, a son of Gaetano Crocco.

Consider an element of fluid in the flow field subjected to translational and rotational motion: because stagnation pressure loss and entropy generation can be viewed as essentially the same thing, there are three popular forms for writing Crocco's theorem:

1. Stagnation pressure: $𝐮\times 𝝎=v\mathrm{\nabla }{p}_0$
2. Entropy (the following form holds for plane steady flows): $T\frac{ds}{dn}=\frac{d{h}_0}{dn}+u\omega$
3. Momentum: $\frac{\partial 𝐮}{\partial t}+\mathrm{\nabla }(\frac{u^2}{2}+h)=𝐮\times 𝝎+T\mathrm{\nabla }s+𝐠,$

In the above equations, $𝐮$ is the flow velocity vector, $\omega$ is the vorticity, $v$ is the specific volume, $p_0$ is the stagnation pressure, $T$ is temperature, $s$ is specific entropy, $h$ is specific enthalpy, $𝐠$ is specific body force, and $n$ is the direction normal to the streamlines. All quantities considered (entropy, enthalpy, and body force) are specific, in the sense of "per unit mass".