Crocco's theorem

Crocco's theorem is a fluid dynamics theorem relating the flow velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. Crocco's 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:

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 × ω = v ∇ p 0
2. Entropy (the following form holds for plane steady flows): T d s d n = d h 0 d n + u ω
3. Momentum: ∂ u ∂ t + ∇ ( u 2 2 + h ) = u × ω + T ∇ s + g ,

In the above equations, u is the flow velocity vector, ω 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".