Crocco's Theorem

Crocco's theorem is a fluid dynamics theorem relating the velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. The theorem was first written by Italian scientist Luigi Crocco, a son of Gaetano Crocco.

Because stagnation pressure loss and entropy generation can be viewed as essentially the same thing, there are two popular forms for writing Crocco's theorem:

1. Stagnation pressure\[ V \times \omega = \frac{1}{\rho} \nabla p_0 \]
2. Entropy\[ T \frac{ds}{dn} = \frac{dh_0}{dn} + ||V||\omega \]
3. quantity of movement\[ \frac{\partial \mathbf{v}}{\partial t} + \nabla \left(\frac{\mathbf{v}^2}{2} + h \right) = \mathbf{v} \times \mathbf{\omega} + T \nabla s + \mathbf{f},\]

In the above equations, \( V \) is velocity, \( \omega \) is vorticity, \( \rho \) is density, \( p_0 \) is stagnation pressure, \( T \) is temperature, \( s \) is entropy, and \( n \) is the direction normal to the streamlines.