Crocco's Theorem

Crocco's theorem is an aerodynamic 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:

:$\frac=T \nabla\,s-\nabla \,h$

However, usually this theorem is connected with the name of Italian scientist Luigi Crocco,Crocco L
Eine neue Stromfunktion für die Erforschung der Bewegung der Gase mit Rotation
ZAMM, Vol. 17, Issue 1, pp. 1–7, 1937. DOI: 10.1002/zamm.19370170103. Crocco writes the theorem in the form $\scriptstyle\mathrm\,\mathbf u\times\mathbf u=T\mathrm\,S$ for perfect gas (the last formula on page 2). 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:

# Stagnation pressure: $\mathbf u \times \boldsymbol \omega =v \nabla p_0$ Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopædia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11)
# Entropy (the following form holds for plane steady flows): $T \frac = \frac +u \omega$ Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY (eq. (7.33)).
# Momentum: $\frac + \nabla \left(\frac + h \right) = \mathbf u \times \boldsymbol \omega + T \nabla s + \mathbf,$

In the above equations, $\mathbf u$ 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, $\mathbf$ 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".

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Fluid dynamics
Aerodynamics