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:^[1]

$\frac{D\mathbf u}{Dt}=T \nabla\,s-\nabla \,h$

However, usually this theorem is connected with the name of Italian scientist Luigi Crocco,^[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:

1. Stagnation pressure: $\mathbf u \times \boldsymbol \omega =v \nabla p_0$ ^[3]
2. Entropy (the following form holds for plane steady flows): $T \frac{ds}{dn} = \frac{dh_0}{dn} +u \omega$ ^[4]
3. Momentum: $\frac{\partial \mathbf u}{\partial t} + \nabla \left(\frac{u^2}{2} + h \right) = u \times \boldsymbol \omega + T \nabla s + \mathbf{g},$

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, $\mathbf{g}$ 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".

References

1. Friedmann A. An essay on hydrodynamics of compressible fluid (Опыт гидромеханики сжимаемой жидкости), Petrograd, 1922, 516 p., reprinted in 1934 under the editorship of Nikolai Kochin (see the first formula on page 198 of the reprint).
2. 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 $\scriptstyle\mathrm{rot}\,\mathbf u\times\mathbf u=T\mathrm{grad}\,S$ for perfect gas (the last formula on page 2).
3. Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopaedia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11)
4. Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY (eq. (7.33)).