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Crocco's theorem

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

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: ${\displaystyle \mathbf {u} \times {\boldsymbol {\omega }}=v\nabla p_{0}}$ ^[3]
2. Entropy (the following form holds for plane steady flows): ${\displaystyle T{\frac {ds}{dn}}={\frac {dh_{0}}{dn}}+u\omega }$ ^[4]
3. Momentum: ${\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \left({\frac {u^{2}}{2}}+h\right)=\mathbf {u} \times {\boldsymbol {\omega }}+T\nabla s+\mathbf {g} ,}$

In the above equations, ${\displaystyle \mathbf {u} }$ is the flow velocity vector, ${\displaystyle \omega }$ is the vorticity, ${\displaystyle v}$ is the specific volume, ${\displaystyle p_{0}}$ is the stagnation pressure, ${\displaystyle T}$istemperature, ${\displaystyle s}$ is specific entropy, ${\displaystyle h}$ is specific enthalpy, ${\displaystyle \mathbf {g} }$ is specific body force, and ${\displaystyle n}$ is the direction normal to the streamlines. All quantities considered (entropy, enthalpy, and body force) are specific, in the sense of "per unit mass".

1. ^ Friedmann A. An essay on hydrodynamics of compressible fluid (Опыт гидромеханики сжимаемой жидкости), Petrograd, 1922, 516 p., reprinted Archived 2016-03-03 at the Wayback Machine in 1934 under the editorship of Nikolai Kochin (see the first formula on page 198 of the reprint).

^ 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 ${\displaystyle \scriptstyle \mathrm {rot} \,\mathbf {u} \times \mathbf {u} =T\mathrm {grad} \,S}$ for perfect gas (the last formula on page 2).