Editing Euler equations (fluid dynamics) (section)

===Friedmann form and Crocco form===
{{See also|Crocco's theorem}}
By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:

<math display="block">v \nabla p = -T \nabla s + \nabla h,</math>

in the convective form of Euler momentum equation, one arrives to:

<math display="block">\frac{D\mathbf u}{Dt}=T \nabla\,s-\nabla \,h.</math>

[[Alexander Friedmann|Friedmann]] deduced this equation for the particular case of a [[perfect gas]] and published it in 1922.{{sfn|Friedmann|1934|p=198|loc=Eq 91}} However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it.

On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains:

<math display="block">\frac{\partial\mathbf{u}}{\partial t} + \frac{1}{2} \nabla\left(u^2\right) + (\nabla \times \mathbf{u}) \times \mathbf{u} + \frac{\nabla p}{\rho} = \mathbf{g},</math>

and by defining the specific total enthalpy:

<math display="block">h^t = h + \frac{1}{2}u^2,</math>

one arrives to the [[Crocco's theorem|Crocco–Vazsonyi form]]{{sfn|Henderson|2000|p=177|loc=par. 2.12 Crocco's theorem}} (Crocco, 1937) of the Euler momentum equation:

<math display="block">\frac{\partial \mathbf{ u}}{\partial t} + (\nabla \times \mathbf u) \times  \mathbf u  - T \nabla s + \nabla h^t = \mathbf{g}.</math>

In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form:

<math display="block">\begin{align}
   \mathbf{u} \times \nabla \times \mathbf{u} + T\nabla s - \nabla h^t &= \mathbf{g},\\
     \mathbf{u} \cdot \nabla s &= 0, \\
   \mathbf{u} \cdot \nabla h^t &= 0.
\end{align}</math>

Finally if the flow is also isothermal:

<math display="block">T \nabla s = \nabla (T s),</math>

by defining the specific total [[Gibbs free energy]]:

<math display="block"> g^t \equiv h^t + Ts,</math>

the Crocco's form can be reduced to:

<math display="block">\begin{align}
  \mathbf{u} \times \nabla \times \mathbf{u} - \nabla g^t &= \mathbf{g},\\
                              \mathbf{u} \cdot \nabla g^t &= 0.
\end{align}</math>

From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.