Maybe you are comparing the Crocco relation and the Bernoulli relation?

Crocco requires only an ideal fluid model (stationary condition can be removed if the velocity depends on a potential) and works with compressible flow while Bernoulli requires:

1) curl v=0

4) entropy is homogeneous ;

5) rho(x, t)=rho0

I agree about the conditions for H=constant, strictly related to Crocco's theorem. The condition is possible by imposing ideal flow condition in the governing equation of total energy. Assuming that requirements for both equations are satisfied, how can I explain the different results I get by imposing the same variation of the v^2/2 term?

Could you elaborate further your question? We know from Crocco that for an irrotational flow we have

Grad H = T Grad s

where H= v^2/2 +h + psi. Furthermore, we have grad v^2/2 = v.grad v - (curl v) x v.

Therefore, you see that grad v^2/2 can be the same for infinite values of the the terms in the RHS provided that their difference is the same.

Let's assume we have an ideal flow that is steady, irrotational, isoentropic and incompressible.

So hypothesis are satisfied for both equations.

I know that it has to be:

gamma/(gamma-1) p/rho + v^2 /2 = constant.

p/rho + v^2 /2 = constant.

Suppose that my term v^2 / 2 has a variation of +10.

Using the first equation, I should get a variation of gamma/(gamma-1) p/rho equal to -10 to keep the first member constant. This is equivalent to say that p/rho should variate of a quantity -10/((gamma)/(gamma-1)). 

Using the second equation, I should get a variation of p/rho equal to -10.

So I get two different variations of the same quantity, which is impossible.

Are there conflictual hypothesis in the two equations?

I am not sure to understand but if you consider the second equation then you have adopted the hypothesis of incompressible flow. Hence, there is no state equation p=rho*R*T as introduced for compressible flows...Crocco does not require the introduction of a model for the gas as you write in the first equation. And as you can see, the two equations cannot state the same integral...

I attached a document where in Sec.1.6 you can find how I present this issue to my students

Thank you for the answer.

Even if you said you didn't fully understand the question, you provided the answer I was searching for.

The first equation is in fact obtained by applying the equation of state of the gas to the H=const. derived from Crocco's equation (with isoentropic and irrotational flow hypothesis).

As you said, there is no need to introduce a model for the gas, so it's not possible to apply an equation of state to an uncompressible flow.