I thnk you can compare your work with this page. It's chock-full of rendered equations, ending with this conclusion:

Edit: another, somewhat longer & deeper analysis here

I thnk you can compare your work with this page. It's chock-full of rendered equations, ending with this conclusion:

where $v$ is the flow velocity and the Lorentz factor $\gamma$ is given by

$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

The time component of eq. $(9.6)$ is implied by the other three. In the case of isentropic flow, that is, when $\sigma/n=const.$, and assuming the flow to be steady, the spatial components of eq. $(9.6)$ give

$$\gamma(v\cdot grad)(\gamma\omega v/n)+c^2grad(\omega/n)=0$$

Scalar multiplication by $v$ leads to

$$(v\cdot grad)(\gamma\omega/n)=0$$

which implies that along any streamline the quantity

$$\gamma\omega/n=const.$$

This is the relativistic version of Bernoulli's equation.

Another, somewhat longer and deeper analysis can be found here.