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\end{equation}  you get   \begin{equation}  \frac{1}{2}\rho v^2 \frac{1}{2}v^2  - \Phi_G +\int{\frac{dP}{\rho}} = \rm{cst} \end{equation}  To evaluate the last term on the RHS let's assum we are dealing with a monatomic ideal gas. Then  \begin{equation}