Steady rotating-frame Bernoulli integral ($\frac{1}{2}|\mathbf{v}{|}^2+p/\rho +\Phi -\frac{1}{2}{\Omega }^2{r}_{\perp }^2$)

For steady flow of constant-density fluid in a uniformly rotating frame with conservative body forces, the displayed quantity is constant along each relative streamline. The centrifugal potential contributes the negative quadratic term, and the Coriolis force contributes no work because it is perpendicular to the relative velocity. The result follows by dotting the rotating Euler equations for an inviscid fluid with that velocity.