Rotating-flow pressure potential ($\psi =p/\rho +\Phi -\frac{1}{2}|\mathbf{\Omega }\times \mathbf{r}{|}^2$)

For a uniformly rotating incompressible fluid of constant mass density, pressure and conservative body forces combine as the gradient of this scalar. The centrifugal term has a negative sign. Pressure enforces incompressible flow, whereas the velocity-dependent Coriolis force remains explicit and cannot be included in a scalar position potential. Adding a spatially constant function of time to $\psi$ does not change the motion.