The centripetal acceleration may be written in terms of a potential function \( \phi_c \)
\[
\phi_c = \frac{|\boldsymbol{\Omega}|^2 |\mathbf{r}_\perp|^2}{2} = \frac{|\boldsymbol{\Omega} \times \mathbf{r}|^2}{2} \Rightarrow \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}) = -\nabla \phi_c
\]
Since tthe centrifugal force can be written as a potential, it can be combined with the force potential of the momentum equation \(\rho \frac{d\mathbf{u}}{dt} = -\nabla p + \rho \nabla \phi + \mathcal{F}(\mathbf{u})\) to give a total force potential
\[
\Phi = \phi + \phi_c
\]
For the Coriolis acceleration \( 2\boldsymbol{\Omega} \times \mathbf{u}_R \), it is the only new term involving the fluid velocity, and it causes the main structural change in the momentum equation when moving to a rotating frame. The centrifugal force only modifies the force potential.
If spatial gradients are perceived identically in rotating and nonrotating frames, the momentum equation for an observer in a uniformly rotating coordinate frame becomes
\[
\rho \left[ \frac{d\mathbf{u}}{dt} + 2\boldsymbol{\Omega} \times \mathbf{u} \right] = -\nabla p + \rho \nabla \Phi + \mathcal{F}
\]
where \( \mathbf{u} \) is the velocity in the rotating frame
Although \( p \), \( \Phi \), and their gradients are frame-independent, the function \( \mathcal{F} \) (representing other forces) depends on the velocity field.
For a Newtonian fluid it is simple to show
\[
\mathcal{F}(\mathbf{u}_I) = \mathcal{F}(\mathbf{u}_R)
\]