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Let $\mathbf{v}$ denote the velocity relative to the uniformly rotating frame, and write $r_{\perp }^2=x^2+y^2$. For rotation $\mathbf{\Omega }=\Omega {\mathbf{e}}_z$, the Euler equations for an inviscid fluid in that frame areThe negative quadratic term is the centrifugal potential, whose negative gradient is the outward centrifugal acceleration per unit mass. In steady flow, dotting the equation with $\mathbf{v}$ eliminates the Coriolis force, since $\mathbf{v}\cdot (\mathbf{\Omega }\times \mathbf{v})=0$. The remaining acceleration term is $\mathbf{v}\cdot \nabla (|\mathbf{v}{|}^2/2)$. ThusBy part (b), the steady rotating-frame Bernoulli integral isSteadiness here is measured in the rotating frame, including the body-force potential used in that frame.