= Solution
In an inertial frame, a <circular orbit> at radius $R$ has radial <acceleration> $-R\Omega(R)^2\mathbf e_R$. Neglecting fluid and magnetic forces gives
$$
-\rho R\Omega(R)^2=-\rho\frac{d\Phi}{dR},\qquad\boxed{\frac{d\Phi}{dR}=R\Omega(R)^2.}
$$
In a local rotating frame with $\Omega_0=\Omega(R_0)$, the fluid is instantaneously at rest at the reference radius $R_0$. The radial <centrifugal force> $+\rho R_0\Omega_0^2$ then balances the inward <gravitational force>. Away from $R_0$, the fluid's nonzero rotating-frame <velocity> must also be retained; the entire differentially rotating disk is not at rest in that one frame.
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