= Solution
For a steady axisymmetric flow $\mathbf u_0=r\Omega(r)\widehat{\boldsymbol\phi}$, <incompressible flow> is automatic. The only material <acceleration> is centripetal, $-r\Omega^2\widehat{\mathbf r}$. The radial <Euler equations> therefore require
$$
\boxed{\frac{dp_0}{dr}=\rho r\Omega(r)^2.}
$$
With no <viscosity> there is no tangential shear <stress> to redistribute <angular momentum>. Any sufficiently regular $\Omega(r)$ can therefore be a steady inviscid profile, with the radial <pressure> adjusted by integrating this equation. This is a statement of equilibrium, not of stability: some such profiles are centrifugally unstable.
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