Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 6 / a / Solution

For a steady axisymmetric flow ${\mathbf{u}}_0=r\Omega (r)\hat{\mathbf{\varphi }}$, incompressible flow is automatic. The only material acceleration is centripetal, $-r{\Omega }^2\hat{\mathbf{r}}$. The radial Euler equations therefore requireWith 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.