Uniform-vorticity circular flow in an annulus ($u_{\varphi }=V(4a^2/r-r)/(3a)$)

Circular streamlines and constant vorticity imply $u_{\varphi }=\omega r/2+C/r$. Zero velocity at $2a$ and positive azimuthal velocity $V$ at $a$ giveThe radial Euler equations for an inviscid fluid give $p_r=\rho u_{\varphi }^2/r$, yielding outer-minus-inner pressure $(15-16\log 2)\rho V^2/18$. The pressure difference is independent of circulation orientation.