= Uniform-vorticity circular flow in an annulus
{title2=$u_\phi=V(4a^2/r-r)/(3a)$}
Circular streamlines and constant <vorticity> imply $u_\phi=\omega r/2+C/r$. Zero velocity at $2a$ and positive azimuthal velocity $V$ at $a$ give
$$
u_\phi=\frac V{3a}(4a^2/r-r),\qquad \omega=-\frac{2V}{3a}.
$$
The radial <Euler equations for an inviscid fluid> give $p_r=\rho u_\phi^2/r$, yielding outer-minus-inner pressure $(15-16\log2)\rho V^2/18$. The pressure difference is independent of circulation orientation.
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