  In this paper, we deal with the long standing open problem of characterising
rotationally symmetric solutions to $\Delta u = -2$, when Dirichlet boundary
conditions are imposed on a ring-shaped planar domain. From a physical
perspective, the solution represents the velocity of a homogeneous
incompressible fluid, flowing in steady parallel streamlines through a hollow
cylindrical pipe and obeying a no-slip condition. In contrast with Serrin's
classical result, we show that the simplest possible set of overdetermining
conditions, namely the prescription of locally constant Neumann boundary data,
is not sufficient to obtain a complete characterisation of the solutions. A
further requirement on the number of maximum points arises in our analysis as a
necessary and sufficient condition for the rotational symmetry. In
fluid-dynamical terms, our results imply that if the wall shear stress is
constant on some connected component of the pipe's wall, then the velocity of
the fluid must attain its maximal value only at finitely many streamlines,
unless the hollow pipe itself consists of a couple of concentric cylindrical
round tubes. A major difficulty in the analysis of this problem comes from the
lack of monotonicity of the model solutions, which makes the moving plane
method ineffective. To remedy this issue, we introduce some new arguments in
the spirit of comparison geometry, that we believe of independent interest.
