  Steady incompressible potential flows of an inviscid or viscous fluid are
considered in infinite N-dimensional cylinders with tangential boundary
conditions. We show that such flows, if away from stagnation, are constant and
parallel to the direction of the cylinder. This means equivalently that a
harmonic function whose gradient is bounded away from zero in an infinite
cylinder with Neumann boundary conditions is an affine function. The proof of
this rigidity result uses a combination of ODE and PDE arguments, respectively
for the streamlines of the flow and the harmonic potential function.
