We consider the planar Taylor-Couette system for the steady motion of a viscous incompressible fluid in the region between two concentric disks, the inner one being at rest and the outer one rotating with constant angular speed. We study the uniqueness and multiplicity of solutions to the forced system in different classes. For any angular velocity, we prove that the classical Taylor-Couette flow is the unique smooth solution displaying rotational symmetry. Instead, we show that infinitely many solutions arise, even for arbitrarily small angular velocities, in a larger class of incomplete solutions that we introduce. By prescribing the transversal flux, the unique solvability of the Taylor-Couette system is recovered among rotationally invariant incomplete solutions. Finally, we study the behavior of these solutions as the radius of the outer disk goes to infinity, connecting our results with the celebrated Stokes paradox. This is a joint work with Filippo Gazzola (Politecnico di Milano) and Jiří Neustupa (Institute of Mathematics of the Czech Academy of Sciences). 

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