  In this paper we investigate the uniqueness of solutions of the steady planar
Navier-Stokes equations with different boundary conditions in the exterior
domain. For a class of incompressible flow with constant vorticity, we prove
the uniqueness of the solution under the enhanced Navier boundary conditions.
At the same time, some counterexamples are given to show that the uniqueness of
the solution fails under the Navier boundary conditions. For the general
incompressible flow with Dirichlet boundary condition, we prove various
sufficient conditions for the uniqueness of the solution.
