  We investigate analytically and numerically the existence of stationary
solutions converging to zero at infinity for the incompressible Navier-Stokes
equations in a two-dimensional exterior domain. More precisely, we find the
asymptotic behaviour for such solutions in the case where the net force on the
boundary of the domain is non-zero. In contrast to the three dimensional case,
where the asymptotic behaviour is given by a scale invariant solution, the
asymptote in the two-dimensional case is not scale invariant and has a wake. We
provide an asymptotic expansion for the velocity field at infinity, which shows
that, within a wake of width $|\boldsymbol{x}|^{2/3}$, the velocity decays like
$|\boldsymbol{x}|^{-1/3}$, whereas outside the wake, it decays like
$|\boldsymbol{x}|^{-2/3}$. We check numerically that this behaviour is accurate
at least up to second order and demonstrate how to use this information to
significantly improve the numerical simulations. Finally, in order to check the
compatibility of the present results with rigorous results for the case of zero
net force, we consider a family of boundary conditions on the body which
interpolate between the non-zero and the zero net force case.
