  We propose and analyse an augmented mixed finite element method for the Oseen
equations written in terms of velocity, vorticity, and pressure with
non-constant viscosity and homogeneous Dirichlet boundary condition for the
velocity. The weak formulation includes least-squares terms arising from the
constitutive equation and from the incompressibility condition, and we show
that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the
arguments of the continuous analysis, the stability and solvability of the
discrete problem are established. The method is suited for any Stokes inf-sup
stable finite element pair for velocity and pressure, while for vorticity any
generic discrete space (of arbitrary order) can be used. A priori and a
posteriori error estimates are derived using two specific families of discrete
subspaces. Finally, we provide a set of numerical tests illustrating the
behaviour of the scheme, verifying the theoretical convergence rates, and
showing the performance of the adaptive algorithm guided by residual a
posteriori error estimation.
