  We propose and analyse an augmented mixed finite element method for the
Navier--Stokes equations written in terms of velocity, vorticity, and pressure
with non-constant viscosity and no-slip boundary conditions. The weak
formulation includes least-squares terms arising from the constitutive equation
and from the incompressibility condition. The theoretical and practical
implications of using augmentation is discussed in detail. In addition, we use
fixed--point strategies to show the existence and uniqueness of continuous and
discrete solutions under the assumption of sufficiently small data. The method
is constructed using any compatible finite element pair for velocity and
pressure as dictated by Stokes inf-sup stability, while for vorticity any
generic discrete space (of arbitrary order) can be used. We establish optimal a
priori error estimates. Finally, we provide a set of numerical tests in 2D and
3D illustrating the behaviour of the scheme as well as verifying the
theoretical convergence rates.
