  In this paper we propose and analyse a new formulation and pointwise
divergence-free mixed finite element methods for the numerical approximation of
Darcy--Brinkman equations in vorticity--velocity--pressure form, coupled with a
transport equation for thermal energy with viscous dissipative effect and mixed
Navier-type boundary conditions. The solvability analysis of the continuous and
discrete problems is significantly more involved than usual as it hinges on
Banach spaces needed to properly control the advective and dissipative terms in
the non-isothermal energy balance equation. We proceed by decoupling the set of
equations and use the Banach fixed-point theorem in combination with the
abstract theory for perturbed saddle-point problems. Some of the necessary
estimates are straightforward modifications of well-known results, while other
technical tools require a more elaborated analysis. The velocity is
approximated by Raviart--Thomas elements, the vorticity uses N\'ed\'elec spaces
of the first kind, the pressure is approximated by piecewise polynomials, and
the temperature by continuous and piecewise polynomials of one degree higher
than pressure. Special care is needed to establish discrete inf-sup conditions
since the curl of the discrete vorticity is not necessarily contained in the
discrete velocity space, therefore suggesting to use two different
Raviart--Thomas interpolants. A discrete fixed-point argument is used to show
well-posedness of the Galerkin scheme. Error estimates in appropriate norms are
derived, and a few representative numerical examples in 2D and 3D and with
mixed boundary conditions are provided.
