  Several generalizations of the well-known fluid model of Braginskii (Rev. of
Plasma Phys., 1965) are considered. We use the Landau collisional operator and
the moment method of Grad. We focus on the 21-moment model that is analogous to
the Braginskii model, and we also consider a 22-moment model. Both models are
formulated for general multi-species plasmas with arbitrary masses and
temperatures, where all the fluid moments are described by their evolution
equations. The 21-moment model contains two "heat flux vectors" (3rd and
5th-order moments) and two "viscosity-tensors" (2nd and 4th-order moments). The
Braginskii model is then obtained as a particular case of a one ion-electron
plasma with similar temperatures, with de-coupled heat fluxes and
viscosity-tensors expressed in a quasi-static approximation. We provide all the
numerical values of the Braginskii model in a fully analytic form (together
with the 4th and 5th-order moments). For multi-species plasmas, the model makes
calculation of transport coefficients straightforward. Formulation in fluid
moments (instead of Hermite moments) is also suitable for implementation into
existing numerical codes. It is emphasized that it is the quasi-static
approximation which makes some Braginskii coefficients divergent in a
weakly-collisional regime. Importantly, we show that the heat fluxes and
viscosity-tensors are coupled even in the linear approximation, and that the
fully contracted (scalar) perturbations of the 4th-order moment, which are
accounted for in the 22-moment model, modify the energy exchange rates. We also
provide several Appendices, which can be useful as a guide for deriving the
Braginskii model with the moment method of Grad.
