  There are several reasons to extend the presentation of Navier-Stokes
equations to multicomponent systems. Many technological applications are based
on physical phenomena that are present neither in pure elements nor in binary
mixtures. Whereas Fourier's law must already be generalized in binaries, it is
only with more than two components that Fick's law breaks down in its simple
form. The emergence of dissipative phenomena affects also the inertial
confinement fusion configurations, designed as prototypes for the future fusion
nuclear plants hopefully replacing the fission ones.
  This important topic can be described in much simpler terms than in many
textbooks since the publication of the formalism put forward recently by Snider
in \textit{Phys. Rev. E} \textbf{82}, 051201 (2010). In a very natural way, it
replaces the linearly dependent atomic fractions by the independent set of
partial densities. Then, the Chapman-Enskog procedure is hardly more
complicated for multicomponent mixtures than for pure elements. Moreover, the
recent proposal of a convergent kinetic equation by Baalrud and Daligault in
\textit{Phys. Plasmas} \textbf{26}, 082106 (2019) demonstrates that Boltzmann
equation with the potential of mean force is a far better choice in situations
close to equilibrium, as described by the Navier-Stokes equations, than Landau
or Lenard-Balescu equations.
  In our comprehensive presentation, we emphasize the physical arguments behind
Chapman-Enskog derivation and keep the mathematics as simple as possible. This
excludes as a technical non-essential aspect the solution of the linearized
Boltzmann equation through an expansion in Hermite polynomials. We discuss the
link with the second principle of Thermodynamics of entropy increase, and what
can be learned from this exposition.
