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Chapman-Enskog derivation of multicomponent Navier-Stokes equations

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.