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Mixtures of relativistic gases in gravitational fields: combined Chapman-Enskog and Grad method and the Onsager relations

In this work we study a r-species mixture of gases within the relativistic kinetic theory point of view. We use the relativistic covariant full Boltzmann equation and we incorporate the Schwarzschild metric. The method of solution of the Boltzmann equation is a combination of the Chapman-Enskog and Grad representations. The thermodynamic fluxes are expressed as functions of the thermodynamic forces so that the generalized expressions for the Navier-Stokes, Fick and Fourier laws are obtained. The constitutive equations for the diffusion and heat fluxes of the mixture are functions of thermal and diffusion forces which depend on the acceleration and the gravitational potential gradient. While this dependence is of relativistic nature for the thermal force, this is not the case for the diffusion forces. We show also that the matrix of the diffusion coefficients is symmetric and the thermal-diffusion coefficient is equal to the diffusion-thermal one, proving the Onsager reciprocity relations. The entropy flux of the mixture is also expressed in terms of the thermal and diffusion forces, so that its dependence on the acceleration and gravitational potential gradient is also determined.