The change of entropy density is given by
as it is
where we have a convective term of the entropy flow.
According to Gibbs' relation follows for the movement of the mass center
and with somewhere else derived terms the entropy balance equation
we get for the entropy flow
the entropy production
The first term in the entropy production is due to the heat conduction, the second due to the diffusion, the third due to the field velocity gradient with viscous flow and the fourth emerges due to the chemical reactions, described with affinities..
The heat flow is given by
where the first term describes the "pure" heat conduction, the second describes the heath transfer due to diffusion - Soret effect.
The diffusion flow is described by
where the first term arises due to the thermodiffusion and the second due to the "pure" diffusion. The quantities
are the kinetic coefficients connected with the heat conduction, diffusion and crossover effects coefficients.