Thermodynamic Fluxes and Relaxation Processes

The laws of conservation determine the equations of fluid motion which, however, contain a few unknown quantities discussed below. 

1 Note, that a set of internal variables with labels, which take a continuous set of values, can be considered. Grmela (1985) and Jongschaap (1991) have generalised the above-written relations for this case. They showed that the values of the distribution function itself W(p, t) in the problem of dynamics of dumbbells (see Appendix F), for example, can be considered as a set of internal variables, whereas the arguments of the function play the role of the label a with a continuous set of values p. 

Expression for production of entropy (8.18) can be now compared with the general results of non-equilibrium thermodynamics, which are known for both non-stationary and stationary cases. It is obvious, that last term in the right-hand side of relation (8.18) corresponds to a non-stationary case and includes the equation of change of internal variables that is relaxation equation. The first two terms in formula (8.18) correspond to a stationary case and should be considered as the products of thermodynamic fluxes and thermodynamic forces (it is possible with any multipliers). When the internal variables are absent, we should write a relation between the fluxes and forces in the form 

At small gradients, the right parts of these relations can be expanded in a power series. In linear approximation of a parity for the anisotropic environment one gets 

Here one can take advantage of the Onsager principle, that is equate factors at cross members. 

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