On the natural structure of thermodynamic potentials and fluxes in the theory of chemically non-reacting binary mixtures

A theory describing the behavior of chemically non-reacting binary mixtures can be based on a detailed formulation of the governing equations for the individual components of the mixture or on treating the mixture as a single homogenized continuous medium. We argue that if we accept that both approaches can be used to describe the behavior of the given mixture, then the requirement on the equivalence of these approaches places restrictions on the possible structure of the internal energy, entropy, Helmholtz potential, and also of the diffusive, energy, and entropy fluxes. (The equivalence of the approaches is understood in the sense that the quantities used in one approach can be interpreted in terms of the quantities used in the other approach and vice versa. Further, both approaches must lead to the same predictions concerning the evolution of the physical system under consideration). In the case of a general chemically non-reacting binary mixture of components at the same temperature, we show that these restrictions can indeed be obtained by purely algebraic manipulations. An important outcome of this analysis is, for example, a general form of the evolution equation for the diffusive flux. The restrictions can be further exploited in the specification of thermodynamically consistent constitutive relations for quantities such as the interaction (drag) force or the Cauchy stress tensor. As an example of the application of the current framework, we derive, among others, a generalization of Fick's law and we recover several non-trivial results obtained by other techniques. The qualitative features of the derived generalization of Fick's law are demonstrated by a numerical experiment.