Volume 28, Issue 1

Our main objective is to describe non-Fickean thermodiffusion in binary fluids within the framework of three recent theories of non-equilibrium thermodynamics, namely Extended Irreversible Thermodynamics (EIT), GENERIC (General Equation for the Non-Equilibrium Reversible Irreversible Coupling) and Thermodynamics with Internal Variables (IVT). In the first part presented in this paper, we develop the EIT description. For pedagogical reasons, we start from the simplest situation to end with the most intricate one. Therefore, we first examine the simple problem of mass diffusion at uniform temperature. Then we study heat transport in a one-component fluid before considering the more complex coupled heat and mass transfer. In Part II developed in the accompanying paper, we follow the same hierarchy of situations from the point of view of GENERIC. Finally, in Part III, we present the point of view of the thermodynamic theory of internal variables. Similarities and differences between EIT, GENERIC and IVT are stressed. In the present work, we have taken advantage of the problem of heat conduction to revisit the notion of caloric.

All the time evolution equations describing approaches to thermodynamic equilibrium states on all levels of description share a common structure. This structure has been collected in an abstract equation called GENERIC. A time evolution equation describing a particular physical system is obtained as a particular realization of GENERIC. In this paper we work out a realization describing the time evolution of a mixture of two fluids in which both non-Fickean diffusion and heat conduction take place.

Classical thermodynamics of irreversible processes is easily extended to describe non-equilibrium processes which are not directly involved in the conservation laws of mass, momentum and energy. These non-equilibrium processes can be understood as the evolution in time of some internal variables. Thermodynamics with internal variables is here used to describe unusual transport phenomena taking place in binary fluid mixtures and suspensions of particles. We also revisit the unusual heat transport of superfluids. A comparison is made with the description of these unusual phenomena given by Extended Irreversible Thermodynamics (Part I) and the GENERIC framework (Part II).

Helium II is modelled as a mixture, within macroscopic hydrodynamics, with two partial pressures and a single temperature. The temperature-dependency of concentrations of the superfluid and the normal fluid is accounted for by regarding the mixture as reacting. Moreover, a force of interaction between the superfluid and the normal fluid which traces back to Landau is considered. Accounting for dissipative processes turns out to be a standard application of classical descriptions of viscosity and heat conduction. A detailed analysis of the thermodynamic restrictions is developed. The effciency of the resulting model is tested by revisiting the main phenomena in helium II and evaluating the expression of the wave speed of sound. The mass production has no effect on the wave speed, while the force of interaction affects the propagation of second sound. Also, a simple non-local model is given for the description of persistent currents.