Isothermal nonstandard diffusion in a two-component fluid mixture: A Hamiltonian approach

A nonlinear formalism describing non-Fickian diffusion in a two-component mixture at uniform temperature is presented. It is based on a Hamiltonian description and is achieved by introducing two potentials: the thermodynamic free energy potential and the so-called dissipative potential. The evolution equations are expressed in terms of Poisson's brackets whose generating potentials are the two above-mentioned potentials. In view of a better understanding, the paper is presented in a rather pedagogical way, starting with the simple Hamiltonian description of classical hydrodynamics before examining respectively classical Fickian diffusion and non-Fickian diffusion in ordinary fluids including non-local effects. The formalism rests on the introduction of a "flux" variable in the spirit of Extended Irreversible Thermodynamics (EIT). It is seen that this extra variable, which complements the classical variables (mass density, mass concentration and barycentric velocity), is related to the relative velocities: of both constituents, The evolution equations for the whole set of variables are derived in a systematic manner and the correlation between mechanical properties and diffusion is emphasized. The present work generalizes earlier nonclassical models and opens the way to more complicated situations involving for example thermal effects, phase change, multicomponent mixtures and diffusion in polymeric systems.