Balance and constitutive equations for diffusion in mixtures of fluids

The paper deals with the balance equations and constitutive models for mixtures of reacting fluids and addresses particular attention to the modelling of diffusion. Once the balance equations are established, both in the Eulerian and in the Lagrangian description, the second law of thermodynamics is exploited by letting the constitutive functions of the single constituents depend on temperature, mass density, their gradients, and the stretching tensor pertaining to the constituents. The results provide some generalizations of known relations about pressure and the law of mass action. Next the balance equations for the whole mixture are considered and the exploitation of the corresponding entropy inequality is provided thus showing that the modelling with single constituents is more appropriate. The balance equations for the diffusion fluxes are determined also in the case that a single constituent is considered to select the reference field as is natural in the case of a solute–solvent mixture. The balance equation so established allows the Maxwell–Stefan model of molecular friction to be incorporated as the interaction force between constituents. Next, the restriction to binary mixtures, in stationary conditions, provides a simple model for the diffusion flux that, in appropriate limit approximations, leads to the classical Fick’s law and the wellknown Nernst–Planck law.