Nonlinear diffusion equations in fluid mixtures

Abstract

The whole set of balance equations for chemically-reacting fluid mixtures is established. The diffusion flux relative to the barycentric reference is shown to satisfy a first-order, non-linear differential equation. This in turn means that the diffusion flux is given by a balance equation, not by a constitutive assumption at the outset. Next, by way of application, limiting properties of the differential equation are shown to provide Fick's law and the Nernst-Planck equation. Moreover, known generalized forces of the literature prove to be obtained by appropriate constitutive assumptions on the stresses and the interaction forces. The entropy inequality is exploited by letting the constitutive functions of any constituent depend on temperature, mass density and their gradients thus accounting for nonlocality effects. Among the results, the generalization of the classical law of mass action is provided. The balance equation for the diffusion flux makes the system of equations for diffusion hyperbolic, provided heat conduction and viscosity are disregarded. This is ascertained by the analysis of discontinuity waves of order 2 (acceleration waves). The wave speed is derived explicitly in the case of binary mixtures.

A unified framework for Navier–Stokes Cahn–Hilliard models with non-matching densities

“…We note however that it was first derived in 1975 by Müller31 and revisited in e.g. Ref 29. and Ref 30…”

Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids

“…The requirement distinguishes from the compressible situation by the occurrence of the hydrodynamic pressure p, see e.g. Morro [20]. We take the following model for the mass transfer:…”

“…Since then, the topic has become more mature with the important contributions of Truesdell [29,30] and Truesdell and Toupin [32]. More complete overviews of rational mixture theory are provided by Green and Naghdi [14], Müller [21], Müller and Ruggeri [22], Bowen [4,5], Truesdell [31], Morro [20], and others.…”