Anomalous Diffusion in Miscible Incompressible Fluids

Abstract

In this work we present an application of Linear Irreversible Thermodynamics to the problem of diffusion among miscible incompressible fluids. First, we suppose that the mixture density is a function of the volume fractions. This hypothesis allows us to reformulate the interface diffusion problem in terms of transport equations for the concentrations or the mass-volume fractions, the momentum balance equations for the diffusion flux and the energy balance equations for the relative internal energies, respectively. The transport equations so obtained are generalized equations of the classical diffusion equations obtained by means of fluctuating hydrodynamics and the H model of critical dynamics.

Afterwards we apply linear irreversible thermodynamics in order to get the entropy production and then assume a linear relation between fluxes and forces, obtaining the constitutive equations for the diffusion flux, the heat flux, the stress tensor for each component. Finally, from this choice we compare our results with those originated from fluctuating hydrodynamics and continuum theory. We also show that the constitutive equation for diffusion flux is a Maxwell-Cattaneo's type relaxation equation and therefore a generalization of Fick's law. Also the stress tensor constitutive equation generalizes the Korteweg de Vries stress tensor equations and we present how to arrive at a telegrapher type hyperbolic equation for the concentrations or the mass-volume fractions.