Non-classical Diffusion, Thermo-diffusion and Suspensions

Abstract

Although under normal circumstances the classical Fick’s law gives an excellent description of matter diffusion, it is however not appropriate for describing some particular features, like those concerned with inertial effects, strong non- homogeneities, or couplings between mass transport and viscous or thermal effects. A simple non-Fickian diffusion model including relaxation is presented in Sect. 13.1. A modified diffusion equation taking into account density gradients and generalizing Cahn–Hilliard’s approach is also derived. Inertial effects are of importance some stochastic processes, as correlated random walks, discussed in Sect. 13.2. Furthermore, the coupling between diffusion flux and viscous pressure (a coupling between quantities of different tensorial order which is excluded by the classical theory of non-equilibrium thermodynamics) leads to several non-Fickian processes which are relevant in glasses and polymer solutions as shown in Sect. 13.3. Section 13.4 is devoted to the analysis of reaction–diffusion processes, where hyperbolic diffusion is combined with chemical reactions. All these aspects have considerable practical outputs: to mention only a few examples, shear-induced diffusion is at the basis of chromatographic techniques of separation of macromolecules and, on the other hand, it is important in macromolecular processing, where homogeneity is required. Reaction–diffusion coupling plays a central role in the study of spreading of epidemics, propagation of forest fires, human migration, among others. In the above sections, the temperature was assumed to be uniform. This restriction is relaxed in Sect. 13.5 wherein the coupling between matter and heat transport is investigated. Section 13.6 is concerned with the motion of suspensions of solid particles in fluids. A last Sect. 13.7 is devoted to inertial effects in fast solidification of binary alloys and their influence on the morphology of the solidification fronts. Other problems of interest, as shear-induced polymer diffusion or Taylor dispersion – diffusion combined with a velocity gradient – are dealt with in detail in the companion book Thermodynamics of Fluids under Flow (Jou et al. 2001).