Flow and non-Fickian mass transport in immiscible blends of two rheologically different polymers

The aim of this paper is to derive a new mathematical model that investigates, under isothermal conditions, the coupling arising among non-Fickian diffusion, flow, and the deformation of the internal structure in a ternary mixture composed of a simple fluid and a binary immiscible blend of two rheologically different polymers. Along with the overall linear momentum density vector and the global scalar mass density, the diffusion mass flux density vector and the scalar mass fraction of the simple fluid, three symmetric second-rank tensor variables are adopted to track down the blend microstructural dynamic changes: two are contravariant polymer conformation tensors, and one is a covariant interface area tensor. The general equation for the nonequilibrium reversible and irreversible coupling (GENERIC) is used to derive, on the mesoscopic level of description, a set of coupled and nonlinear time evolution and constitutive equations for the selected state variables. As a special case, a new reduced model for the structure-controlled flow-free diffusion is derived, and the non-Fickian character of diffusion is shown. Using the method of characteristics, the paper also examines the physics of propagation of both linear dispersive and nonlinear hyperbolic waves produced by the disturbances in the simple fluid concentration. Explicit new formulas for the characteristic speed, phase velocity, and attenuation are derived.