Multiscale modelling analysis and computations of complex heterogeneous multiphase systems

Abstract

In this thesis, we analytically and computationally investigate various aspects related to the multiphase-multicomponent interfacial processes and reactive transport in homogeneous domains and heterogeneous periodic perforated media. More precisely, we perform formal homogenization arguments to the microscopic Cahn-Hilliard type equations governed the dynamics in binary and ternary mixtures, in the presence of two or more phases. We additionally consider the coupling of the Cahn-Hilliard type species diffusion to fluid flow, a coupling which gives rise to more complex systems since a Navier-Stokes momentum balance is involved. Each particular model can be formally derived by an Energetic Variational Approach, that combines the classical idea of gradient flows for free energy minimization as a direct consequence of the second law of thermodynamics, together with the Least Action and Maximum Dissipation Principles. Moreover, as an extension of the already established two-scale convergence approach, we investigate further a reiterated homogenization procedure over three separated scales of periodic oscillations. Finally, we examine the General Equations for Non-Equilibrium Reversible-Irreversible Coupling commonly known by the abbreviation GENERIC, an extended two-generator variational framework, which was initially developed in order to model the rheological properties of complex fluids, far from thermodynamic equilibrium.