Analysis of a Cahn-Hilliard model for a three-phase flow problem

In this study, we present a rigorous model for analyzing the behavior of three immiscible components within incompressible viscous flows. By combining the Cahn-Hilliard free energy approach with the Stokes equation, we address both thermodynamic and hydrodynamic aspects of the fluid mixture. For a particular consistent selection of the bulk-free energy function, we enhance the Stokes equation to account for capillary forces and surface tension effects, and the Cahn-Hilliard equation with the convection effects. At the pore scale, we show the existence of weak solution in a bounded and sufficiently smooth domain Omega subset of Rd,d=2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}<^>d, d= 2, 3 $$\end{document}. Using two-scale convergence techniques, we derive an upscaled model that effectively characterizes the averaged behavior of the entire system.