A micro-scale diffused interface model with Flory-Huggins logarithmic potential in a porous medium

A diffused interface model describing the evolution of two conterminous incompressible fluids in a porous medium is discussed. The system consists of the Cahn-Hilliard equation with Flory-Huggins logarithmic potential, coupled via surface tension term with the evolutionary Stokes equation at the pore scale. An evolving diffused interface of finite thickness, depending on the scale parameter $\epsilon$ separates the fluids. The model is studied in a bounded domain $\mathrm{\Omega }$ with a sufficiently smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ in ${\mathbb{R}}^d$ for $d=2$, $3$. At first, we investigate the existence of the system at the micro-scale and derive the essential $\mathit{\text{a-priori}}$ estimates. Then, using the two-scale convergence approach and unfolding operator technique, we obtain the homogenized model for the microscopic one.