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

Abstract

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 $\hat{I}\text{c◯}$ with a sufficiently smooth boundary $\partial \hat{I}\text{c◯}$ 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 \textit{a-priori} estimates. Then, using the two-scale convergence approach and unfolding operator technique, we obtain the homogenized model for the microscopic one.

Upon further review, we have identified significant errors in our methodology that affect the validity of our results. We are currently working on rectifying these issues, but we are uncertain when we will be able to submit a corrected version. To avoid misleading readers, we kindly request the withdrawal of our paper