A Mathematical Model for Two‐Phase Flow in Confined Environments: Numerical Solution and Validation

This study presents a numerical framework for modeling two‐phase flow in confined environments, focusing on the interplay between capillary and viscous forces. The model integrates the Cahn‐Hilliard and Navier‐Stokes (CH‐NS) equations, utilizing a diffuse‐interface approach to capture interfacial dynamics without the limitations of sharp‐interface models. Implemented in the finite element platform FEniCS , the framework incorporates Dirichlet boundary conditions to model a fully non‐wetting phase. The validation of the proposed model is achieved through two applications: The retraction of an oil droplet from a capillary tube and the drainage of water‐wet microfluidic chips. Numerical results align with experimental data, demonstrating the framework's ability to replicate interfacial behaviors, including capillary‐driven dynamics and fingering phenomena. This work provides a versatile computational tool for studying immiscible fluid flow, offering potential for advancements in fundamental research on microfluidics, enhanced oil recovery, and remediation of contaminated soil.