A numerical study of two-phase flows in complex domain with a generalized Navier slip and penetration boundary condition on permeable boundaries

Abstract

A phase-field model consisting of the Cahn-Hilliard and Navier-Stokes equations with a generalized Navier slip and penetration boundary condition is proposed to simulate the behavior of two-phase flows through permeable surfaces. The proposed boundary condition is a generalization of the generalized Navier boundary condition to penetrable boundaries, enabling the simulation of significant scientific problems such as gas penetration through polymer films and oil filtration through porous materials. To address the challenges imposed by the new boundary condition to conventional numerical schemes, we develop a new numerical algorithm by using a finite element method that naturally incorporates the boundary condition into the weak formulation. The algorithm solves a semi-implicit system for the Cahn-Hilliard equation and a fully implicit system for the Navier-Stokes equations. Complex geometries required in the applications are handled by using body-conforming unstructured meshes. Furthermore, an adaptive mesh refinement strategy based on a gradient-jump error indicator is devised to accelerate the simulation process while obtaining a reliable solution on an optimally refined mesh. Extensive numerical experiments, including two practical applications, are conducted to validate the effectiveness and efficiency of the proposed approach. The first application involves bubble penetration through a polymer film, encompassing processes such as bouncing, spreading, pinning, slipping, and penetrating. The numerical results show qualitative agreement with experimental results. In the second application, we examine the robustness of the algorithm by testing different physical parameters with high contrast for the displacement and infiltration of two-phase flows in a complex pore structure.