Mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces

Abstract

Establishing multi-component phase field models on dynamic surfaces, and exploring the similarities and differences in phase field evolution between evolving and static surfaces, are interesting studies in multi-phase flow problems. This paper focuses on mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces. In the modeling process, the evolution of the energy functional, componential mass conservation, and point-wise mass conservation (hyperplane link condition) are considered. On an evolving surface, the energy dissipation law does not hold because the surface velocity can be viewed as an external force in the system. Meanwhile, due to the velocity, the surface area may change, the componential mass conservation property and the hyperplane link condition can not be satisfied simultaneously. From the point of view of preserving these two physical properties of mass conservation, three types of N-component Cahn-Hilliard model are established on the evolving surface: componential mass conservation, point-wise mass conservation, and both componential and point-wise mass conservation. For the numerical simulation, the evolving surface finite element method is considered for the space-time discretization of the proposed model. In order to obtain a linear, decoupled, high-accurate, and stable scheme for long time numerical simulations, the stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. Through several numerical examples, the rationality of the model and the efficiency of the numerical method are shown. Additionally, three- and four-component phase separation phenomena are shown to investigate the N-component Cahn-Hilliard dynamic on various evolving surfaces.