Phase-field modeling and consistent energy-stable simulation of binary creeping flows in contact with solid

Abstract

In this work, we present an efficient and practical model for describing two-phase creeping fluid flows in contact with a solid substrate. In the framework of phase-field method, a ternary Cahn–Hilliard model is modified by adding a term reflecting the wetting condition of liquid phase on the liquid–solid interface. The contact angle dynamics can be implicitly achieved by solving the phase-field equations and the explicit treatment on liquid–solid boundary is absent. Therefore, various discretization methods in space can be naturally adopted. To update the creeping flows in arbitrary domains, we herein consider the incompressible Darcy equations with a penalty term. The coupled binary fluid system theoretically satisfies the energy dissipation law with respect to a total energy functional. An energy dissipation-preserving time-marching scheme is constructed based on the auxiliary variable approach. Furthermore, a simple correction technique is utilized to improve the consistency between original and numerical values. We analytically prove that the proposed scheme still satisfies the energy law in its discrete version. Extensive numerical experiments are performed to validate the accuracy, consistent stability, and capability of our method.