Two-phase flow with moving contact lines (MCLs) is an unsolved problem in fluid dynamics. It is challenging to solve the problem numerically due to its intrinsic multiscale property that the microscopic slip must be taken into account in a macroscopic model. It is even more difficult when the solid substrate has microscopic inhomogeneity or roughness. In this paper, we propose a novel unified numerical framework for two-phase flows with MCLs. The framework cover some typical sharp-interface models for MCLs and can deal with the contact angle hysteresis (CAH) naturally. We prove that all the models, including the nonlinear Cox model and a CAH model, are thermodynamically consistent in the sense that an energy dissipation relation is satisfied. We further derive a new variational formula which leads to a stable and consistent numerical method independent of the choice of the slip length and the contact line frictions. This enables us to efficiently solve the macroscopic models for MCLs without resolving very small scale flow field in the vicinity of the contact line. We prove the well-posedness of the fully decoupled scheme which is based on a stabilized extended finite element discretization and a level-set representation for the free interface. Numerical examples are given to show the efficiency of the numerical framework.  

　　Publication:  

　　SIAM Journal on Scientific Computing, Volume 45, Issue 6 (2023)  

　　http://dx.doi.org/10.1137/23M1546816  

　　Author:  

　　Xianmin Xu  

　　LSEC, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China.  

　　Email: xmxu@lsec.cc.ac.cn