浸润是自然界和日常生活中常见的有趣物理现象。因其在工业生产(如印染、石油开采、自清洁材料制造等)和生物过程中众多的应用，浸润现象最近是物理、表面化学和材料等领域多学科交叉的研究热点问题。因涉及粗糙非匀质固壁边界条件、移动接触线及表面活性剂的运动, 浸润现象一般非常复杂。粗糙边界和移动接触线对浸润现象的影响，具有多尺度性质；而表面活性剂与两相流体耦合，则具有多物理的特性。这种多尺度多物理性质使得对浸润现象进行数值模拟十分困难。目前，对浸润现象数值方法的研究尚不充分。本项目中，我们通过发展粗糙表面上稳态浸润现象、带移动接触线的两相流问题、以及曲面偏微分方程与两相流耦合的高效数值方法，构造统一的计算方法，用以模拟复杂的浸润现象。同时，本项目也发展研究相关问题的新的数值分析技术。这些计算方法和数值分析技术可用于高效求解其他复杂多尺度自由界面问题。

The study of wetting phenomena is of critical importance for many applications and has attracted much interest in physics, chemistry and material sciences, stimulated by the development of surface engineering and the studies on the super- hydrophobicity property in a variety of natural and artificial objects. Due to close relations with rough boundary conditions, three-phase moving contact lines and surfactant(surface active agent) movement, wetting is a complicated multi-scale multi-physics problem. This makes it very challenging to simulate complex wetting phenomena. In this project, we will develop a unified numerical method for stationary and dynamic wetting on rough surfaces with surfactant effects. For that purpose, we will develop efficient numerical methods and new analysis techniques for rough boundary problems, moving contact line problems and surface partial differential equations. These methods and analysis techniques are expected to be useful in some other multi-scale free boundary problems.