Numerical Methods for Free Boundary Problems: Two-Phase Flows and Contact Line Dynamics

拟议的研究将发展各种自由边界问题的数值方法。该研究将利用变分/有限元方法，稳定性/能量估计，和自动网格技术，在分析多物理流问题，表现出自由界面和移动接触线。该项目将把自由边界问题理论推向新的和具有挑战性的应用领域。具体目标是：（A）通过变分前沿跟踪、水平集方法、扩散界面或这些方法的组合来研究域表示和变形;（B）开发和分析二维电介质上电润湿（EWOD）驱动的流体液滴的有效离散公式（有接触线钉扎），探讨了含时界面运动的适定性问题;（C）探索EWOD中电场的降阶建模和更高保真度模型;（D）开发用于三维电润湿的自适应相场方法，其利用接触线钉扎将静电耦合到多相流体流;（E）为两相液滴与固体基质相互作用开发一种模型和数值方法，精确处理移动接触线问题的精细尺度细节;以及（F）先进的多阶段网格划分技术，以鲁棒和自动的方式处理3-D问题，并注意并行实现问题。本计画将探讨时间相依区域变形问题的适定性，并研究具有非光滑动力学（例如接触线钉扎）的多相流的数学性质。这项工作的更广泛影响来自于它与许多涉及移动边界/界面的物理/工业过程的联系。实例包括将保护层施加到衬底的工业涂层流;由电场驱动的微流体装置中的流体流（在生物医学领域中很重要）;流体中的刚体运动（颗粒流）;脂质生物膜的动力学（在生物学中的应用）;从刚性支撑物剥离胶带。该研究将使系统级建模和模拟在宏观长度和时间尺度的各种应用，如电润湿两相流，液滴冲击过程（油漆/冷却表面），和涂层的固体薄膜，其中可以包括精细尺度流体动力学和化学。此外，该项目将创建新的方法，用于自动生成复杂形状的网格，有效地捕捉移动边界。研究的成果之一将是一个自动网格工具（即代码），将通过PI的网站向公众提供。 最后，将开发一门关于形状优化（带PDE约束）的课程，为研究生提供连续体模型优化方面的专业知识。

The proposed research will develop numerical methods for a variety of free boundary problems. The research will take advantage of variational/finite element methods, stability/energy estimates, and automatic meshing technology in analyzing multi-physics flow problems that exhibit free interfaces and moving contact lines. The project will push the theory of free boundary problems into new and challenging application areas. Specific objectives are: (A) investigate domain representation and deformation by variational front-tracking, level set methods, diffuse interface, or a combination of these approaches; (B) develop and analyze efficient discrete formulations of Electro-Wetting On Dielectric (EWOD) driven fluid droplets in 2-D (with contact line pinning) and explore well-posedness questions of the time-dependent interface motion; (C) explore reduced order modeling of the electric field in EWOD and higher fidelity models; (D) develop an adaptive phase-field method for electro-wetting in 3-D that couples electro-statics to multi-phase fluid flow with contact line pinning; (E) develop a model and numerical method for 2-phase droplets interacting with a solid substrate that accurately handles the fine-scale details of the moving contact line problem; and (F) advance multi-phase meshing technology to handle 3-D problems in a robust and automatic way with attention given to parallel implementation issues. This project will investigate well-posedness of time dependent domain-deforming problems and study mathematical properties of multi-phase flows with non-smooth dynamics (e.g. contact line pinning). The broader impact of the work arises from its connection with many physical/industrial processes that involve moving boundaries/interfaces. Examples include industrial coating flows that apply a protective layer to a substrate; fluid flows in micro-fluidic devices driven by electric fields (important in the bio-medical field); motion of rigid bodies in a fluid (particulate flows); dynamics of lipid bio-membranes (applications in biology); the peeling of adhesive tape from a rigid support. The research will enable system level modeling and simulation at macroscopic length and time scales for a variety of applications, such as electrowetting 2-phase flow, droplet impacting processes (painting/cooling of surfaces), and coating of solids by films, which can include fine-scale fluid dynamics and chemistry. In addition, the project will create new methods for automatic grid generation of complex shapes that efficiently capture moving boundaries. One outcome of the research will be an automatic meshing tool (i.e. code) which will be made available to the public through the PI's web-site. Finally, a course on shape optimization (with PDE-constraints) will be developed that gives graduate students expertise in optimization with continuum models.