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.
Fluids are ever-present in nature and industrial processes. Some examples are ocean currents, air flow in the atmosphere, and air or water flows past airplanes or boats. For this project, we mainly looked at the motion of water droplets on surfaces. These problems exhibit surface tension, a well-known effect that dramatically affects droplet motion. An example is water droplets that "bead up" on a surface. Understanding the details of how droplets spread on surfaces can be applied in painting or industrial coating flows that apply a protective layer to a surface. Also, being able to model how small droplets are "pumped" in micro-fluidic devices can be useful in the bio-medical field for controlling a "lab-on-a-chip." But first we need a model of the physics that is able to predict how well the technology or device works, and we must be able to simulate the model accurately. This requires a computer running a code that well approximates the model in some sense. The projectâ€™s main purpose was to develop numerical methods, or procedures that can be run on a computer, that faithfully reflect the underlying physical model of droplets moving on surfaces with surface tension effects and other physical interactions. More specifically, we developed models and numerical simulations for moving droplets by electric fields (electrowetting-driven droplets). This included the way surface tension interacts with surfaces in interesting ways. For instance, droplets may become "stuck" to a surface and not move even if they are acted on by a force such as gravity (you have probably witnessed this). We were able to model and simulate this effect as well, and it is crucial for making the model realistic. The big advantage of our computer simulations is they allow for efficient fine-tuning and optimization of the technologies described above (as opposed to manually running experiments). The details for how the models are formulated and of the numerical methods can be found in the associated papers. The research led to the creation of free software capable of running the models and methods we developed. The software package is known as FELICITY (Finite ELement Implementation and Computational Interface Tool for You), which also includes a tool for generating computational representations of 3-D objects (meshes). This was necessary to represent the shape of droplets on a computer during a simulation. We also developed a course on shape optimization to disseminate advanced research tools to graduate students. And we developed a complete set of professional notes for an undergraduate course on numerical analysis; this is a fundamental course for understanding how the numerical simulations we developed actually work.