Collaborative Research: Models, Algorithms, and Simulations for Two-Phase Ferrofluid Flows in Contact with a Solid Surface

由于两相磁流体流动可以通过外加磁场控制，因此在科学和工程领域有着广泛的应用。两相磁流体混合物的流动通常涉及流体动力学、界面运动、内禀粒子自旋、磁化场和感应磁场的耦合。预计所提出的研究将不仅导致可靠的适定模型，有效的数值算法的两相铁磁流体系统，但也扩大了适用性的数学模型，分析和代码，目前感兴趣的各种物理问题，通过数值模拟。 拟议的研究将产生高效，最佳收敛，易于实现，能量稳定的数值算法，以及公开可用的有限元代码。它将为本科生和研究生提供充足和宝贵的机会，接受数学建模，计算数学和机械工程领域的多学科培训，并为科学和工程应用开发最先进的数学模型和数值算法。从这项合作工作开始，研究人员计划将开发的模型，方法和软件包传播给更多的工程师和科学家，以解决他们的现实问题，在专业会议和座谈会上介绍研究，并在国内举办特别会议/小型讨论会，本项目的智力价值在于开发合适的数学模型和设计的挑战。高效和精确的计划高度非线性两相铁磁流体系统，耦合自由界面，流体力学，磁化场，感应磁场，和内禀粒子自旋。由于两相铁磁流体系统中存在复杂的非线性耦合，因此很少有人致力于解决两相铁磁流体系统的诱导建模/数值模拟挑战。所提出的模型是一致的，从而保持能量耗散规律。利用有限元方法在处理复杂边界方面的优势，所提出的数值格式具有高效、精确、易于实现、能量稳定等优点，并具有允许大时间步长和受控误差界来精确捕捉界面奇异性的优点。此外，模型、数值算法和模拟将有助于更好地理解各种具有实际意义的物理问题，例如基于铁磁流体的新材料。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Being controllable by applied magnetic fields, two-phase ferrofluid flows have found diverse applications in various science and engineering fields. Flows of two-phase ferrofluid mixtures usually involve the coupling of hydrodynamics, interface motion, intrinsic particle spins, magnetization field, and induced magnetic field. It is expected that the proposed research will not only lead to reliable well-posed models, efficient numerical algorithms for the two-phase ferrofluid system, but also extend the applicability of mathematical models, analyses and codes to various physical problems of current interest through numerical simulations. The proposed research will produce efficient, optimally convergent, easily implemented, energy stable numerical algorithms, as well as publicly available finite element codes. It will provide ample and valuable opportunities for undergraduate and graduate students to receive multidisciplinary training in the areas of mathematical modeling, computational mathematics, and mechanical engineering, and to develop state-of-the-art mathematical models and numerical algorithms for science and engineering applications. Starting from this collaborative work, the investigators plan to disseminate the developed models, methods and software packages to more engineers and scientists for solving their realistic problems, present the research in professional conferences and colloquia, and organize special sessions/minisymposiums in domestic/international conferences for related works.The intellectual merit of this project lies in the challenge to develop suitable mathematical models and design efficient and accurate schemes for the highly nonlinear two-phase ferrofluid system, which couples free interfaces, hydrodynamics, magnetization field, induced magnetic field, and intrinsic particle spins. Very few efforts have been made to address the induced modeling/numerical challenges of the two-phase ferrofluid system due to the complicated nonlinear couplings among these constituents. The proposed models are thermodynamically consistent and thus preserve the energy dissipation laws. With strengths of the finite element methods in dealing with the complex boundary, the proposed numerical schemes are efficient, accurate, easily implemented, and energy stable with some discrete energy dissipation laws which have advantages of allowing the large time step and controlled error bound to capture the interfacial singularities accurately. In addition, the models, numerical algorithms and simulations will contribute to better understandings of various physical problems of practical interests, such as new materials based on ferro fluids.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.