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

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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.