The Fast Exact Closure for Jeffery’s equation with diffusion |
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| Authors: | Stephen Montgomery-Smith David Jack Douglas E Smith |
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| Institution: | 1. Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;2. Department of Mechanical Engineering, Baylor University, Waco, TX 76798, USA;3. Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA;1. Karlsruhe Institute of Technology (KIT), Institute of Vehicle System Technology, Karlsruhe, Germany;2. Institute of Textile Technology and Process Engineering (ITV), Denkendorf, Germany;3. Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), St. Augustin, Germany;4. University of Stuttgart, Institute of Aircraft Design (IFB), Stuttgart, Germany;5. Fraunhofer Institute for Chemical Technology (ICT), Pfinztal, Germany;1. Department of Computer Science, University of California, Los Angeles, CA 90095, United States;2. Department of Radiological Sciences, University of California, Los Angeles, CA 90095, United States;3. Computer Science Department, University of Chicago, Chicago, IL 60637, United States;1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA;2. Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, NY, 10010, USA;1. DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan;2. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan;3. Department of Computational Science and Engineering, Yonsei University, Seoul, South Korea |
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| Abstract: | Jeffery’s equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold’s numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a ‘closure’ that approximates the distribution function’s fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery’s equation. Numerical examples for dense fiber suspensions are provided with both a Folgar–Tucker (1984) 3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) 9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations. |
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