Continuum kinetic theories provide an important tool for the analysis and simulation of particle suspensions. When those particles are anisotropic, the addition of a particle orientation vector to the kinetic description yields a $2d-1$ dimensional theory which becomes intractable to simulate, especially in three dimensions or near states where the particles are highly aligned. Coarse-grained theories that track only moments of the particle distribution functions provide a more efficient simulation framework, but require closure assumptions. For the particular case where the particles are apolar, the Bingham closure has been found to agree well with the underlying kinetic theory; yet the closure is non-trivial to compute, requiring the solution of an often nearly-singular nonlinear equation at every spatial discretization point at every timestep. In this paper, we present a robust, accurate, and efficient numerical scheme for evaluating the Bingham closure, with a controllable error/efficiency tradeoff. To demonstrate the utility of the method, we carry out high-resolution simulations of a coarse-grained continuum model for a suspension of active particles in parameter regimes inaccessible to kinetic theories. Analysis of these simulations reveals that inaccurately computing the closure can act to effectively limit spatial resolution in the coarse-grained fields. Pushing these simulations to the high spatial resolutions enabled by our method reveals a coupling between vorticity and topological defects in the suspension director field, as well as signatures of energy transfer between scales in this active fluid model.

翻译：连续介质动理论为颗粒悬浮液的分析与模拟提供了重要工具。当这些颗粒为各向异性时，将颗粒取向向量引入动理论描述会得到一个$2d-1$维的理论，该理论在模拟中变得棘手，尤其是在三维空间或颗粒高度取向的状态附近。仅追踪颗粒分布函数矩的粗粒化理论提供了更高效的模拟框架，但需要闭合假设。对于颗粒无极性这一特定情况，Bingham闭合已被发现与底层动理论吻合良好；然而该闭合的计算并不简单，需要在每个时间步的每个空间离散点上求解一个通常近乎奇异的非线性方程。本文提出了一种稳健、精确且高效的数值方案来评估Bingham闭合，具有可控的误差/效率权衡。为展示该方法的应用价值，我们在动理论无法企及的参数区域内，对活性颗粒悬浮液的粗粒化连续介质模型进行了高分辨率模拟。对这些模拟的分析表明，不精确计算闭合效应会有效限制粗粒化场的空间分辨率。将模拟推进至本方法所能达到的高空间分辨率，揭示了悬浮液指向场中涡度与拓扑缺陷之间的耦合，以及该活性流体模型中跨尺度能量传递的特征。