In this work, a new solver, FPSolve, is developed to study fiber orientation kinetics using the Fokker–Planck (FP) equation. The solver employs the finite-volume method. The FP equation is discretized on unstructured cubed-sphere grids using the centered differencing scheme (CDS) or a blend of the CDS and the upwind differencing scheme. Time integration is performed using a second-order two-stage explicit Runge–Kutta scheme. Different shape factors and rotational diffusion coefficients are implemented to study suspensions in dilute to semiconcentrated regimes. The verification of the solver is performed for the fiber orientation in simple shear flow up to a Peclet number of . Grid independence analysis is presented to show the second-order accuracy of FPSolve. It is demonstrated that the solver does not need stabilization by upwinding. Simulations for semiconcentrated suspensions are performed using the model of Ferec et al. (2014). Time-accurate solutions of the FP equation with explicit time stepping for this model are presented for the first time.

中文翻译：

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用于求解纤维取向动力学 Fokker-Planck 方程的有限体积框架

在这项工作中，开发了一种新的求解器 FPSolve，用于使用福克-普朗克 (FP) 方程研究纤维取向动力学。求解器采用有限体积法。使用中心差分方案 (CDS) 或 CDS 与迎风差分方案的混合，在非结构化立方球网格上对 FP 方程进行离散化。使用二阶两阶段显式龙格-库塔方案执行时间积分。采用不同的形状因子和旋转扩散系数来研究稀释到半浓缩状态下的悬浮液。解算器的验证针对佩克莱特数以下的简单剪切流中的纤维取向进行。网格独立性分析显示了 FPSolve 的二阶精度。结果表明，求解器不需要逆风稳定。使用 Ferec 等人的模型进行半浓缩悬浮液的模拟。（2014）。首次提出了该模型具有显式时间步长的 FP 方程的时间精确解。