Simulations and Models for Sedimentation at Small Reynolds Numbers

该奖项支持对各种低雷诺数沉淀流进行同步的数学和数值研究。这些系统由于单个固体粒子下落时产生的流体流动的长程性质而变得复杂，需要高效的粒子模拟算法，将不同依赖关系的可变非线性系数引入模型连续偏微分方程组，并导致相互作用粒子的统计力学中的非平凡分布。在这项工作中，专门用于稀释壁面边界沉积的高效算法将被用来提供定量数据，以便与现有的和新开发的动力学和流体模型进行比较。这些模拟将被概括为包括倾斜通道、多分散性、非球形颗粒和小的非零雷诺数的影响。固体颗粒在流体中的沉积之间的相互作用在许多自然和工业过程中是重要的，包括河流中的泥沙和空气中的粉尘的落下运动，以及在使用离心力或重力分离的应用中，例如对蛋白质、矿物、废水和可回收塑料的应用。虽然描述单个颗粒和悬浮流体之间相互作用的数学是众所周知的，但基于这些规则的大规模流动的模拟在计算上是令人望而却步的。使用宏观模型要好得多，因为它们忽略了单个粒子的详细运动，所以求解成本较低。不幸的是，现有的沉积宏观模型的有效范围非常有限。这项工作将考虑有限类的沉积系统，在这些系统中，由于简化的几何和描述粒子运动的数学规则的限制，有效地计算大量粒子之间的相互作用仍然是可能的。结果将被用来帮助确定改进的宏观模型，以描述这些受限制的类别和更一般的泥沙流。最终，这项工作将导致改进的方程，以模拟自然的泥沙流，并更好地设计基于泥沙的分离过程。

This award supports simultaneous mathematical and numerical study ofvarious low Reynolds number sedimentation flows. These systems arecomplicated by the long range nature of the fluid flows generated byindividual solid particles as they fall, requiring efficientalgorithms for particle simulations, introducing variable nonlinearcoefficients of different dependencies into model continuum partialdifferential equations, and leading to nontrivial distributions in thestatistical mechanics of the interacting particles. In this work,efficient algorithms specialized to dilute wall-bounded sedimentation,previously developed to investigate the scaling of velocityfluctuations in monodisperse sedimentation, will be used to providequantitative data for comparison with existing and newly-developedkinetic and fluid models. The simulations will be generalized toinclude effects due to inclined channels, polydispersity,non-spherical particles, and small non-zero Reynolds numbers.Interactions between solid particles sedimenting in a fluid areimportant in many natural and industrial processes, including thesettling motions of silt in rivers and dust in the air, and inapplications which use centrifugal or gravitational separation such asthose for proteins, minerals, waste water, and recyclable plastics.While the mathematics describing the interactions between individualparticles and the suspending fluid are well known, simulations oflarge scale flows based on these rules are computationallyprohibitive. It would be far preferable to use macroscopic modelswhich are less costly to solve because they neglect the detailedmotion of individual particles. Unfortunately, existing macroscopicmodels for sedimentation have very limited ranges of validity. Thiswork will consider restricted classes of sedimentation systems inwhich efficient computation of the interactions between large numbersof particles remains possible because of simplified geometries andlimits for the mathematical rules describing particle motion. Resultswill be used to help identify improved macroscopic models to describeboth these restricted classes and more general sedimentation flows.Ultimately, this work will lead to improved equations for modelingsedimentation flows in nature and for better design of separationprocesses based on sedimentation.