Simulations and Models for Sedimentation at Small Reynolds Numbers

该奖项支持同时对各种低雷诺数沉降流进行数学和数值研究。这些系统是复杂的，由单个固体颗粒产生的流体流动的长范围性质，因为他们下降，需要有效的粒子模拟算法，引入不同依赖的变量非线性系数到模型连续体偏微分方程，并导致非平凡分布在相互作用的粒子的统计力学。在这项工作中，专门用于稀释壁面沉积的高效算法，先前开发用于研究单分散沉积中速度波动的缩放，将用于提供定量数据，用于与现有和新开发的动力学和流体模型进行比较。模拟将被推广到包括由于倾斜通道，多分散性，非球形颗粒和小非零雷诺数造成的影响。流体中固体颗粒之间的相互作用在许多自然和工业过程中都很重要，包括河流中泥沙的沉降运动和空气中灰尘的沉降运动，以及使用离心或重力分离的应用，如蛋白质、矿物质、废水和可回收塑料的分离。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模拟在计算上是禁止的。更可取的方法是使用宏观模型，因为它们忽略了单个粒子的详细运动，求解成本更低。不幸的是，现有的宏观沉积模型的有效性范围非常有限。这项工作将考虑受限制的沉积系统，在这些系统中，由于简化的几何形状和描述粒子运动的数学规则的限制，大量粒子之间相互作用的有效计算仍然是可能的。结果将用于帮助确定改进的宏观模型来描述这些受限类别和更一般的沉积流。最终，这项工作将导致改进的方程，以模拟自然界中的沉积流动，并更好地设计基于沉积的分离过程。

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.