Collaborative Research: Physical, Mathematical, and Engineering Problems in Slow Granular Flow

这项研究工作解决了一系列的基本和应用问题，在缓慢流动的粒状材料。 它被组织成四个项目，部分原因是它们在颗粒材料领域的重要性，但也因为它们提出了更广泛意义的有趣的数学和科学问题。 第一个项目攻击一个基本的物理问题：如何包括细观力学效应的连续描述的颗粒流，特别是速度波动的影响。 第二个项目涉及数学周围的多维连续模型的颗粒流，特别是从不适定的偏微分方程中提取数学上严格的信息的问题。 第三个项目提出了扩展Jenike的径向解决方案，在轴对称漏斗流锥形漏斗一般横截面。 第四个项目涉及细颗粒材料的流动，其中间隙气体显著影响流动。该研究项目包括建模、分析、数值模拟和实验。本研究项目的核心是关于颗粒材料流动的基本问题：“慢流动颗粒材料的哪些行为可以用连续介质公式来理解？这个问题可以被看作是一种尝试，以调和连续模型，用于工业设计和工程问题解决，与离散模型，介绍了了解小到中等规模的物理实验的结果。 连续介质模型是非常可取的，因为它们更易于处理的分析和计算比粒子动力学模拟，直接处理离散的流动。 从这个基本问题中产生的一个重要问题，也是本项目要解决的问题，就是连续统描述应该采取的形式。 这个问题一直是工程文献中争论的主题，因为Janssen在1895年证明了一个颗粒材料柱中的应力不会随着深度无限增加，而是渐近地接近一个常数。该研究计划的意义远远超出了数学和物理学中颗粒材料的范围。 该项目得到了长期工业合作的支持。

This research effort addresses a spectrum of fundamental and applied problems in the slow flow of granular materials. It is organized into four projects, chosen partly because of their importance in the field of granular materials, but also because they raise intriguing mathematical and scientific issues of broader significance. The first project attacks a fundamental physical problem: How to include micromechanical effects in a continuum description of granular flow, especially the effect of velocity fluctuations. The second project concerns the mathematics surrounding multidimensional continuum models for granular flow, specifically the issue of extracting mathematically rigorous information from ill-posed partial differential equations. The third project proposes to extend Jenike's radial solution for flows in axisymmetric hoppers to conical hoppers with a general cross section. The fourth project deals with flows of fine granular materials, where the interstitial gas significantly affects the flow. The research program involves coordinated efforts in modeling, analysis, numerical simulations, and experiment.At the heart of this research project is a basic question concerning the flow of granular materials: "What behaviors of slowly flowing granular material can be understood in terms of a continuum formulation?" This question may be viewed as an attempt to reconcile continuum models, used in industrial design and engineering problem solving, with discrete models, introduced to understand the results of small- to medium-scale physical experiments. Continuum models are highly desirable since they are much more tractable analytically and computationally than particle dynamics simulations, which treat the discreteness of the flow directly. An important issue that arises from this basic question, and which is addressed in this project, is the form that a continuum description should take. This issue has been the subject of debate in the engineering literature ever since Janssen in 1895 demonstrated that stresses in a column of granular material do not increase indefinitely with depth, but approach asymptotically to a constant. The research program has significance well beyond the context of granular materials in mathematics as well as physics. The project is supported by a long-standing industrial collaboration.