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

这项研究工作解决了颗粒材料缓慢流动中的一系列基本和应用问题。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耐人寻味的数学和科学问题。第一个项目解决了一个基本的物理问题：如何在颗粒流的连续描述中包括微观机械效应，特别是速度波动的影响。第二个项目涉及围绕颗粒流的多维连续介质模型的数学，特别是从不适定的偏微分方程组中提取数学上严格的信息的问题。第三个项目建议将Jenike关于轴对称漏斗内流动的径向解推广到具有一般截面的圆锥形漏斗。第四个项目涉及细粒物质的流动，其中间隙气体对流动有显著影响。该研究项目涉及建模、分析、数值模拟和实验的协调工作。该研究项目的核心是一个关于颗粒物质流动的基本问题：“缓慢流动的颗粒物质的行为可以用连续统公式来理解？”这个问题可以被看作是试图调和工业设计和工程问题解决中使用的连续介质模型与离散模型，这些离散模型被引入来理解中小型物理实验的结果。连续介质模型是非常理想的，因为它们在分析和计算上比粒子动力学模拟要容易得多，粒子动力学模拟直接处理流动的离散性。由这个基本问题引起的一个重要问题，也是这个项目中解决的问题，是连续统描述应该采用的形式。自从1895年Janssen证明颗粒材料柱中的应力不会随着深度无限增加，而是渐近于一个常数以来，这个问题一直是工程学文献中争论的主题。这项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畴。该项目得到了长期的行业合作的支持。

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.