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的一系列基础和应用问题。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具有更广泛意义的有趣的数学和科学问题。第一个项目解决了一个基本的物理问题：如何在颗粒流动的连续描述中包括微力学效应，特别是速度波动的影响。第二个项目涉及围绕颗粒流动的多维连续体模型的数学，特别是从病态偏微分方程中提取数学严格信息的问题。第三个项目提出将Jenike的轴对称漏斗流动径向解扩展到具有一般截面的锥形漏斗。第四个项目涉及细颗粒材料的流动，其中间隙气体对流动有显著影响。该研究计划包括在建模、分析、数值模拟和实验方面的协调努力。这个研究项目的核心是一个关于颗粒材料流动的基本问题：“缓慢流动的颗粒材料的哪些行为可以用连续体公式来理解？”这个问题可以看作是一种调和连续模型的尝试，连续模型用于工业设计和工程问题的解决，而离散模型用于理解中小型物理实验的结果。连续体模型是非常可取的，因为它们比直接处理流动离散性的粒子动力学模拟在分析和计算上更容易处理。从这个基本问题中产生的一个重要问题是连续体描述应该采取的形式，这个问题在这个项目中得到了解决。自1895年杨森证明颗粒状材料柱中的应力不随深度无限增加，而是渐近地接近一个常数以来，这个问题一直是工程文献中争论的主题。该研究项目的意义远远超出了颗粒材料在数学和物理学中的范围。该项目得到了长期的工业合作的支持。

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.