Collaborative Proposal: 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。 它分为四个项目，选择这些项目的部分原因是它们在颗粒材料领域的重要性，而且还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：如何将微机械效应纳入颗粒流的连续描述中，特别是速度波动的影响。 第二个项目涉及围绕颗粒流多维连续介质模型的数学，特别是从不适定偏微分方程中提取数学严格信息的问题。 第三个项目建议将 Jenike 针对轴对称料斗中的流动的径向解决方案扩展到具有一般横截面的锥形料斗。 第四个项目涉及细颗粒材料的流动，其中间隙气体显着影响流动。该研究项目涉及建模、分析、数值模拟和实验方面的协调工作。该研究项目的核心是一个有关颗粒材料流动的基本问题：“可以用连续介质公式来理解缓慢流动的颗粒材料的哪些行为？” 这个问题可以被视为协调工业设计和工程问题解决中使用的连续模型与离散模型的尝试，离散模型是为了理解中小型物理实验的结果而引入的。 连续体模型非常受欢迎，因为它们在分析和计算上比直接处理流动离散性的粒子动力学模拟更容易处理。 从这个基本问题中产生的一个重要问题是连续统描述应该采取的形式，这个问题在这个项目中得到解决。 自从 Janssen 在 1895 年证明颗粒材料柱中的应力不会随着深度无限增加，而是渐近地趋近于常数以来，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.