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

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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.