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Ill-posed partial differential equations are one of the mathematical challenges which arise in granular flow. Physically, ill-posedness appears at the point when sheared material begins to crack and to move as separate pieces: in technical terms, when a shear band begins to form. A mathematical model for this appears in [A1] in the references , and a key initial-value problem (the Riemann problem) for the model is solved in [A2] in the references . In a slightly different direction, the role of imperfections in shear banding is analyzed in [A3] in the references .
Another challenge of granular flow concerns predicting stress fluctuations. Although engineering models deal exclusively with the average behavior of materials, recent experiments by Behringer and collaborators ([E4] in the references ) have emphasized that fluctuations from the average may be substantial, or even crucial. The simple probabilistic model analyzed in [A4] in the references is a first attempt to understand these phenomena theoretically.
We are attacking these problems with coordinated efforts in mathematical analysis, numerical computation, and physical experiments. On our joint NSF grant in granular flow at Duke, I am the PI for work in analysis; John Trangenstein is the PI for work in computations; and Bob Behringer is the PI for work in experiments. (Other collaborators include Michael Shearer, Xabier Garaizar, and Bruce Pitman.)
The computational team has developed code in two distinct directions: (i) a continuum code that includes front tracking ([C1] in the references ) and adaptive mesh refinement at a shear band ([C3] in the references ) and (ii) a molecular-dynamics code that follows the motion of every grain in the sample ([C2] in the references). The latter problem is relevant in the fundamental problem of predicting the macroscopic constitutive behavior from a knowledge of the micromechanics of individual grains. Although MD calculations provide complete information about the flow, the calculations are too lengthy for practical situations. Therefore we are planning to merge the two approaches in the future: i.e., to use MD at the finest level of an adaptive mesh refinement code and continuum equations at all higher levels.
As mentioned above, experiments in stress fluctuations are reported in [E4] in the references . Other experiments on granular flow are reported in [E1 and E2] in the references . Of particular interest is [E1], where porosity waves were first reported. Incidentally, a survey of experiments in granular flow appears in [E3] in the references .
Another component of my research has grown out of my teaching; specifically out of a graduate-level seminar-type course that I have developed over the past few years. Several months before the semester begins, I ask the science and engineering faculty at Duke for research problems on which they would like mathematical help. In discussions with the (math) students who are likely to take the course, I choose one of the suggested problems to be the focus of the course. In the first half of the semester, I give lectures on techniques of applied math and on the general scientific background of the problem. In the second half of the semester (and beyond), students use these techniques, under my supervision, to research the problem.
In the spring of 1995 we studied problems in industrial math; this course is described in an article in the Notices . In the spring of 1996 we studied lithotripsy (a medical treatment for breaking up kidney stones by ultrasonic shock waves, surgery is not required); this course was written up in the Duke newspaper . Lithotripsy raises a lot of interesting fluid-mechanics questions, some of them related to the "hot" topic of sonoluminescence. As an outgrowth of the course, I have started a collaboration with the two professors of Mechanical Engineering who originally proposed this area as the focus.
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Last modified: 17 September 1996