However, in the nonlinear equations describing granular flow, the wedge of ill-posed directions rotates as the solution varies, and this may control growth. Although transients are unpredictable, nevertheless at large times the solution tends to a well defined steady-state. So far we have been unable to predict this behavior reliably. I hope that the model problem will shed light on this question. Below are annotated references to work in various other directions in granular flow.
Industry uses continuum models because they are simpler.
However, a granular medium consists of many individual particles, and the separation between microscales and macroscales is incomparably smaller than in a fluid. Thus, a fundamental investigation of granular flow must include its discrete nature. In the near future we plan to study numerically solutions of the steady-state equations in this geometry. Although continuum models for granular materials deal exclusively with the average behavior of materials, recent experiments by Behringer and collaborators have emphasized that fluctuations from the average may be substantial. Although MD calculations provide complete information about the flow, the calculations are too lengthy for practical situations.