Instabilities in Granular Flows

The flow and handling of granular materials is of major importance to many industries. Yet despite efforts over several decades, the modeling of such flows has achieved only modest success. Dense gravity-driven flows in hoppers have been often modeled as elastic-plastic continua, for example. In this picture, the granular material flows as a plastic with a frictional yield condition, and deforms as an elastic solid otherwise. This model has been used to analyze mass flows, where the material is flowing throughout the hopper, but has failed to provide adequate agreement with experiment in the prediction of quantities such as discharge rate, for example [17]. Despite such shortcomings, Jenike’s [6] construction of steady-state incompressible rigid-plastic radial solutions in hoppers with simple geometries has been of considerable importance in hopper design [10]. These are solutions for quasistatic flow (inertial effects are neglected) where grains travel radially in conical or wedge-shaped hoppers. Only recently have numerical solutions of steady-state quasistatic flows in more complicated geometries been produced [5]. However, it is now clear that there are serious mathematical difficulties with the equations for time-dependent incompressible rigid-plastic flow (IRPF). In many instances, the equations for such flows have been shown to be ill-posed i.e. they possess instabilities with arbitrarily short wavelengths (Schaeffer [12], Valanis and Peters [18]). Hence, it is problematic to interpret the steady-state rigid-plastic flow equations as long-time solutions of the time-dependent equations. Additionally, both the steady-state and time-dependent equations have physical shortcomings. There are several features of hopper flows where the particle size is important, indicating that it may not be appropriate to model such dense granular flows as continua.