A void-based description of compaction and segregation in flowing granular materials

Guided by the kinematical treatment of vacancies in theories for solid-state diffusion, we develop a theory for compaction and segregation in flowing granular materials. This theory leads to a partial differential equation for the macroscopic motion of the material coupled to a system of partial differential equations for the volume fractions of the individual particle types. When segregation is ignored, so that the focus is compaction, the latter system is replaced by a scalar partial differential equation that closely resembles equations arising in theories of traffic flow. To illustrate the manner in which the theory describes compaction and segregation, we present three explicit solutions. In particular, for an arbitrary loosely packed mixture of small and large particles in a fixed container under the influence of gravity, we show that a layer of large particles forms at the free surface and grows with time, while a closely packed mixture of large and small particles forms and grows from the base of the container; the final solution, attained in a finite time, consists of a layer of closely packed large particles above a closely packed mixed state. At the level of everyday experience, this solution at least qualitatively explains why in a container of mixed nuts, Brazil nuts are generally found at the top.