A hierarchy of particle-size segregation models: From polydisperse mixtures to depth-averaged theories

Particle size segregation in granular avalanches occurs due to inter-particle percolation and squeeze expulsion. The general theory for a polydisperse mixture yields a segregation equation for each grain size class. For a three constituent mixture of large, small and medium sized particles there are three segregation equations, one of which can be eliminated, since the concentrations of all the species necessarily sums to unity. The remaining two coupled parabolic equations can be solved using a standard Galerkin finite element method. Numerical solutions show that small particles percolate to the base of the avalanche, large particles are squeezed to the surface and the medium sized grains are sandwiched in between. For certain choices of the segregation parameters it is possible to generate instabilities that lead to saw-tooth segregation in the three-phase mixture, but these die off as the grains separate into bi-disperse sub-mixtures. In general, all the grains contribute to the segregation process and develop an inversely graded particle size distribution, that coarsens upwards. This is known as reverse distribution grading. Sometimes, however, the fine particles may not segregate readily, leading to reverse coarse tail grading. For a bi-disperse mixture, the general theory yields one independent segregation equation, which always seeks to drive the particles into an inversely graded state. However, when the bulk flow shears small particles over the top of large, a breaking size-segregation wave is created. Such waves are important close to flow fronts, because they allow large particles that are over-run to rise up to the surface again and be recirculated. Computing the evolving particle-size distribution in a three-dimensional flow is still a challenge. However, it is possible to obtain a simplified representation by integrating the segregation equation through the avalanche depth. This fits naturally into the depth-averaged framework of avalanche models and opens the door to fully couple calculations to study levee formation and segregation induced flow fingering.