Multi-component particle size segregation in shallow granular avalanches

A general continuum theory for particle size segregation and diffusive remixing in polydisperse granular avalanches is formulated using mixture theory. Comparisons are drawn to existing segregation theories for bi-disperse mixtures and the case of a ternary mixture of large, medium and small particles is investigated. In this case the general theory reduces to a system of two coupled parabolic segregation remixing equations, that have a single diffusion coefficient and three parameters which control the segregation rates between each pair of constituents. Considerable insight into many problems where the effect of diffusive remixing is small is provided by the non-diffusive case. Here the equations reduce to a system of two first order conservation laws, whose wave speeds are real for a very wide class of segregation parameters. In this regime the system is guaranteed to be non-strictly hyperbolic for all admissible concentrations. If the segregation rates do not increase monotonically with the grain size ratio, it is possible to enter another region of parameter space, where the equations may either be hyperbolic or elliptic dependent on the segregation rates and the local particle concentrations. Even if the solution is initially hyperbolic everywhere regions of ellipticity may develop during the evolution of the problem. Such regions in a time-dependent problem necessarily lead to short wavelength Hadamard instability and ill-posedness. A linear stability analysis is used to show that the diffusive remixing terms are sufficient to regularize the theory and prevent unbounded growth rates at high wave numbers. Numerical solutions for the time-dependent segregation of an initially almost homogeneously mixed state are performed using a standard Galerkin finite element method. The diffuse solutions may be linearly stable or unstable dependent on the initial concentrations. In the linearly unstable region “sawtooth” concentration stripes form that trap and focus the medium sized grains. The large and small particles still percolate through the avalanche and separate out at the surface and base of the flow due to the no flux boundary conditions. As these regions grow, the unstable striped region is annihilated. The theory is used to investigate inverse distribution grading and reverse coarse tail grading in multi-component mixtures. These terms are commonly used by geologists to describe particle size distributions in which either the whole grain size population coarsens upwards, or, just the coarsest clasts are inversely graded and a fine grained matrix is found everywhere. An exact solution is constructed for the steady segregation of a ternary mixture as it flows down an inclined slope from an initially homogeneously mixed inflow. It shows that for distribution grading, the particles segregate out into three inversely graded sharply segregated layers sufficiently far downstream, with the largest particles on top, the fines at the bottom and the me