Longitudinal stripe formation in bidisperse granular free-surface flows with secondary vortices

Rapid granular free-surface flows on inclined planes can develop secondary vortices aligned with the dominant flow direction. The reason for their formation remains a subject of research, but plausible mechanisms include instabilities driven by (i) dilatation/compressibility, (ii) normal stress differences and (iii) a self-induced Raleigh–Taylor instability caused by segregation of large–dense and small–light particles. In this paper, a set of novel experiments are performed with large and small particles (of the same bulk density), which form longitudinal stripes due to a combination of secondary recirculation and particle-size segregation. A conceptual model is formulated, in which large particles concentrate in the downwelling sections, small particles concentrate in the upwelling sections and a breaking-size-segregation wave separates the two pure phases from one another. In each secondary vortex, the breaking waves allow the large and small particles to continuously recirculate. Assuming that a series of counter-rotating vortices exist, it is shown that this internal cross-slope structure emerges naturally from solving the gravity-shear-driven segregation-advection equations. When viewed from above, this generates a series of alternating bands of large and small particles, that are sharply separated from one another and are aligned with the downslope direction. Each complete stripe (measured from centre to centre of each large band) is formed by two counter-rotating secondary vortices. Despite the apparent order of the steady-state stripes, it is shown that the individual large and small particle paths form complex interpenetrating co-rotating sub-vortices as they avalanche downslope.