Transversely segregated longitudinal stripes in bidisperse granular flows

Transversely segregated longitudinal flow bands, or stripes, can form in rapid thin dense granular flows due to the formation of a series of counter-rotating secondary vortices whose axes of rotation are aligned with the flow direction (Pearse et al. 2026 J. Fluid Mech. 1032 , A4). The formation of the secondary vortices is still a subject of active research, but it is likely due to a combination of dilatation and second-normal-stress differences. This paper assumes the existence of the secondary flow and investigates how large and small particles are able to segregate to form stable compositional stripes. In the absence of diffusion, the secondary flow rotates the sharp interface between large particles (above) and small particles (below), around to the vertical, where it breaks to form a series of pure large- and pure small-particle bands that extend through the flow depth, and are separated by a series of breaking-size-segregation waves. Diffusion can compete against this overturning effect. In particular, there is a region of parameter space where the diffusion and the secondary recirculation balance to create a stable wavy interface between large and small particles, which can suppress the development of surface compositional differences. In this regime, a depth-averaged theory for segregation is derived that captures the three-dimensional time-dependent development of the particle-size distribution from a uniformly inversely graded inflow into a series of transversely segregated stripes. The solution is in good quantitative agreement with the evolution of the surface particle-size distribution observed in experiments.