Granular fingering instability: A first attempt to access the most unstable mode

Mixtures of grains of different sizes tend to segregate as they avalanche downslope, with large particles rising to the near surface regions which move faster. As a result, large particles tend to be preferentially transported to flow front where they can accumulate by being over-run and resegregated to the surface. If the large particles are also more frictional, the flow becomes unstable and breaks-up in a series of fingers: the so-called granular fingering instability. This instability is observed in a wide variety of systems, from geophysical mass flows, such as pyroclastic flows, to small-scale experiments relevant to industry. Key features of the fingering pattern are predicted by a particle-size segregation model, coupled with a depth-averaged avalanche model, in which a viscous term play a vitally important role in making the equations well-posed. We carry out a detailed numerical stability analysis to investigate what sets the wavelength of the fingers.