  To understand the process of pattern formation in a low-density granular
flow, we propose a simple particle model. This model considers spherical
particles moving over an inclined flat surface based on three forces: gravity
as the driving force, repulsive force due to particle collision, and drag force
as the particle-- interaction through the ambient fluid. Numerical simulations
of this model are conducted in two different types of two-dimensional planes,
i.e., the monolayer was treated. In the horizontal plane parallel to the slope,
particles aggregate at the moving front of the granular flow; and subsequently,
flow instability occurs as a wavy pattern. This flow pattern is caused by the
interparticle interaction arising from the drag force. Additionally, a vortex
convection of particles is formed inside the aggregations. Meanwhile, in the
vertical plane on the slope, particle aggregation is also found at the moving
front of the granular flow. The aggregation resembles a head--tail structure,
where the frontal angle against the slope approaches 60 degree from a larger
angle as time progresses. Comparing the numerical result by varying the
particle size, the qualitative dynamics of the granular flow are independent of
size. To elucidate this reason, we perform a nondimensionalization for this
model. The result indicates that our model can be simplified to dimensionless
equations with one dimensionless parameter that represents the ratio of the
gravity term to the excluded volume effect.
