Granular matter and the time-dependent viscous eikonal equation

Deposition of granular matter under gravity can be described by the well-known two-layer model for a standing and a rolling layer. Matter from sources enters the rolling layer which flows along the gradient of the standing layer and finally enters the standing layer via interaction of the two layers. From this system of two coupled hyperbolic partial differential equations a time-dependent viscous eikonal equation is derived as a limiting case for weak sources, a thin rolling layer and fast convection of the rolling layer. This equation, supplied with boundary conditions, describes the deposition of dry sand from evenly distributed sources onto a flat table with a vertical rim of variable height. The stationary problem can also be seen as an application of the method of vanishing viscosity to the eikonal equation. For certain types of interaction between the two layers the resulting eikonal equation can be transformed into a linear equation. This transformation yields additional insight into the problem.

► We describe the deposition of granular matter by a rolling/standing layer model.
► Convection in the rolling layer acts like diffusion in the standing layer.
► The time-dependent viscous eikonal equation is the limiting case of the two-layer model.
► We discuss convergence to equilibrium using a Cole–Hopf transformation.
► We give solution formulas for the stationary boundary value problems.