Dynamics of Heap Formation in Grains: A Continuum Model

The formation dynamics of granular heaps in a vibrating bed is studied by a simple continuum model using the heap profile as the dynamical variable. Vibration increases the local height but is counter-balanced by the nonlinear coupling, which tends to suppress the growth of the height. The time evolution of the heap towards the steady state is obtained and it is demonstrated explicitly that the analytic result of the steady state profile is stable. Phenomena such as heap formation, downward and upward heaps can be reproduced. Our results agree with the experimentally observed change of downward to upward steady heaps as the vibration strength is increased. The effective current in the model is also computed and can be identified with the associated convection in the heap. The formation dynamics closely obey an exponential approach towards the steady state heap. Characteristic formation times are found to increase with the vibrational strength and the dependence on the system size is also explored.