  Granular flows due to simultaneous vertical and horizontal excitations of a
flat-bottomed cylindrical pan are investigated using event-driven molecular
dynamics simulations. In agreement with recent experimental results, we observe
a transition from a solid-like state, to a fluidized state in which circulatory
flow occurs simultaneously in the radial and tangential directions. By going
beyond the range of conditions explored experimentally, we find that each of
these circulations reverse their direction as a function of the control
parameters of the motion. We numerically evaluate the dynamical phase diagram
for this system and show, using a simple model, that the solid-fluid transition
can be understood in terms of a critical value of the radial acceleration of
the pan bottom; and that the circulation reversals are controlled by the phase
shift relating the horizontal and vertical components of the vibrations. We
also discuss the crucial role played by the geometry of the boundary
conditions, and point out a relationship of the circulation observed here and
the flows generated in vibratory conveyors.
