Free surface from convective flow in a vibrated granular medium

Vertical vibration of a granular medium with greater acceleration than g creates a pair of convection rolls with opposite rotation propagating along its edges. For 2D monodisperse disks, the top free surface bends downward for inclined walls with angles $\theta < 60^{\circ }$ with respect to the horizontal axis. It remains flat for $\theta = 60^{\circ }$ , identical to rotational invariance of the hexagonal close packing, and bends upward for $\theta > 60^{\circ }$ . Grain boundaries of alternating 5- and 7-fold disclinations emerge at the center of a container to separate two hexagonal lattices, on the left and right halves of a container, with different orientation. When the opening angle of inclined walls $\phi = 180^{\circ } - 2\,\theta $ is equal to an integral multiple of $60^{\circ }$ , namely $\theta = 30^{\circ }, 60^{\circ }$ and $90^{\circ }$ , such grain boundaries disappear. For 2D bidisperse disks, the top free surface is still smooth but is less deformed from a flat profile. Hexagonal close packing exists only locally in small plaquettes without grain boundaries for all angles θ. In hydrodynamic description the monodisperse free surface is well described by a contour of the constant pressure calculated from the Navier-Stokes equation.