Traffic Equations and Granular Convection
  We investigate both numerically and analytically the convective instability of granular materials by two dimensional traffic equations. In the absence of vibrations the traffic equations assume two distinctive classes of fixed bed solutions with either a spatially uniform or nonuniform density profile. The former one exists only when the function V(\rho) that monitors the relaxation of grains assumes a cut off at the closed packed density, \rho_c, with V(\rho_c)=0, while the latter one exists for any form of V. Since there is little difference between the uniform and nonuniform solution deep inside the bed, the convective instability of the bulk may be studied by focusing on the stability of the uniform solution. In the presence of vibrations, we find that the uniform solution bifurcates into a bouncing solution, which then undergoes a supercritical bifurcation to the convective instability. We determine the onset of convection as a function of control parameters and confirm this picture by solving the traffic equations numerically, which reveals bouncing solutions, two convective rolls, and four convective rolls. Further, convective patterns change as the aspect ratio changes: in a vertically long container, the rolls move toward the surface, and in a horizontally long container, the rolls move toward the walls. We compare these results with those reported previously with a different continuum model by Hayakawa, Yue and Hong[Phys. Rev. Lett. 75,2328, 1995]. Finally, we also present a derivation of the traffic equations from Enskoq equation. 
Daniel C. Hong and Su Yue