TRAFFIC EQUATIONS AND GRANULAR CONVECTION

We investigate both numerically and analytically the convective instab ility of granular materials by two-dimensional traffic equations. In t he absence of vibrations traffic equations assume two distinctive clas ses of fixed bed solutions with either a spatially uniform or nonunifo rm density profile. The former one exists only when the function V(rho ) that monitors the relaxation of grains assumes a cutoff at the close d packed density, rho(c), with V(rho(c)) = 0, while the latter one exi sts for any form of V. Since there is little difference between the un iform and nonuniform solution deep inside the bed, the convective: ins tability of the bulk may be studied by focusing on the stability of th e uniform solution. In the presence of vibrations, we find that the un iform solution bifurcates into a bouncing solution, which then undergo es a; supercritical bifurcation to the convective instability. We dete rmine the onset of convection as a function of control parameters-and confirm this picture by solving the traffic equations numerically, whi ch reveals bouncing solutions, two convective rolls, and four convecti ve rolls. Further, convective patterns change as the aspect ratio chan ges: in a vertically long container, the rolls move toward the surface , and in a horizontally long container, the rolls move toward the side walls. We compare these results with the those reported previously wi th a different continuum model by Hayakawa, Yue, and Hong [Phys. Rev. Lett. 75, 2328 (1995)]. Finally, we also present a derivation of the t raffic equations from Enskoq equation. [S1063-651X(98)09410-0].