Hydrodynamic Theory for Reverse Brazil Nut Segregation and the   Non-monotonic Ascension Dynamics
  Based on the Boltzmann-Enskog kinetic theory, we develop a hydrodynamic theory for the well known (reverse) Brazil nut segregation in a vibro-fluidized granular mixture. Using an analogy with standard fluid mechanics, we have recently suggested a novel mechanism of segregation in granular mixtures based on a {\it competition between buoyancy and geometric forces}: the Archimedean buoyancy force, a pseudo-thermal buoyancy force due to the difference between the energies of two granular species, and two geometric forces, one compressive and the other-one tensile in nature, due to the size-difference. For a mixture of perfectly hard-particles with elastic collisions, the pseudo-thermal buoyancy force is zero but the intruder has to overcome the net compressive geometric force to rise. For this case, the geometric force competes with the standard Archimedean buoyancy force to yield a threshold density-ratio, $R_{\rho 1}=\rho_l/\rho_s < 1$, above which the {\it lighter intruder sinks}, thereby signalling the {\it onset} of the {\it reverse buoyancy} effect. For a mixture of dissipative particles, the non-zero pseudo-thermal buoyancy force gives rise to another threshold density-ratio, $R_{\rho 2}$ ($> R_{\rho 1}$), above which the intruder rises again. Focussing on the {\it tracer} limit of intruders in a dense binary mixture, we find that the rise-time of the intruder could vary {\it non-monotonically} with the density-ratio. For a given size-ratio, there is a threshold density-ratio for the intruder at which it takes the maximum time to rise, and above(/below) which it rises faster, implying that {\it the heavier (and larger) the intruder, the faster it ascends}. Our theory offers a unified description for the (reverse) Brazil-nut segregation and the non-monotonic ascension dynamics of Brazil-nuts. 
Meheboob Alam, Leonardo Trujillo and Hans J. Herrmann