# Hydrodynamic theory for reverse brazil nut segregation and the non-monotonic ascension dynamics

## Abstract

Based on the Boltzmann–Enskog kinetic theory, we develop a hydrodynamic theory for the well known (reverse) Brazil nut segregation in a vibrofluidized granular mixture. Under strong shaking conditions, the granular mixture behaves in some ways like a fluid and the kinetic theory constitutive models are appropriate to close the continuum balance equations for mass, momentum and granular energy. Using this analogy with standard fluid mechanics, we have recently suggested a novel mechanism of segregation in granular mixtures based on a *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* _{ρ 1} = ρ_{ l }/ρ_{ s } < 1, above which the *lighter intruder sinks*, thereby signalling the *onset* of the *reverse buoyancy* effect. For a mixture of dissipative particles, on the other hand, the non-zero pseudo-thermal buoyancy force gives rise to another threshold density-ratio, *R* _{ρ 2} (> *R* _{ρ 1}), above which the intruder rises again. Focussing on the *tracer* limit of intruders in a dense binary mixture, we study the dynamics of an intruder in a vibrofluidized system, with the effect of the base-plate excitation being taken into account through a 'mean-field' assumption. We find that the rise-time of the intruder could vary *nonmonotonically* 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 *the heavier (and larger) the intruder, the faster it ascends*. The peak on the rise-time curve decreases in height and shifts to a lower density-ratio as we increase the pseudo-thermal buoyancy force. The rise (/sink) time *diverges* near the threshold density-ratio for reverse-segregation. Our theory offers a *unified* description for the (reverse) Brazil-nut segregation and the nonmonotonic ascension dynamics of Brazil-nuts.

## Key words

Granular mixture Brazil-nut segregation Reverse buoyancy Nonmonotonic rise velocity## Preview

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