Buoyancy driven convection in vertically shaken granular matter: experiment, numerics, and theory

Abstract

Buoyancy driven granular convection is studied for a shallow, vertically shaken granular bed in a quasi 2D container. Starting from the granular Leidenfrost state, in which a dense particle cluster floats on top of a dilute gaseous layer of fast particles (Meerson et al. in Phys Rev Lett 91:024301, 2003; Eshuis et al. in Phys Rev Lett 95:258001, 2005), we witness the emergence of counter-rotating convection rolls when the shaking strength is increased above a critical level. This resembles the classical onset of convection-at a critical value of the Rayleigh number-in a fluid heated from below. The same transition, even quantitatively, is seen in molecular dynamics simulations, and explained by a hydrodynamic-like model in which the granular material is treated as a continuum. The critical shaking strength for the onset of granular convection is accurately reproduced

Discrete and continuum descriptions of shaken granular matter

“…The overall picture is of a continuously fluidized bed of grains which goes from homogeneous Leidenfrost configurations to increasingly velocity correlated convective states, until flows are strong enough to sustain the density inhomogeneous buoyancy-driven convective state. From a bifurcation theory perspective, the convection transition can be understood as a pattern formation phase-transition, with the emergence of convective cells with a critical length-scale independent of the domain size, which is consistent with previously realized hydrodynamic stability analysis of the Leidenfrost state [80].…”

“…Nevertheless, as the shaking strength is increased, the granular Leidenfrost state looses its horizontal homogeneity and gives rise to a buoyancy-driven convective state, still a steady state but with v 0. Granular hydrodynamics was recently proven accurate in predicting the critical modes of this transition [80], confirming its validity for this particular state.…”

“…In a subsequent study, carried out in a slightly deeper, quasi-2D setup, Eshuis et al increased further the energy injection until the Leidenfrost state lost its stability and gave rise to a buoyancy driven convective state [71]. It was then shown that this transition can be captured by a linear stability analysis over a granular hydrodynamics model, with good agreement on the critical points for both experiments and simulations [80]. They took as initial stable state the vertical density profile, obtained from simulations and experiments, and studied its stability against a modulation of the hydrodynamic fields with a given wave number k x .…”

“…What other factors may affect λ * c is not studied further here, although we notice that previously realized stability analysis of the granular hydrodynamic equations have found an expression for λ * c as a function of the constitutive relations, which are in themselves dependent on the particle properties [80].…”

“…If the vibration strength is increased, the Leidenfrost state evolves to a buoyancy-driven convective state [150], in analogy to RayleighBernard convection. Recently, it was shown that granular hydrodynamics is able to quantitatively capture the critical points of this instability, by performing a linear stability analysis of perturbations over the Leidenfrost state [150,80].…”