Dynamic properties in a collisional model for confined granular fluids. A review

Abstract

Granular systems confined in a shallow box and driven by vertical vibration provide a simple geometry to study fluidized granular media. Grains gain kinetic energy vertically through collisions with the walls and redistribute it horizontally via interparticle collisions. The $Δ$-model has been proposed as a simplified description of this setup. In this model, a fixed velocity increment $Δ$ is added to the normal component of the relative velocity at collisions, effectively integrating out the vertical motion while preserving collisional energy injection. This compensates for inelastic losses and yields stable homogeneous steady states amenable to kinetic theory. An Enskog kinetic equation is formulated and analyzed to obtain the stationary temperature and equation of state. The Chapman--Enskog method is then applied to derive the Navier--Stokes transport coefficients and study inhomogeneous states. The theory is extended to granular mixtures with different masses, sizes, restitution coefficients, or $Δ$ values, leading to nonequipartition of energy even in homogeneous states. The resulting hydrodynamic equations, with transport coefficients obtained in the low-density regime, show unconditional stability of the homogeneous state and violation of Onsager reciprocity. Theoretical predictions agree well with molecular dynamics and direct simulation Monte Carlo results.

Figures (1)

• Figure S1: Fig: Conceptual motivation of the $\Delta$-model. (a) Quasi two-dimensional setup, where spherical grains are placed in a vertically vibrating shallow box. Grains can collide with the vibrating walls and among themselves. (b) Lateral view of the system. Grain collisions with the top and bottom walls inject energy into the vertical degrees of freedom, which is later transferred to the horizontal ones via grain-grain oblique collisions. (c) Top view of the quasi two-dimensional system. As the height of the box is larger than the particle diameters, the can partially overlap at collisions when seeing from above. (d) In the $\Delta$-model, the vertical motion is abstracted out keeping its effect on injecting energy into the horizontal degrees of freedom. If particles reach the collision with a small relative velocity, the net effect is to gain energy, but if their normal relative velocity is large, inelasticity overcomes the injection and the collision is dissipative. Note that in the $\Delta$-model, particles move only in $x$ and $y$, implying that there is no overlap and collisions take place when the distance is exactly equal to the particle diameter.