Dynamic Properties in a Collisional Model for Confined Granular Fluids: A Review

Granular systems confined in a shallow box and subjected to vertical vibration provide an attractive geometry for studying fluidized granular media. In this configuration, grains acquire kinetic energy in the vertical direction through collisions with the confining walls, and this energy is subsequently transferred to the horizontal degrees of freedom via interparticle collisions. In recent years, the so-called Δ-model has been introduced as a simplified yet effective description of the dynamics of granular systems in such geometries. This review presents the results obtained from kinetic theory for the granular Δ-model. To model the energy transfer mechanism, a fixed velocity increment Δ is added to the normal component of the relative velocity during collisions. In this way, the vertical motion is effectively integrated out while retaining the collisional energy injection characteristic of the confined setup. This mechanism compensates for the energy loss due to inelastic collisions and leads to stable homogeneous steady states that can be analyzed within the framework of kinetic theory. The Enskog kinetic equation is formulated for this model and first analyzed in homogeneous steady states, yielding the stationary temperature and the equation of state. The dynamics of inhomogeneous states is then investigated using the Chapman-Enskog method, from which the Navier-Stokes transport coefficients are derived. The theory is further extended to granular mixtures, in which particles may differ in mass, size, restitution coefficient, or in the value of Δ. In this case, the phenomenology becomes richer; for example, energy equipartition is violated even in homogeneous steady states. The mixture dynamics is studied through the corresponding Navier-Stokes equations, and the associated transport coefficients are obtained in the low-density regime. The analysis of the hydrodynamic equations shows that, in agreement with simulations, the homogeneous state is linearly stable. Moreover, the intrinsically nonequilibrium nature of the model leads to the violation of Onsager reciprocity relations in granular mixtures. The theoretical predictions exhibit in general good agreement with both molecular dynamics simulations and direct simulation Monte Carlo results.