(72b) Fluid-Mediated Sources to Granular Temperature in Homogeneous Fluidization

When considering the dynamics of rapid particle-laden flows, a hydrodynamic description for the solids phase is often adopted. Drawing from a strong analogy with molecular fluids, Enskog kinetic theory has served as a valuable starting point for the modelling of particle-laden suspensions. Traditionally, one considers the classic Chapman-Enskog expansion of the Boltzmann equation to arrive at a Navier-Stokes equation with detailed expressions for the transport coefficients. This approach encompasses the classic two-fluid model. In many cases, the role of a surrounding viscous fluid is neglected. However, it has become increasingly clear that fluid ow through an assembly of particles leads to a distribution of hydrodynamic forces with significant variance. Variance in drag force, arising from neighbor-induced perturbations to the flow field, acts as sources and sinks to the granular temperature balance. Here, we describe drag force disturbances as a stochastic quantity that follows an Ornstein-Uhlenbeck process. For the case of homogeneous fluidization, we derive the sources and sinks to granular temperature by analytically solving the Fokker-Planck equation in force-velocity phase space and integrating over the relevant quadrants. The obtained solutions may be utilized for comparison with PR-DNS benchmark data as well as employed within the classic two-fluid model.