Flows of solid particles are known to exhibit a clustering instability—dynamic microstructures characterized by a dense region of highly concentrated particles surrounded by a dilute region with relatively few particles—that has no counterpart in molecular fluids. Clustering is pervasive in rapid flows. Its presence impacts momentum, heat, and mass transfer, analogous to how turbulence affects single-phase flows. Yet predicting clustering is challenging, again analogous to the prediction of turbulent flows. In this review, we focus on three key areas: (a) state-of-the-art mathematical tools used to study clustering, with an emphasis on kinetic theory–based continuum models, which are critical to the prediction of the larger systems found in nature and industry, (b) mechanisms that give rise to clustering, most of which are explained via linear stability analyses of kinetic theory–based models, and (c) a critical review of validation studies of kinetic theory–based models to highlight the accuracies and limitations of such theories.

Transport properties of a suspension of solid particles in a viscous gas are studied. The dissipation in such systems arises from two sources: inelasticity in particle collisions and viscous dissipation due to the effect of the gas phase on the particles. Here we consider a simplified case in which the mean relative velocity between the gas and solid phases is taken to be zero, such that "thermal drag" is the only remaining gas-solid interaction. Unlike the previous, more general, treatment of the drag force [Garzó et al., J. Fluid Mech. 712, 129 (2012)]JFLSA70022-112010.1017/jfm.2012.404, here we take into account contributions to the (scaled) transport coefficients η^{*} (shear viscosity), κ^{*} (thermal conductivity), and μ^{*} (Dufour-like coefficient) coming from the temperature dependence of the (dimensionless) friction coefficient γ^{*} characterizing the amplitude of the drag force. At moderate densities, the thermal drag model (which is based on the Enskog kinetic equation) is solved by means of the Chapman-Enskog method and the Navier-Stokes transport coefficients are determined in terms of the coefficient of restitution, the solid volume fraction, and the friction coefficient. The results indicate that the effect of the gas phase on η^{*} and μ^{*} is non-negligible (especially in the case of relatively dilute systems) while the form of κ^{*} is the same as the one obtained in the dry granular limit. Finally, as an application of these results, a linear stability analysis of the hydrodynamic equations is carried out to analyze the conditions for stability of the homogeneous cooling state. A comparison with direct numerical simulations shows a good agreement for conditions of practical interest.

SignificanceThe quantitative ability of a kinetic-theory-based, two-fluid model is demonstrated in a clustering (unstable) gas-solid system via highly resolved simulations. Unlike previous works, this assessment is validated against ideal computational fluid dynamics-discrete element method data to minimize sources of discrepancy. Overall, good agreement in mean-slip velocities is observed with relative errors less than 20% over a mean solids concentration range of 0.02-0.25. Local concentration gradient distributions are also studied, showing a distinct shift toward higher gradients at higher mean solids concentrations which is proposed as the bottleneck in obtaining grid-independence rather than the cluster length scale.

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