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.

A hydrodynamic description for an $s$-component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

The Enskog kinetic theory is used as a starting point to model a suspension of solid particles in a viscous gas. Unlike previous efforts for similar suspensions, the gas-phase contribution to the instantaneous particle acceleration appearing in the Enskog equation is modelled using a Langevin equation, which can be applied to a wide parameter space (e.g. high Reynolds number). Attention here is limited to low Reynolds number flow, however, in order to assess the influence of the gas phase on the constitutive relations, which was assumed to be negligible in a previous analytical treatment. The Chapman–Enskog method is used to derive the constitutive relations needed for the conservation of mass, momentum and granular energy. The results indicate that the Langevin model for instantaneous gas–solid force matches the form of the previous analytical treatment, indicating the promise of this method for regions of the parameter space outside of those attainable by analytical methods (e.g. higher Reynolds number). The results also indicate that the effect of the gas phase on the constitutive relations for the solid-phase shear viscosity and Dufour coefficient is non-negligible, particularly in relatively dilute systems. Moreover, unlike their granular (no gas phase) counterparts, the shear viscosity in gas–solid systems is found to be zero in the dilute limit and the Dufour coefficient is found to be non-zero in the elastic limit.

The linear integral equations defining the Navier-Stokes (NS) transport coefficients for polydisperse granular mixtures of smooth inelastic hard disks or spheres are solved by using the leading terms in a Sonine polynomial expansion. Explicit expressions for all the NS transport coefficients are given in terms of the sizes, masses, compositions, density, and restitution coefficients. In addition, the cooling rate is also evaluated to first order in the gradients. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Finally, a detailed comparison between the derivation of the current theory and previous theories for mixtures is made, with attention paid to the implication of the various treatments employed to date.

For rapidly flowing granular mixtures, existing kinetic-theory descriptions based on an assumed form of the velocity distribution function typically contain one of two simplifying assumptions: a Maxwellian velocity distribution or an equipartition of energy. In the current work, the influence of non-equipartition effects is explored in the context of two flow types: flow in which species segregation does not occur (namely, simple shear flow) and a segregating flow. For the former case, a comparison between existing kinetic theories and molecular-dynamics simulations of a binary system indicates that the incorporation of a non-Maxwellian velocity distribution is critical for reliable stress predictions, as is consistent with previous findings. However, the predictions are fairly insensitive to the equipartition versus non-equipartition treatment, despite the presence of a significant non-equipartition of energy. Nevertheless, an analysis of the diffusion equation for a segregating flow indicates that the presence of a non-equipartition of energy gives rise to additional components of the driving forces associated with size segregation. These additional components involve gradients of the species temperature, whereas theories based on an equipartition assumption only involve gradients in the mixture temperature. Molecular-dynamics simulations of the segregating flow, in conjunction with kinetic theory of binary systems, show that the non-equipartition effects are non-negligible for systems characterized by moderate values of mass differences and restitution coefficients. These simulations also reveal that the more massive particle may exhibit a lower species temperature than its lighter counterpart, contrary to previous observations in non-segregating systems. A physical explanation for this behaviour is provided.