Assessing a Hydrodynamic Description for Instabilities in Highly Dissipative, Freely Cooling Granular Gases

An intriguing phenomenon displayed by granular flows and predicted by kinetic-theory-based models is the instability known as particle "clustering," which refers to the tendency of dissipative grains to form transient, loose regions of relatively high concentration. In this work, we assess a modified-Sonine approximation recently proposed [Garz\'o et al., Physica A 376, 94 (2007)] for a granular gas via an examination of system stability. In particular, we determine the critical length scale associated with the onset of two types of instabilities -vortices and clusters- via stability analyses of the Navier-Stokes-order hydrodynamic equations by using the expressions of the transport coefficients obtained from both the standard and the modified-Sonine approximations. We examine the impact of both Sonine approximations over a range of solids fraction \phi <0.2 for small restitution coefficients e=0.25--0.4, where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations, of which a small percentage were not considered due to inelastic collapse. Results show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a (highdissipation) qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size is briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.Comment: 4 figures; to be published in Phys. Rev.

Enskog Theory for Polydisperse Granular Mixtures II. Sonine Polynomial Approximation

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.Comment: 26 pages, to be published in Phys. Rev.

Enskog Theory for Polydisperse Granular Mixtures. I. Navier-Stokes order Transport

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 portion 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.Comment: 36 pages, to be published in Phys. Rev.

Discrete-Element-Method-Based Determination of Particle-Level Inputs for the Continuum Theory of Flows with Moderately Cohesive Particles

In this work, the cohesion-specific inputs for a recent continuum theory for cohesive particles are estimated for moderately cohesive particles that form larger agglomerates via discrete element method (DEM) simulations of an oscillating shear flow. In prior work, these inputs (critical velocities of agglomeration and breakage and collision cylinder diameters) were determined for lightly cohesive particles via the DEM of simple shear flow—i.e., a system dominated by singlets and doublets. Here, the DEM is again used to extract the continuum theory inputs, as experimental measurements are infeasible (i.e., collisions between particles of a diameter of <100 μm). However, simulations of simple shear flow are no longer feasible since the rate of agglomeration grows uncontrollably at higher cohesion levels. Instead, oscillating shear flow DEM simulations are used here to circumvent this issue, allowing for the continuum theory inputs of larger agglomerate sizes to be determined efficiently. The resulting inputs determined from oscillating shear flow are then used as inputs for continuum predictions of an unbounded, gas–solid riser flow. Although the theory has been previously applied to gas–solid flows of lightly cohesive particles, an extension to the theory is needed since moderately cohesive particles give rise to larger agglomerates (that still readily break). Specifically, the wider distribution of agglomerate sizes necessitates the use of polydisperse kinetic-theory-based closures for the terms in the solids momentum and granular energy balances. The corresponding continuum predictions of entrainment rate and agglomerate size distribution were compared against DEM simulations of the same system with good results. The DEM simulations were again used for validation, as it is currently extremely challenging to detect agglomerate sizes and the number of fractions in an experimental riser flow

Toward general regime maps for cohesive‐particle flows: Force versus energy‐based descriptions and relevant dimensionless groups

Much confusion exists on whether force‐ or energy‐based descriptions of cohesive‐particle interactions are more appropriate. We hypothesize a force‐based description is appropriate when enduring‐contacts dominate and an energy‐based description when contacts are brief in nature. Specifically, momentum is transferred through force‐chains when enduring‐contacts dominate and particles need to overcome a cohesive force to induce relative motion, whereas particles experiencing brief contacts transfer momentum through collisions and must overcome cohesion‐enhanced energy losses to avoid agglomeration. This hypothesis is tested via an attempt to collapse the dimensionless, dependent variable characterizing a given system against two dimensionless numbers: A generalized bond number, BoG–ratio of maximum cohesive force to the force driving flow, and a new Agglomerate number, Ag–ratio of critical cohesive energy to the granular energy. A gamut of experimental and simulation systems (fluidized bed, hopper, etc.), and cohesion sources (van der Waals, humidity, etc.), are considered. For enduring‐contact systems, collapse occurs with BoG but not Ag, and vice versa for brief‐contact systems, thereby providing support for the hypothesis. An apparent discrepancy with past work is resolved, and new insight into Geldart’s classification is gleaned.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/169329/1/aic17337.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/169329/2/aic17337_am.pd