Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

Abstract

In this work we quantitatively assess, via instabilities, a Navier-Stokes-order (smallKnudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers (Kn u ) at the time of cluster detection reveals Kn u 1 due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a smallKn (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier-Stokes-order (small-Kn) theories are surprisingly accurate outside their expected range of validity.

The surprising relevance of a continuum description to granular clusters

“…However, for granular materials, this 'Burnett' expansion must also account for non-trivial effects of particle inelasticity (Sela & Goldhirsch 1998;Kumaran 2006). Mitrano et al (2014) suggest that such a laborious approach might be avoided, while preserving clusters as a natural instability of the hydrodynamic equations. To show this, they juxtapose results from simulations and a numerical integration of the hydrodynamic equations that Garzó & Dufty (1999) derived for small Kn (figure 1).…”

“…In short, continuum equations at 0.06 0.08 0.10 0.12 0.14 0.16 0.18 the Navier-Stokes order initiate realistic granular clusters. This surprises Mitrano et al (2014) for two reasons. First, vortices produce velocity correlations contradicting the molecular chaos assumption.…”

“…The article by Mitrano et al (2014) addresses the pivotal question of whether particle suspensions can be modelled by continuum equations, despite the inevitable formation of clusters. Whether particles interact with atmospheric turbulence (Chun et al 2005) or flow as a suspension through chemical reactors (Pepiot & Desjardins 2012), they invariably produce clusters that theory cannot ignore.…”

“…Yet, a persistent question is whether this can capture cluster formation, particularly when particles cross (Desjardins, Fox & Villedieu 2008). Therefore, because clusters also form in a granular 'gas' of inelastic particles colliding in a vacuum, it is instructive to test, as Mitrano et al (2014) did, whether hydrodynamic equations are relevant to the particle phase without a fluid.…”

Scale-up and multiphase reaction engineering

“…However, proper credit was not given to many in academia that still hold the reaction engineering banner high and continue to develop better foundations for future reactor scale-up by focusing on a specific scale of scrutiny or a specific process. For [108,109], and Li at Beijing [110,111,112 ,113]). There are also exciting developments in novel, nature inspired contactors (Coppens at London [114][115][116]117 ]), in membrane reactors (Tsotsis at USC [118,119]) and microreactors (Jensen at MIT [120][121][122]).…”