J. Fluid Mech. (2014), vol. 738, R2, doi:10.1017/jfm.2013.602 Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime Peter P. Mitrano 1 , John R. Zenk 1 , Sofiane Benyahia 2 , Janine E. Galvin 2 , Steven R. Dahl 1 and Christine M. Hrenya 1, † 1 Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309, USA 2 National Energy Technology Laboratory, Morgantown, WV 26507, USA (Received 26 September 2013; revised 5 November 2013; accepted 9 November 2013) In this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small- Knudsen-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 small- Kn (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. Key words: granular media, kinetic theory, nonlinear instability 1. Introduction Since Maxwell first asserted, via a series of stability analyses, that the rings of Saturn consist of discrete particles rather than a collection of thin, solid ringlets, the stability of particulate systems has been studied (Goldreich & Tremaine 1982). Flow instabilities stemming from dissipative particle interactions have implications ranging † Email address for correspondence: hrenya@colorado.edu c  Cambridge University Press 2013 738 R2-1