Granular Matter (2007) 10:47–52 DOI 10.1007/s10035-007-0057-z Dense inclined flows of inelastic spheres James T. Jenkins Received: 1 February 2007 / Published online: 7 August 2007 © Springer-Verlag 2007 Abstract We outline an extension of the hydrodynamic equations for dense flows of identical, inelastic spheres that incorporates an additional length scale in the expression for the collisional rate of dissipation. This length scale is identi- fied with the length of a particle chain. In steady, fully devel- oped inclined flows, the resulting theory predicts that at a given angle of inclination a range of flow depths is possible, that such flows possess a region of uniform volume fraction, and that this volume fraction decreases as the angle of incli- nation increases. The balance of particle fluctuation energy, integrated through the depth of a flow, results in a relation between the mean flow velocity, the depth, and the angle of inclination that collapses experimental data taken over a range of inclination angle. Keywords Dense grain flow · Inclined flow · Inelastic spheres 1 Introduction Numerical simulations of steady, fully developed flows of inelastic spheres down bumpy inclines [1] indicate a range of flow depths is possible at a fixed angle of inclination, that such flows possess a core in which the volume fraction is constant, and that this volume fraction decreases as the angle of inclination increases. Similar observations have been made in simulations of flows of inelastic disks [2–4]. Also, Pouliquen [5], when interpreting his experimental results on steady, fully developed inclined flows over a bumpy base, J. T. Jenkins (B ) Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA e-mail: jtj2@cornell.edu URL: http://www.tam.cornell.edu observed that the relationship between the average flow rate, depth of flow, and angle of inclination could be simplified by incorporating the dependence on the angle of inclination through the relation between it and the depth at which the flow stopped. Numerical simulations [6] support this observation. Hydrodynamic equations derived using methods of the kinetic theory [7, 8] have had some success in predicting the observed features of collisional shearing flows in dilute and moderately dense situations [9–11]. However, the assump- tion that the particles interact through uncorrelated, instan- taneous, binary collisions, upon which such derivations are usually based, can fail in dense shearing flows, especially those subjected to gravity. In such flows, repeated collisions and/or enduring contacts between the particles can occur throughout much of the flow. Recently, attempts have been made to incorporate correlated interactions [12, 13] and enduring contacts [14–19] into theories for dense, inclined flows. All such theories have the capability of describing some of the features of dense, inclined flows observed in experi- ments and simulations. Some require more assumptions than others and, as a consequence, might be regarded as less fun- damental. However, with the possible exception of a non- linear kinetic theory recently proposed by Kumaran [20] for frictional spheres, no existing theory has the capability of pre- dicting three key features of steady, fully developed, dense, inclined flows: the possibility of flows at a single angle of inclination over a large range of depths, a uniform solids fraction through most of the depth that decreases with angle of inclination [1, 2]; and the collapse of the data relating mean flow velocity and depth of flow at different angles of incli- nation with an appropriate scaling [6]. Here we extend the theory for dense, inclined flows of circular disks that predicts these features [21] to spheres. The result is a slight extension of the simplest theory for 123