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Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions

Published online by Cambridge University Press:  19 May 2010

MARK J. WOODHOUSE  and
ANDREW J. HOGG
MARK J. WOODHOUSE*
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
ANDREW J. HOGG
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Present address: School of Mathematics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address for correspondence: mark.woodhouse@manchester.ac.uk
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Abstract

The linear stability of steady solutions for a rapid granular flow down an inclined chute, modelled using a kinetic theory continuum model, is analysed. The previous studies of Forterre & Pouliquen (J. Fluid Mech., vol. 467, 2002, p. 361) and Mitarai & Nakanishi (J. Fluid Mech., vol. 507, 2004, p. 309) are extended by considering fully three-dimensional perturbations, allowing variations in both the cross-slope and downslope directions, as well as normal to the base. Our results demonstrate the existence of three qualitatively different unstable perturbations, each of which can be the most rapidly growing instability for different steady flows. By considering the linear stability of many steady solutions along macroscopic flow curves, we show that linear stability occurs in only a small part of parameter space, and furthermore the regions of linear instability do not correlate with density inversion of the underlying steady solutions. Our results suggest that inelastic clustering is the dominant instability mechanism.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 461 - 488
Copyright
Copyright © Cambridge University Press 2010

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