Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T17:18:32.058Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alam, M. 2006 Streamwise structures and density patterns in rapid granular Couette flow: a linear stability analysis. J. Fluid Mech. 553, 1–32.CrossRefGoogle Scholar
Alam, M., Arakeri, V. H., Nott, P. R., Goddard, J. D. & Herrmann, H. J. 2005 Instability-induced ordering, universal unfolding and the role of gravity in granular Couette flow. J. Fluid Mech. 523, 277–306.Google Scholar
Alam, M. & Nott, P. R. 1998 Stability of plane Couette flow of a granular material. J. Fluid Mech. 377, 99–136.CrossRefGoogle Scholar
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. & Sorensen, D. 1999 LAPACK Users' Guide, 3rd edn. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Conway, S. L. & Glasser, B. J. 2004 Density waves and coherent structures in granular Couette flows. Phys. Fluids 16, (3), 509–529.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Forterre, Y. & Pouliquen, O. 2001 Longitudinal vortices in granular flows. Phys. Rev. Lett. 86, (26), 5886–5889.Google Scholar
Forterre, Y. & Pouliquen, O. 2002 Stability analysis of rapid granular chute flows: formation of longitudinal vortices. J. Fluid Mech. 467, 361–387.CrossRefGoogle Scholar
Goldhirsch, I. & Zanetti, G. 1993 Clustering instability in dissipative gases. Phys. Rev. Lett. 70, (11), 1619–1622.Google Scholar
Golub, G. H. & Van Loan, C. F. 1989 Matrix Computations, 2nd edn. The Johns Hopkins University Press.Google Scholar
Hopkins, M. A. & Louge, M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, (1), 47–57.Google Scholar
Jenkins, J. T. 2007 Dense inclined flows of inelastic spheres. Granul. Matter 10, 47–52.Google Scholar
Kumaran, V. 2008 Dense granular flow down an inclined plane: from kinetic theory to granular dynamics. J. Fluid Mech. 599, 121–168.Google Scholar
Liss, E. D., Conway, S. L. & Glasser, B. J. 2002 Density waves in gravity-driven granular flow through a channel. Phys. Fluids 14, (9), 3309–3326.Google Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256.CrossRefGoogle Scholar
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86, (2), 376–413.Google Scholar
McNamara, S. & Young, W. R. 1994 Inelastic collapse in two dimensions. Phys. Rev. E 50, (1), 28–31.Google Scholar
Mitarai, N. & Nakanishi, H. 2001 Instability of dilute granular flows on rough slope. J. Phys. Soc. Japan 70, (10), 2809–2812.CrossRefGoogle Scholar
Mitarai, N. & Nakanishi, H. 2004 Linear stability analysis of rapid granular flow down a slope and density wave formation. J. Fluid Mech. 507, 309–334.CrossRefGoogle Scholar
Moler, C. B. & Stewart, G. W. 1973 An algorithm for generalised matrix eigenvalue problems. SIAM J. Numer. Anal. 10, (2), 241–256.CrossRefGoogle Scholar
Richman, M. W. 1988 Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240.Google Scholar
Savage, S. B. 1992 Instability of unbounded uniform granular shear flow. J. Fluid Mech. 241, 109–123.CrossRefGoogle Scholar
Schmid, P. J. & Kytömaa, H. K. 1994 Transient and asymptotic stability of granular shear flow. J. Fluid Mech. 264, 255–275.CrossRefGoogle Scholar
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74.CrossRefGoogle Scholar
Tan, M.-L. & Goldhirsch, I. 1997 Intercluster interactions in rapid granular shear flows. Phys. Fluids 9, (4), 856–869.Google Scholar
Wang, C.-H., Jackson, R. & Sundaresan, S. 1996 Stability of bounded rapid shear flows of a granular material. J. Fluid Mech. 308, 31–62.CrossRefGoogle Scholar
Wang, C.-H., Jackson, R. & Sundaresan, S. 1997 Instabilities of fully developed rapid flow of a granular material in a channel. J. Fluid Mech. 342, 179–197.CrossRefGoogle Scholar
Woodhouse, M. J. 2008 Rapid granular shear flows. PhD thesis, School of Mathematics, University of Bristol, Bristol.Google Scholar
Woodhouse, M. J., Hogg, A. J. & Sellar, A. A. 2010 Rapid granular flows down inclined planar chutes. Part 1. Steady flows, multiple solutions and existence domains. J. Fluid Mech. (in press).Google Scholar