Skip to main content Accessibility help

Application of the compressible $I$ -dependent rheology to chute and shear flow instabilities

  • J. S. Fannon (a1), I. R. Moyles (a1) (a2) and A. C. Fowler (a1) (a3)


We consider the instability properties of dense granular flow in inclined plane and plane shear geometries as tests for the compressible inertial-dependent rheology. The model, which is a recent generalisation of the incompressible $\unicode[STIX]{x1D707}(I)$ rheology, constitutes a hydrodynamical description of dense granular flow which allows for variability in the solids volume fraction. We perform a full linear stability analysis of the model and compare its predictions to existing experimental data for glass beads on an inclined plane and discrete element simulations of plane shear in the absence of gravity. In the case of the former, we demonstrate that the compressible model can quantitatively predict the instability properties observed experimentally, and, in particular, we find that it performs better than its incompressible counterpart. For the latter, the qualitative behaviour of the plane shear instability is also well captured by the compressible model.


Corresponding author

Email address for correspondence:


Hide All
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, 277306.
Alam, M. & Nott, P. R. 1997 The influence of friction on the stability of unbounded granular shear flow. J. Fluid Mech. 343, 267301.
Alam, M. & Nott, P. R. 1998 Stability of plane Couette flow of a granular material. J. Fluid Mech. 377, 99136.
Ancey, C. 2001 Dry granular flows down an inclined channel: experimental investigations on the frictional-collisional regime. Phys. Rev. E 65 (1), 011304.
Baker, J. L., Barker, T. & Gray, J. M. N. T. 2016a A two-dimensional depth-averaged rheology for dense granular avalanches. J. Fluid Mech. 787, 367395.
Baker, J. L., Johnson, C. G. & Gray, J. M. N. T. 2016b Segregation-induced finger formation in granular free-surface flows. J. Fluid Mech. 809, 168212.
Barker, T. & Gray, J. M. N. T. 2017 Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow. J. Fluid Mech. 828, 532.
Barker, T., Schaeffer, D. G., Bohorquez, P. & Gray, J. M. N. T. 2015 Well-posed and ill-posed behaviour of the 𝜇(I) rheology for granular flow. J. Fluid Mech. 779, 794818.
Barker, T., Schaeffer, D. G., Shearer, M. & Gray, J. M. N. T. 2017 Well-posed continuum equations for granular flow with compressibility and 𝜇(I)-rheology. Proc. R. Soc. Lond. A 473 (2201), 20160846.
Berzi, D. & Jenkins, J. T. 2015 Steady shearing flows of deformable, inelastic spheres. Soft Matt. 11, 47994808.
Bouzid, M., Trulsson, M., Claudin, P., Clément, E. & Andreotti, B. 2013 Nonlocal rheology of granular flows across yield conditions. Phys. Rev. Lett. 111, 238301.
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover Publications.
Brodu, N., Delannay, R., Valance, A. & Richard, P. 2015 New patterns in high-speed granular flows. J. Fluid Mech. 769, 218228.
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover Publications.
Da Cruz, F., Emam, S., Prochnow, M., Roux, J.-N. & Chevoir, F. 2005 Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72 (2), 021309.
Daerr, A. 2001 Dynamical equilibrium of avalanches on a rough plane. Phys. Fluids 13 (7), 21152124.
Delannay, R., Valance, A., Mangeney, A., Roche, O. & Richard, P. 2017 Granular and particle-laden flows: from laboratory experiments to field observations. J. Phys. D: Appl. Phys. 50 (5), 053001.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Edwards, A. N. & Gray, J. M. N. T. 2015 Erosion–deposition waves in shallow granular free-surface flows. J. Fluid Mech. 762, 3567.
Fenistein, D. & van Hecke, M. 2003 Kinematics: wide shear zones in granular bulk flow. Nature 425 (6955), 256.
Forterre, Y. 2006 Kapiza waves as a test for three-dimensional granular flow rheology. J. Fluid Mech. 563, 123132.
Forterre, Y. & Pouliquen, O. 2002 Stability analysis of rapid granular chute flows: formation of longitudinal vortices. J. Fluid Mech. 467, 361387.
Forterre, Y. & Pouliquen, O. 2003 Long-surface-wave instability in dense granular flows. J. Fluid Mech. 486, 2150.
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 124.
Fowler, A. 2011 Mathematical Geoscience. Springer Science & Business Media.
Fowler, A. C. 1997 Mathematical Models in the Applied Sciences. Cambridge University Press.
Goddard, J. D. & Lee, J. 2017 On the stability of the 𝜇(I) rheology for granular flow. J. Fluid Mech. 833, 302331.
Goddard, J. D. & Lee, J. 2018 Regularization by compressibility of the 𝜇(I) model of dense granular flow. Phys. Fluids 30 (7), 073302.
Gray, J. M. N. T. 2001 Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441, 129.
Gray, J. M. N. T. & Edwards, A. N. 2014 A depth-averaged rheology for shallow granular free-surface flows. J. Fluid Mech. 755, 503534.
Gray, J. M. N. T., Gajjar, P. & Kokelaar, P. 2015 Particle-size segregation in dense granular avalanches. C. R. Phys. 16 (1), 7385.
Gray, J. M. N. T., Tai, Y.-C. & Noelle, S. 2003 Shock waves, dead zones and particle-free regions in rapid granular free-surface flows. J. Fluid Mech. 491, 161181.
Henann, D. L. & Kamrin, K. 2013 A predictive, size-dependent continuum model for dense granular flows. Proc. Natl Acad. Sci. USA 110 (17), 67306735.
Henann, D. L. & Kamrin, K. 2016 A finite element implementation of the nonlocal granular rheology. Intl J. Numer. Meth. Engng 108 (4), 273302.
Heyman, J., Delannay, R., Tabuteau, H. & Valance, A. 2017 Compressibility regularizes the 𝜇(I)-rheology for dense granular flows. J. Fluid Mech. 830, 553568.
Higham, N. J.(Ed.) 2015 The Princeton Companion to Applied Mathematics. Princeton University Press.
Jackson, R. 1983 Some mathematical and physical aspects of continuum models for the motion of granular materials. In Theory of Dispersed Multiphase Flow (ed. Meyer, R. E.), pp. 291337. Academic Press.
Jaeger, H. M., Nagel, S. R. & Behringer, R. P. 1996 Granular solids, liquids, and gases. Rev. Mod. Phys. 68 (4), 12591273.
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187202.
Jop, P. 2015 Rheological properties of dense granular flows. C. R. Phys. 16 (1), 6272.
Jop, P., Forterre, Y. & Pouliquen, O. 2005 Crucial role of sidewalls in granular surface flows: consequences for the rheology. J. Fluid Mech. 541, 167192.
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094), 727730.
Kamrin, K. & Henann, D. L. 2015 Nonlocal modeling of granular flows down inclines. Soft Matt. 11 (1), 179185.
Kamrin, K. & Koval, G. 2012 Nonlocal constitutive relation for steady granular flow. Phys. Rev. Lett. 108 (17), 178301.
Koval, G., Roux, J.-N., Corfdir, A. & Chevoir, F. 2009 Annular shear of cohesionless granular materials: from the inertial to quasistatic regime. Phys. Rev. E 79 (2), 021306.
Kumaran, V. & Maheshwari, S. 2012 Transition due to base roughness in a dense granular flow down an inclined plane. Phys. Fluids 24 (5), 053302.
Malik, M., Alam, M. & Dey, J. 2006 Nonmodal energy growth and optimal perturbations in compressible plane Couette flow. Phys. Fluids 18 (3), 034103.
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86 (2), 376413.
Mandal, S. & Khakhar, D. V. 2016 A study of the rheology of planar granular flow of dumbbells using discrete element method simulations. Phys. Fluids 28 (10), 103301.
Midi, G. D. R. 2004 On dense granular flows. Eur. Phys. J. E 14 (4), 341365.
Moyles, I. & Wetton, B. 2015 Fingering phenomena in immiscible displacement in porous media flow. J. Engng Maths 90 (1), 83104.
Mueth, D. M., Debregeas, G. F., Karczmar, G. S., Eng, P. J., Nagel, S. R. & Jaeger, H. M. 2000 Signatures of granular microstructure in dense shear flows. Nature 406 (6794), 385389.
Müller, S. B. & Kleiser, L. 2008 Viscous and inviscid spatial stability analysis of compressible swirling mixing layers. Phys. Fluids 20 (11), 114103.
Pihler-Puzović, D. & Mullin, T. 2013 The timescales of granular segregation in horizontally shaken monolayers. Proc. R. Soc. Lond. A 469 (2157), 20130203.
Pouliquen, O. 1999 Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11 (3), 542548.
Pouliquen, O., Cassar, C., Jop, P., Forterre, Y. & Nicolas, M. 2006 Flow of dense granular material: towards simple constitutive laws. J. Stat. Mech.-Theory E 2006 (07), P07020.
Pouliquen, O. & Forterre, Y. 2002 Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J. Fluid Mech. 453, 133151.
Pouliquen, O. & Forterre, Y. 2009 A non-local rheology for dense granular flows. Phil. Trans. R. Soc. Lond. A 367 (1909), 50915107.
Rao, K. K., Nott, P. R. & Sundaresan, S. 2008 An Introduction to Granular Flow. Cambridge University Press.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Schofield, A. & Wroth, P. 1968 Critical State Soil Mechanics. McGraw-Hill.
Sun, J. & Sundaresan, S. 2011 A constitutive model with microstructure evolution for flow of rate-independent granular materials. J. Fluid Mech. 682, 590616.
Tejchman, J. & Gudehus, G. 2001 Shearing of a narrow granular layer with polar quantities. Intl J. Numer. Anal. Meth. Geomech. 25 (1), 128.
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.
Trulsson, M., Andreotti, B. & Claudin, P. 2012 Transition from the viscous to inertial regime in dense suspensions. Phys. Rev. Lett. 109, 118305.
Trulsson, M., Bouzid, M., Claudin, P. & Andreotti, B. 2013 Dynamic compressibility of dense granular shear flows. Eur. Phys. Lett. 103 (3), 38002.
Varsakelis, C. & Papalexandris, M. V. 2016 Stability of wall bounded, shear flows of dense granular materials: the role of the Couette gap, the wall velocity and the initial concentration. J. Fluid Mech. 791, 384413.
Viroulet, S., Baker, J. L., Rocha, F. M., Johnson, C. G., Kokelaar, B. P. & Gray, J. M. N. T. 2018 The kinematics of bidisperse granular roll waves. J. Fluid Mech. 848, 836875.
Wang, C.-H., Jackson, R. & Sundaresan, S. 1996 Stability of bounded rapid shear flows of a granular material. J. Fluid Mech. 308, 3162.
Woodhouse, M. J., Thornton, A. R., Johnson, C. G., Kokelaar, B. P. & Gray, J. M. N. T. 2012 Segregation-induced fingering instabilities in granular free-surface flows. J. Fluid Mech. 709, 543580.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification

Type Description Title
Supplementary materials

Fannon et al. supplementary material
Fannon et al. supplementary material 1

 Unknown (157 KB)
157 KB


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed