Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2017-05, Vol.473 (2201), p.20160846-20160846
• Main Authors: Barker, T., Schaeffer, D. G., Shearer, M., Gray, J. M. N. T.
• Format: Article
• Language: English
• Subjects: Coefficient of friction; Computational fluid dynamics; Continuum modeling; Continuum Modelling; Coulomb friction; Deformation; Fluid flow; Granular Flow; Ill posed problems; Incompressible flow; Partial differential equations; Rheological properties; Rheology; Soil compressibility; Soil conditions; Soil mechanics; Two dimensional flow; Well posed problems
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2016.0846

Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent μ(I)-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I-dependent rheology. When the I-dependence comes from a specific friction coefficient μ(I), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that μ(I) satisfies certain minimal, physically natural, inequalities.