Well-posed continuum equations for granular flow with compressibility and μ(I)-rheology. (31st May 2017)

• Record Type: Journal Article
• Authors: Barker, T.; Schaeffer, D. G.; Shearer, M.; Gray, J. M. N. T.
• Is Part Of: Proceedings. Volume 473:Number 2201(2017)
• Publication Date: 2017-05-31
• Subjects: granular flow -- continuum modelling -- rheology; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2016.0846
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : 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.