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

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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.
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Funding Information
  • National Science Foundation (DMS-1517291)
  • Engineering and Physical Sciences Research Council (EP/I019189/1)
  • Engineering and Physical Sciences Research Council (EP/K00428X/1)
  • Engineering and Physical Sciences Research Council (EP/M022447/1)
  • Natural Environment Research Council (NE/E003206/1)
  • Natural Environment Research Council (NE/K003011/1)

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