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Compressibility regularizes the 𝜇(I)-rheology for dense granular flows

Published online by Cambridge University Press:  03 October 2017

J. Heyman Open the ORCID record for J. Heyman [Opens in a new window] ,
R. Delannay ,
H. Tabuteau  and
A. Valance
J. Heyman*
Affiliation:
Institut de Physique de Rennes, UMR CNRS 6251, Universit de Rennes 1, Campus de Beaulieu, Btiment 11A, 263 Avenue Gnral Leclerc, 35042 Rennes CEDEX, France
R. Delannay
Affiliation:
Institut de Physique de Rennes, UMR CNRS 6251, Universit de Rennes 1, Campus de Beaulieu, Btiment 11A, 263 Avenue Gnral Leclerc, 35042 Rennes CEDEX, France
H. Tabuteau
Affiliation:
Institut de Physique de Rennes, UMR CNRS 6251, Universit de Rennes 1, Campus de Beaulieu, Btiment 11A, 263 Avenue Gnral Leclerc, 35042 Rennes CEDEX, France
A. Valance
Affiliation:
Institut de Physique de Rennes, UMR CNRS 6251, Universit de Rennes 1, Campus de Beaulieu, Btiment 11A, 263 Avenue Gnral Leclerc, 35042 Rennes CEDEX, France
*
†Email address for correspondence: joris.heyman@univ-rennes1.fr
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Abstract

The $\unicode[STIX]{x1D707}(I)$-rheology was recently proposed as a potential candidate to model the incompressible flow of frictional grains in the dense inertial regime. However, this rheology was shown to be ill-posed in the mathematical sense for a large range of parameters, notably in the low and large inertial number limits (Barker et al., J. Fluid Mech., vol. 779, 2015, pp. 794–818). In this rapid communication, we extend the stability analysis of Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818) to compressible flows. We show that compressibility regularizes the equations, making the problem well-posed for all parameters, with the condition that sufficient dissipation be associated with volume changes. In addition to the usual Coulomb shear friction coefficient $\unicode[STIX]{x1D707}$, we introduce a bulk friction coefficient $\unicode[STIX]{x1D707}_{b}$, associated with volume changes and show that the problem is well-posed if $\unicode[STIX]{x1D707}_{b}>1-7\unicode[STIX]{x1D707}/6$. Moreover, we show that the ill-posed domain defined by Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818) transforms into a domain where the flow is unstable but remains well-posed when compressibility is taken into account. These results suggest the importance of taking into account dynamic compressibility for the modelling of dense granular flows and open new perspectives to investigate the emission and propagation of acoustic waves inside these flows.

JFM classification

Acoustics: Acoustics Complex Fluids: Complex Fluids Granular media: Granular media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 830 , 10 November 2017 , pp. 553 - 568
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Copyright
© 2017 Cambridge University Press 

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