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Analytical solution of the \(\mu (I)-\)rheology for fully developed granular flows in simple configurations

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Abstract

Using the \(\mu (I)\) continuum model recently proposed for dense granular flows, we study theoretically steady and fully developed granular flows in two configurations: a plane shear cell and a channel made of two parallel plates (Poiseuille configuration). In such a description, the granular medium behaves like a fluid whose viscosity is a function of the inertia. In the shear plane geometry our calculation predicts that the height of the shear bands scales with \(U_0^{1/4}P_0^{1/2},\,\mathrm{where }\,U_0\) is the velocity of the moving plate and \(P_0\) the pressure applied at its top. In the Poiseuille configuration, the medium is sheared between the lateral boundaries and a plug flow is located in the center of the channel. The size of the plug flow is found to increase for a decreasing pressure gradient. We show that, for small pressure gradient, the granular material behaves like a Bingham plastic fluid.

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Acknowledgments

We are deeply indebted to J. T. Jenkins and D. Berzi for fruitful discussions. This work is supported by the Région Bretagne (CREATE SAMPLEO). M. T. is supported by the Région Bretagne (ARED Grant).

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Author notes

  1. Merline Tankeo

    Present address: Faculté des Sciences, Département d’informatique, Université de Yaoundé 1, B.P. 812, Yaoundé, Cameroun

Authors and Affiliations

  1. Institut de Physique de Rennes, UMR CNRS 6251, Université de Rennes 1, Bâtiment 11A, 35042 , Rennes, France

    Merline Tankeo & Patrick Richard

  2. LUNAM Université, IFSTTAR, Site de Nantes, Route de Bouaye, CS4, 44344 , Bouguenais Cedex, France

    Patrick Richard

  3. IRISA-CNRS, Université de Rennes 1, 263 Avenue du Général Leclerc, Bât 12, 35042 , Rennes Cedex, France

    Édouard Canot

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  1. Merline Tankeo
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Correspondence to Patrick Richard.

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Tankeo, M., Richard, P. & Canot, É. Analytical solution of the \(\mu (I)-\)rheology for fully developed granular flows in simple configurations. Granular Matter 15, 881–891 (2013). https://doi.org/10.1007/s10035-013-0447-3

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