Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S369 - S395|
|Published online||26 February 2021|
Explicit solutions to a free interface model for the static/flowing transition in thin granular flows
Université Paris-Est, CERMICS (ENPC), F-77455 Marne-la-Vallée Cedex 2, France
2 Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), CNRS, Univ Gustave Eiffel, UPEC, F-77454 Marne-la-Vallée, France
3 Université Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, Seismology Group, 1 Rue Jussieu, 75005 Paris, France
4 ANGE Team, INRIA, CETMEF, Lab. J.-L. Lions, Paris, France
* Corresponding author: firstname.lastname@example.org
Accepted: 22 June 2020
This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows. Our starting point is the asymptotic model derived by Bouchut et al. [Comm. Math. Sci. 14 (2016) 2101–2126] from a free surface incompressible model with viscoplastic rheology including a Drucker–Prager yield stress. This asymptotic model is based on the thin-layer approximation (the flow is thin in the direction normal to the topography compared to its down-slope extension), but the equations are not depth-averaged. In addition to the velocity, the model includes a free surface at the top of the flow and a free time-dependent static/flowing interface at the bottom. In the present work, we simplify this asymptotic model by decoupling the space coordinates, and keeping only the dependence on time and on the normal space coordinate Z. We introduce a time- and Z-dependent source term, assumed here to be given, which represents the opposite of the net force acting on the flowing material, including gravity, pressure gradient, and internal friction. We prove several properties of the resulting simplified model that has a time- and Z-dependent velocity and a time-dependent static/flowing interface as unknowns. The crucial advantage of this simplified model is that it can provide explicit solutions in the inviscid case, for different shapes of the source term. These explicit inviscid solutions exhibit a rich behaviour and qualitatively reproduce some physical features observed in granular flows.
Mathematics Subject Classification: 35R35 / 74N20
Key words: Granular flows / static/flowing transition / Drucker–Prager yield stress / thin-layer flow / interface dynamics
© The authors. Published by EDP Sciences, SMAI 2021
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