Explicit solutions to a free boundary model for the static/flowing transition in granular flows

Christelle Lusso (1) , François Bouchut (2) , Alexandre Ern (1) , Anne Mangeney (3, 4)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 IPGP - Institut de Physique du Globe de Paris
4 ANGE - Numerical Analysis, Geophysics and Ecology

This work is devoted to an analytical description of the dynamics of the static/flowing transition in thin-layer dry granular flows. Our study relies on a free boundary model that has been derived from an incompressible non-Newtonian dynamics with viscoplastic rheology characterized by a Drucker–Prager yield stress, in a non-averaged thin-layer asymptotics with space variable normal to the flow. Here a source term represents the effects of internal friction, gravity and nonhydrostatic pressure. We establish the validity of some explicit solutions to the model with time and space dependent source term, in the inviscid case. These solutions qualitatively reproduce features that are specific of granular flows. We show that the zero of the source term plays a monitoring role for the static/flowing interface. Depending on the time variation of this zero, we can represent different scenarios for the dynamics of the static/flowing interface, from the starting to the arrest of the flow, including a progressive starting, a progressive stopping and a sudden starting of a part of the granular mass.

Mots clés: Granular flows; static/flowing transition; Drucker–Prager yield stress; thin-layer flow; interface dynamics

Domaines: Equations aux dérivées partielles [math.AP]; Géophysique [physics.geo-ph]

Christelle Lusso, François Bouchut, Alexandre Ern, Anne Mangeney. Explicit solutions to a free boundary model for the static/flowing transition in granular flows. 2015. ⟨hal-01180686v1⟩