Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces

Aline Lefebvre-Lepot (1) , Bertrand Maury (2) , Aline Lefebvre (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider here a discrete system of spheres interacting through a lubrication force. This force is dissipative, and singular near contact: it behaves like the reciprocal of interparticle distance. We propose a macroscopic constitutive equation which is built as the natural continuous counterpart of this microscopic lubrication model. This model, which is of the newtonian type, relies on an elongational viscosity, which is proportional to the reciprocal of the local fluid fraction. We then establish the convergence in a weak sense of solutions to the discrete problem towards the solution to the partial differential equation which we identified as the macroscopic constitutive equation.

Mots clés: macroscopic model; Lubrication; fluid-solid interaction; homogeneization

Domaines: Equations aux dérivées partielles [math.AP]

Aline Lefebvre-Lepot, Bertrand Maury, Aline Lefebvre. Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces. Advances in Mathematical Sciences and Applications, 2011, 21 (2), pp.535-557. ⟨hal-00257250v2⟩