# American Institute of Mathematical Sciences

June  2011, 15(4): 935-956. doi: 10.3934/dcdsb.2011.15.935

## On discretization in time in simulations of particulate flows

 1 Ceremade, UMR CNRS 7534, Université Dauphine, Place du Maréchal De Lattre De Tassigny, F-75016 Paris, France 2 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse et CNRS, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France 3 Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, F-67084 Strasbourg Cedex, France

Received  January 2010 Revised  July 2010 Published  March 2011

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases.
Citation: Matthieu Hillairet, Alexei Lozinski, Marcela Szopos. On discretization in time in simulations of particulate flows. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 935-956. doi: 10.3934/dcdsb.2011.15.935
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