On discretization in time in simulations of particulate flows

TL;DR

This paper introduces a robust time discretization scheme for simulating fluid/particle flows, especially effective in lubrication regimes with near-zero particle-wall distances, by approximating trajectories below a threshold.

Contribution

A novel time discretization method that maintains robustness in lubrication regimes and incorporates a new derivation of lubrication forces for particle-wall interactions.

Findings

The scheme effectively handles near-contact scenarios in simulations.

Error estimates are provided both theoretically and numerically.

The derivation of lubrication forces is adaptable to other cases.

Abstract

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.