We address the problem of direct numerical simulation of suspensions of rigid particles in a Stokes flow. We focus on the singular effects due to short-range interactions (lubrication effects) as the particles approach each other. Taking these lubrication effects into account in numerical simulations is a difficult problem: capturing the singularity requires, for example, the use of very fine meshes in the gap between the particles.

We describe in this presentation two methods for taking account of the effects of lubrication in numerical simulations. The first is based on an asymptotic development of the solution in the gap between the particles. It provides accurate results using coarse meshes and without adding new assumptions or models. We then describe a second method, based on a ‘viscous' contact model. From a numerical point of view, it boils down to solving a convex optimisation problem at each time step. It will be shown that dry contacts, with or without friction, can be treated within the same theoretical framework. Finally, the coupling problem between these different contact models will be discussed.