Numerical simulation of rigid particles in Stokes flow: lubrication correction for general shapes of particles

Abstract

We address the problem of numerical simulation of suspensions of rigid particles in a Stokes flow. We focus on the inclusion of the singular short range interaction effects (lubrication effects) in the simulations when the particles come close one to another. As in LefebvreMerletNguyen2015, the key idea is to decompose the velocity and pressure flows in a sum of a singular and a regular part. In this article, the singular part is computed using an explicit asymptotic expansion of the solution when the distance goes to zero. This expansion is similar to the asymptotic expansion proposed in HillairetKelai2015 but is more appropriate for numerical simulations of suspensions. It can be computed for any locally convex (particles convex close to the contact point) and regular shape of particles. Using HillairetKelai2015 as an intermediate result, we prove that the remaining part is regular in the sense that it is bounded independently of the distance. As a consequence, only a small number of degrees of freedom are necessary to obtain accurate results. The method is tested in dimension 2 for clusters of two or three aligned particles with general rigid velocities. We show that, as expected, the convergence is independent of the distance.