2020/08/11 by Richard M. Höfer, Höfer, Richard M., Richard Schubert +1 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Particle Dynamics in Fluid Flows
paper · pdf · doi:10.48550/arxiv.2008.04813
openalex publication_date 2020/08/11 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We investigate the sedimentation of identical inertialess spherical particles\nin a Stokes fluid in the limit of many small particles. It is known that the\npresence of the particles leads to an increase of the effective viscosity of\nthe suspension. By Einstein's formula this effect is of the order of the\nparticle volume fraction \φ. The disturbance of the fluid flow responsible\nfor this increase of viscosity is very singular (like |x|-2).\nNevertheless, for well-prepared initial configurations and \φ\→ 0, we show\nthat the microscopic dynamics is approximated to order \φ2 |\log \φ| by\na macroscopic coupled transport-Stokes system with an effective viscosity\naccording to Einstein's formula. We provide quantitative estimates both for\nconvergence of the densities in the p-Wasserstein distance for all p and\nfor the fluid velocity in Lebesgue spaces in terms of the p-Wasserstein\ndistance of the initial data. Our proof is based on approximations through the\nmethod of reflections and on a generalization of a classical result on\nconvergence to mean-field limits in the infinite Wasserstein metric by Hauray.\n