On the effective viscosity of suspensions

Small particles suspended in a fluid are ubiquitous in nature and technology. It is well-known that the particles change the effective viscosity of the fluid. The problem has been addressed by Einstein in his PhD dissertation in 1906. He obtained a quantitative result known as Einstein's law for the effective viscosity for spherical particles to first order in the volume fraction $\varphi$ of the particles. Rigorous mathematical results have only been obtained in the last years. I will review these results and present recent improvements where we were able to relax the assumptions on the particle configurations considerably.

This covers physically relevant random distributions of particles. A big challenge consists in the analysis of a dynamic version of Einstein's law. Indeed, the interaction between the particles accounting for Einstein's law is very singular ($1/|x{|}^3$ in three dimensions), and we presently do not know how to obtain the corresponding mean field-result for fixed volume fraction $\varphi$ as we lose control over the interparticle distances. Nevertheless, I will present a perturbative result in the case $\varphi \to 0$, that incorporates Einstein's law.

This talk is based on joint works with David Gérard-Varet and Richard Schubert.