Effective viscosity of semi-dilute suspensions

TL;DR

This paper reviews the rheology of rigid particle suspensions in Stokes fluid, introduces new asymptotic expansions for effective viscosity in dilute regimes, and provides an optimal proof of Einstein's viscosity formula.

Contribution

It presents new results on the asymptotic expansion of effective viscosity, including an optimal proof of Einstein's formula and higher-order terms, using diagrammatic renormalization techniques.

Findings

Optimal proof of Einstein's viscosity formula.

Higher-order asymptotic expansion of effective viscosity.

Use of diagrammatic expansions to handle hydrodynamic interactions.

Abstract

This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we turn to our new results on the asymptotic expansion of the effective viscosity in the dilute regime. This includes an optimal proof of Einstein's viscosity formula for the first-order expansion, as well as the continuation of this expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: suitable renormalizations are needed and are captured by means of diagrammatic expansions.