# Analysis Seminar

#### Large-scale rheology of random suspensions in a steady Stokes flow

**Speaker:**
Antoine Gloria, Sorbonne Université

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, February 23, 2023, 11 a.m.

**Synopsis:**

Consider a random suspension of rigid particles in a steady Stokes flow, where the particles’ positions are given by a random stationary ergodic point set. Particles can be passive and neutrally buoyant (they simply hinder the fluid flow), passive and heavier than the fluid (they hinder the flow and sink), or active (they hinder the flow and can propel themselves). In this talk, I will present recent results on these three settings, and make rigorous several predictions of the physics literature from 1905 to today. The presence of passive neutrally-buoyant particles leads to an increase of the large-scale (or effective) viscosity of the fluid, and I will introduce homogenization in this context and justify Einstein’s formula (the expansion of the effective viscosity in the dilute regime). For particles that are heavier than the fluid, the main questions are the definition of the effective sedimentation speed and the scaling of its variance wrt the size of the sedimentation tank (leading to the so-called Caflisch-Luke paradox), which are both subtle due to the long-range character of hydrodynamic interactions. The last and main part of the talk will be dedicated to the large-scale rheology of (neutrally-buoyant) active particles and to the central question of the reduction of the effective viscosity as observed for some swimming bacteria. To this aim I will introduce a simple nonlinear model (inspired by the surveys by Saintillan and Shelley) where the swimming directions of the particles depend locally on the symmetrized velocity gradient of the fluid, establish existence and homogenization, and discuss the increase/reduction of viscosity depending on the active device of the particles (pushers/pullers) in the dilute regime.

This talk is based on joint works with Mitia Duerinckx and Armand Bernou.