Derivation of an effective rheology for dilute suspensions of micro-swimmers

TL;DR

This paper rigorously derives an effective rheology for dilute suspensions of self-propelled micro-swimmers, revealing modifications to viscosity and active stress effects in the fluid dynamics.

Contribution

It provides a novel derivation of an effective Stokes system incorporating active stresses and viscosity changes for dilute micro-swimmer suspensions.

Findings

Effective viscosity matches Einstein's formula.

Presence of an active stress term as a forcing in the fluid.

Analysis of how micro-swimmer properties influence fluid behavior.

Abstract

We provide a rigorous derivation of an effective rheology for dilute viscous suspensions of self-propelled particles. We derive at the limit an effective Stokes system with two additional terms. The first term corresponds to an effective viscosity that matches the famous Einstein's formula $\mu_{eff}=(1+ \frac{5}{2} \lambda)\mu$. The second additional term derived is an effective active stress $\sigma_1$ that appears as a forcing term $\lambda \, \text{div} \, \sigma_1$ in the fluid equation. We then investigate the effects of this term depending on the micro-swimmers nature and their distribution in the fluid.