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

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.

arXiv comment34 pages, 2 figures