# Active matter invasion of a viscous fluid: unstable sheets and a no-flow theorem

## TL;DR

This paper studies how active particles invade viscous fluids, revealing instabilities in particle sheets and proving a no-flow theorem that describes isotropic distributions causing no fluid movement.

## Contribution

It introduces a new instability analysis for active particle invasion and proves a surprising no-flow theorem for isotropic distributions in viscous fluids.

## Key findings

- Aligned pusher sheets are always unstable.
- Puller sheets can be stable or unstable depending on motility.
- Isotropic distributions cause no fluid flow at any time.

## Abstract

We investigate the dynamics of a dilute suspension of hydrodynamically interacting motile or immotile stress-generating swimmers or particles as they invade a surrounding viscous fluid. Colonies of aligned pusher particles are shown to elongate in the direction of particle orientation and undergo a cascade of transverse concentration instabilities, governed at small times by an equation which also describes the Saffman-Taylor instability in a Hele-Shaw cell, or Rayleigh-Taylor instability in two-dimensional flow through a porous medium. Thin sheets of aligned pusher particles are always unstable, while sheets of aligned puller particles can either be stable (immotile particles), or unstable (motile particles) with a growth rate which is non-monotonic in the force dipole strength. We also prove a surprising "no-flow theorem": a distribution initially isotropic in orientation loses isotropy immediately but in such a way that results in no fluid flow everywhere and for all time.