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

Christopher J. Miles, Arthur A. Evans, Michael J. Shelley, and Saverio E. Spagnolie

Phys. Rev. Lett. 122, 098002 – Published 4 March 2019

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 that also describes the Saffman-Taylor instability in a Hele-Shaw cell, or the Rayleigh-Taylor instability in a 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 that is nonmonotonic 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.

Received 14 March 2018
Revised 29 November 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.098002

Physics Subject Headings (PhySH)

Biological self-organization Cell locomotion Collective behavior Flow instability Fluid-particle interactions

Bacteria Collective dynamics Living matter & active matter Microswimmers Self-propelled particles Suspensions

Coarse graining Fokker–Planck equation Phase space dynamics Theories of collective dynamics & active matter

Fluid DynamicsPhysics of Living SystemsPolymers & Soft MatterNonlinear Dynamics

Authors & Affiliations

Christopher J. Miles^*

Department of Physics, University of Michigan, 450 Church St., Ann Arbor, Michigan 48109, USA

Arthur A. Evans

Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, Wisconsin 53706, USA

Michael J. Shelley

Flatiron Institute, Simons Foundation, New York, New York, USA; and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA

Saverio E. Spagnolie^†