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Incompressible active phases at an interface. Part 1. Formulation and axisymmetric odd flows

Published online by Cambridge University Press:  08 November 2022

Leroy L. Jia Open the ORCID record for Leroy L. Jia [Opens in a new window] ,
William T.M. Irvine Open the ORCID record for William T.M. Irvine [Opens in a new window]  and
Michael J. Shelley
Leroy L. Jia*
Affiliation:
Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA
William T.M. Irvine
Affiliation:
James Franck Institute, Enrico Fermi Institute, and Department of Physics, University of Chicago, Chicago, IL 60637, USA
Michael J. Shelley
Affiliation:
Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
*
†Email address for correspondence: ljia@flatironinstitute.org
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Abstract

Inspired by the recent realization of a two-dimensional (2-D) chiral fluid as an active monolayer droplet moving atop a 3-D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2-D linear, incompressible and isotropic fluid, having a viscous shear stress, an active chiral driving stress and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3-D Stokes phase. We pose the dynamics as the solution to a singular integral–differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3-D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current, depending on the subphase depth and the Saffman–Delbrück length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension-one domains embedded in a 3-D medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity. Hall stresses are seen to drive inward radial motion, even in the absence of edge tension.

JFM classification

Complex Fluids: Active matter Low-Reynolds-number flows: Stokesian dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A36
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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