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

Abstract

Inspired by the recent realization of a 2D chiral fluid as an active monolayer droplet moving atop a 3D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2D 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 3D 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 3D 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-Delbruck 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 three-dimensional 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.

References in corpus (5)

• Spontaneous motion in hierarchically assembled active matter
• Motile dislocations knead odd crystals into whorls
• Hyperuniformity and phase enrichment in vortex and rotor assemblies
• Effective Viscosity of a Dilute Suspension of Membrane-bound Inclusions
• The role of monolayer viscosity in Langmuir film hole closure dynamics

Cited by in corpus (6)

• Lorentz Reciprocal Theorem in Fluids with Odd Viscosity
• Analytical solution for the hydrodynamic resistance of a disk in a compressible fluid layer with odd viscosity on a rigid substrate
• Chiral active systems near a substrate: Emergent damping length controlled by fluid friction
• The role of monolayer viscosity in Langmuir film hole closure dynamics
• Dumbbell dimer dynamics in three-dimensional chiral fluids
• Instability of microbial droplets growing on viscous substrates