Collective dynamics of active suspensions on curved viscous interfaces

TL;DR

This paper investigates the collective behavior of self-propelled particles on curved viscous interfaces, combining theoretical analysis and numerical simulations to reveal a finite-wavelength instability influenced by geometry and active stresses.

Contribution

It introduces a novel framework using spin-weighted spherical harmonics to analyze active suspensions on curved surfaces, including stability analysis and nonlinear simulations.

Findings

Identifies a finite-wavelength instability in particle distributions on spherical interfaces.

Shows mode selection depends on vesicle radius and Saffman-Delbrück length.

Confirms instability predictions through nonlinear numerical simulations.

Abstract

Self-propelled particles can navigate complex environments, including viscous fluid interfaces with curved geometries. In this work, we study the emergent dynamics of a suspension of self-propelled particles confined to a stationary curved viscous interface. The evolution of the particle configurations is modeled using the Fokker-Planck equation on the curved surface, formulated using Cartan's moving frame method, and coupled to the bulk and surface Stokes equations with flows driven by an interfacial nematic active stress. Specifically, for a spherical vesicle, the flow field and the distribution of the particles are analyzed theoretically and numerically within the framework of spin-weighted functions and spin-weighted spherical harmonics, which provide a natural geometric description of the probability distribution function on the sphere. A linear stability analysis about the uniform, isotropic state is performed and predicts a finite-wavelength instability, with mode selection arising from the competition between the vesicle radius and the Saffman-Delbr\"uck length. This instability and the associated mode-selection mechanism are also confirmed in nonlinear numerical simulations using a pseudo-spectral method based on spin-weighted spherical harmonics.