Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles
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Journal of Mathematical Fluid Mechanics
Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles Pierre Degond , Sara Merino-Aceituno, Fabien Vergnet and Hui Yu Communicated by I.M. Gamba
Abstract. We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek–Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics–Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek–Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime. Mathematics Subject Classification. 35L60, 35L65, 35P10, 35Q70, 82C22, 82C70, 82C80, 92D50. Keywords. Collective dynamics, Self-organization, Hydrodynamic limit, Alignment interaction, Vicsek model, Low Reynolds number, Jeffery’s equation, Volume exclusion, Stability analysis, Finite inertia, Finite Reynolds number.
1. Introduction Self-organised motion is ubiquitous in nature. It corresponds to the formation of large-scale coherent structures that emerge from the many-interactions between individuals without leader. Well-known examples are bird flocks, fish schools or insect swarms. However, self-organisation also takes place at the microscopic level, for example in bacterial suspensions and sperm dynamics (see e.g. Refs. [7,40] and the reviews [19,29,32]). In these cases, the environment, typically a viscous fluid, plays a key role in the dynamics. In this paper we investigate self-organised motion of self-propelled particles (which we will refer to as ‘swimmers’) in a viscous fluid. The main difficulty in studying these systems comes from the complex mechanical interplay between the swimmers and the fluid. Particularly, highly non-linear interactions occur between neighbouring swimmers through the perturbations that their motions create in the surrounding fluid. While these interactions may be treated through far-field expansions in dilute suspensions [23], they require a much more complex treatment when the density of swimmers is high. Here we assume that, as a result of these swimmer–swimmer interactions, the swimmers align their direction of motion. In view of this, we adopt the Vicsek model for self-propelled particles undergoing local alignment to account for these swimmer–swimmer interactions in a phenomenological way. We then couple this model with the Stokes equation for the surrounding viscous fluid by taking into account the interactions between the swimme
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