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.
Highlights
This can be done following Ref. [10] where repulsion is introduced in the Vicsek model and coarse-grained into the Self-Organised Hydrodynamic model with Repulsion (SOHR)
The fluid is described by Stokes system and the collective motion by the Vicsek model, which represents phenomenologically the interactions between neighbouring agents mediated by the fluid
The coupling is obtained by taking into account the interactions between the agents and the fluid. This involves, Jeffery’s equation that expresses the influence of a viscous fluid on spheroidal particles, on the one hand, and the force exerted by the agents on the fluid due to the dipolar force created by their self-propulsion motion, on the other hand
Summary
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. We adopt the Vicsek model for self-propelled particles undergoing local alignment to account for these swimmer–swimmer interactions in a phenomenological way. We couple this model with the Stokes equation for the surrounding viscous fluid by taking into account the interactions between the swimmers and the fluid. This kinetic equation extends the Doi model [17,18] for liquid crystals (corresponding to passive rod-like or ellipsoidal particle suspensions)
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