Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles

  • Pierre DegondEmail author
  • Sara Merino-Aceituno
  • Fabien Vergnet
  • Hui Yu
Open Access


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.


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 

Mathematics Subject Classification

35L60 35L65 35P10 35Q70 82C22 82C70 82C80 92D50 



PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under Grants No. EP/M006883/1 and EP/P013651/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award No. WM130048 and by the National Science Foundation (NSF) under Grant No. RNMS11-07444 (KI-Net). PD is on leave from CNRS, Institutde Mathématiques de Toulouse, France. S.M.A. was supported by the British Engineering and Physical Research Council under Grant Ref: EP/M006883/1. HY acknowledges the support by Division of Mathematical Sciences [KI-Net NSF RNMS Grant Number 1107444]; DFG Cluster of Excellence Production technologies for high-wage countries [Grant Number DFG STE2063/1-1], [Grant Number HE5386/13,14,15-1]. HY and FV gratefully acknowledges the hospitality of the Department of Mathematics, Imperial College London, where part of this research was conducted. SMA is supported by the WWTF grant (Vienna Science and Technology Fund) Vienna Research Groups for Young Investigators, VRG17-014.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria
  3. 3.Department of MathematicsUniversity of SussexFalmer, BrightonUK
  4. 4.Laboratoire de mathématiques d’Orsay (LMO), Université Paris-Sud, CNRSUniversit Paris-SaclayOrsay CedexFrance
  5. 5.Institut für Geometrie und Praktische MathematikRWTH Aachen UniversityAachenGermany
  6. 6.Mathematical Sciences CenterTsinghua UniversityHaidian DistrictChina

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