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The collective dynamics of self-propelled particles

Published online by Cambridge University Press:  08 January 2008

VISHWAJEET MEHANDIA  and
PRABHU R. NOTT
VISHWAJEET MEHANDIA
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
PRABHU R. NOTT*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
*
†Author to whom correspondence should be addressed: [email protected].
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Abstract

We propose a method for the dynamic simulation of a collection of self-propelled particles in a viscous Newtonian fluid. We restrict attention to particles whose size and velocity are small enough that the fluid motion is in the creeping flow regime. We propose a simple model for a self-propelled particle, and extended the Stokesian Dynamics method to conduct dynamic simulations of a collection of such particles. In our description, each particle is treated as a sphere with an orientation vector p, whose locomotion is driven by the action of a force dipole Sp of constant magnitude S0 at a point slightly displaced from its centre. To simplify the calculation, we place the dipole at the centre of the particle, and introduce a virtual propulsion force Fp to effect propulsion. The magnitude F0 of this force is proportional to S0. The directions of Sp and Fp are determined by p. In isolation, a self-propelled particle moves at a constant velocity u0p, with the speed u0 determined by S0. When it coexists with many such particles, its hydrodynamic interaction with the other particles alters its velocity and, more importantly, its orientation. As a result, the motion of the particle is chaotic. Our simulations are not restricted to low particle concentration, as we implement the full hydrodynamic interactions between the particles, but we restrict the motion of particles to two dimensions to reduce computation. We have studied the statistical properties of a suspension of self-propelled particles for a range of the particle concentration, quantified by the area fraction φa. We find several interesting features in the microstructure and statistics. We find that particles tend to swim in clusters wherein they are in close proximity. Consequently, incorporating the finite size of the particles and the near-field hydrodynamic interactions is of the essence. There is a continuous process of breakage and formation of the clusters. We find that the distributions of particle velocity at low and high φa are qualitatively different; it is close to the normal distribution at high φa, in agreement with experimental measurements. The motion of the particles is diffusive at long time, and the self-diffusivity decreases with increasing φa. The pair correlation function shows a large anisotropic build-up near contact, which decays rapidly with separation. There is also an anisotropic orientation correlation near contact, which decays more slowly with separation. Movies are available with the online version of the paper.

JFM classification

Biological Fluid Dynamics: Micro-organism dynamics Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 239 - 264
Copyright
Copyright © Cambridge University Press 2008

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References

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Mehandia and Nott supplementary movie

Movie 1. Simulation of a suspension of self-propelled particles of concentration Φa=0.05. The arrows indicate the orientation vectors of the particles. Note the frequent occurrence of pairs and larger clusters, in which the particles are in close proximity. The tendency of neighbours to align and swim one behind the other is also evident.

Download Mehandia and Nott supplementary movie(Video)
Video 32 MB

Mehandia and Nott supplementary movie

Movie 1. Simulation of a suspension of self-propelled particles of concentration Φa=0.05. The arrows indicate the orientation vectors of the particles. Note the frequent occurrence of pairs and larger clusters, in which the particles are in close proximity. The tendency of neighbours to align and swim one behind the other is also evident.

Download Mehandia and Nott supplementary movie(Video)
Video 8.5 MB

Mehandia and Nott supplementary movie

Movie 2. Same as movie 1, but with particle concentration Φa=0.1.

Download Mehandia and Nott supplementary movie(Video)
Video 29 MB

Mehandia and Nott supplementary movie

Movie 2. Same as movie 1, but with particle concentration Φa=0.1.

Download Mehandia and Nott supplementary movie(Video)
Video 10.2 MB