Efficient lattice Boltzmann simulations of self-propelled particles with singular forces

Abstract

The motion of microorganisms presents interesting and diffcult problems ranging from mechanisms of propulsion to collective effects. Experimentally, some of the complicating factors, such as death, reproduction, chemotaxis, etc., can be suppressed through genetic manipulation or environmental control. Nonequilibrium statistical mechanics has been used to study simple models, however proceeding analytically is extremely challenging. Thus simulations, where one has total control over and knowledge of the system, are a compelling method for examining models of their behaviour. In this work I present simulations of minimal, self-propelled particles, while ensuring realistic hydrodynamic behaviour using the lattice Boltzmann method (LBM), a well-studied method for simulating fluid flows that scales linearly in computational effort with the system volume. The derivation of the LBM is reviewed, including the addition of forces in a consistent, accurate manner as well as thermal fluctuations that satisfy the fluctuation-dissipation theorem. It is extended to include singular forces via a regularization of the Dirac δ-function. This is implemented and extensively tested for agreement with low Reynolds number hydrodynamics. The regularized singularities are used to develop an effcient algorithm for pointlike particles which move under the influence of an external force, such as gravity, or thermal fluctuations of the fluid. The method is compared to theoretical results and simulations using a well-studied algorithm that resolves the particle, finding good agreement in the dilute limit and significantly reduced computational requirements. Using the singular forces, we then construct a minimal model for self-propelled particles, that may also experience forces or undergo random changes of orientation (modelling the “run-and-tumble” dynamics observed in swimming bacteria such as E. coli). The collective behaviour of these model swimmers is studied in three situations: sedimentation under gravity; in a central, harmonic trap; and in a Poiseuille flow between parallel plates. For sedimentation, the behaviour is not very different from that expected of non-interacting run-and-tumble particles, except that total collapse to the container bottomwhen the weight of the particles equals the propelling force is prevented by the velocity fluctuations caused by the particles’ activity. The trapped particles, for runlengths comparable to the trap size, self-assemble into a pump-like structure, while for short run-lengths an approximately Gaussian distribution seenwithout hydrodynamic interactions, is maintained. In Poiseuille flows we find the particles orient upstream; forweak flows this results in a net upstreamcurrent. We find significant hydrodynamic effects, in the dilute limit, only when there is some mechanism that causes alignment of the particles.