Efficient lattice Boltzmann simulations of self-propelled particles with singular forces
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Date
2010Author
Nash, Rupert William
Metadata
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.