Shape matters: A Brownian microswimmer in a channel

Abstract

We consider the active Brownian particle (ABP) model for a two-dimensional microswimmer with fixed speed, whose direction of swimming changes according to a Brownian process. The probability density for the swimmer evolves according to a Fokker-Planck equation defined on the configuration space, whose structure depends on the swimmer's shape, center of rotation and domain of swimming. We enforce zero probability flux at the boundaries of configuration space. We derive a reduced equation for a swimmer in an infinite channel, in the limit of small rotational diffusivity, and find that the invariant density depends strongly on the swimmer's precise shape and center of rotation. We also give a formula for the mean reversal time: the expected time taken for a swimmer to completely reverse direction in the channel. Using homogenization theory, we find an expression for the effective longitudinal diffusivity of a swimmer in the channel, and show that it is bounded by the mean reversal time.

46 pages, 24 figures. AMSart style

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• Taylor$\unicode{x2013}$Aris dispersion of active particles in oscillatory channel flow
• Derivation of a macroscopic model for Brownian hard needles
• Noise-induced aggregation of swimmers in the Kolmogorov flow
• Optimal run-and-tumble in slit-like confinement
• Variational bounds and nonlinear stability of an active nematic suspension