Locomotion of self-propelled particles such as motile bacteria or phoretic swimmers often takes place in the presence of applied flows and confining boundaries. Interactions of these active swimmers with the flow environment are important for the understanding of many biological processes, including infection by motile bacteria and the formation of biofilms. Recent experimental and theoretical works have shown that active particles in a Poiseuille flow exhibit interesting dynamics including accumulation at the wall and upstream swimming. Compared to the well-studied Taylor dispersion of passive Brownian particles, a theoretical understanding of the transport of active Brownian particles (ABPs) in a pressure-driven flow is relatively less developed. In this paper, employing a small wave-number expansion of the Smoluchowski equation describing the particle distribution, we explicitly derive an effective advection-diffusion equation for the cross-sectional average of the particle number density in Fourier space. We characterize the average drift (specifically upstream swimming) and effective longitudinal dispersion coefficient of active particles in relation to the flow speed, the intrinsic swimming speed of the active particles, their Brownian diffusion, and the degree of confinement. In contrast to passive Brownian particles, both the average drift and the longitudinal dispersivity of ABPs exhibit a nonmonotonic variation as a function of the flow speed. In particular, the dispersion of ABPs includes the classical shear-enhanced (Taylor) dispersion and an active contribution called the swim diffusivity. In the absence of translational diffusion, the classical Taylor dispersion is absent and we observe a giant longitudinal dispersion in the strong flow limit. Our continuum theory is corroborated by a direct Brownian dynamics simulation of the Langevin equations governing the motion of each ABP.

中文翻译：

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活性布朗粒子的上游游动和泰勒分散

自我推动的粒子（如运动细菌或泳客游泳者）的运动通常发生在施加的流量和边界限制的情况下。这些活跃的游泳者与流动环境的相互作用对于理解许多生物学过程（包括运动菌感染和生物膜的形成）很重要。最近的实验和理论工作表明，泊瓦雪流中的活性颗粒表现出有趣的动力学，包括在壁和上游游泳处的积累。与经过充分研究的被动布朗粒子的泰勒色散相比，在压力驱动流中对主动布朗粒子（ABP）传输的理论理解相对较少。在本文中，利用描述颗粒分布的Smoluchowski方程的小波数展开，我们明确得出了一个有效的对流扩散方程，用于傅里叶空间中颗粒数密度的横截面平均值。我们表征了活性颗粒的平均漂移（特别是上游游动）和有效纵向弥散系数与流速，活性颗粒的固有游动速度，布朗扩散和约束程度的关系。与被动布朗粒子相比，ABP的平均漂移和纵向分散性都显示出非单调变化，这是流速的函数。尤其是，ABP的色散包括经典的剪切增强（泰勒）色散和称为游泳扩散性的有效成分。在没有平移扩散的情况下，经典的泰勒色散不存在，我们在强流动极限下观察到巨大的纵向色散。我们对控制每个ABP运动的Langevin方程的直接布朗动力学模拟证实了我们的连续论。