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Active particles exhibit complex transport dynamics in flows through confined geometries such as channels or pores. In this work, we employ a generalized Taylor dispersion (GTD) theory to study the long-time dispersion behavior of active Brownian particles (ABPs) in an oscillatory Poiseuille flow within a planar channel. We quantify the time-averaged longitudinal dispersion coefficient as a function of the flow speed, flow oscillation frequency, and particle activity. In the weak-activity limit, asymptotic analysis shows that activity can either enhance or hinder the dispersion compared to the passive case. For arbitrary activity levels, we numerically solve the GTD equations and validate the results with Brownian dynamics simulations. We show that the dispersion coefficient could vary non-monotonically with both the flow speed and particle activity. Furthermore, the dispersion coefficient shows an oscillatory behavior as a function of the flow oscillation frequency, exhibiting distinct minima and maxima at different frequencies. The observed oscillatory dispersion results from the interplay between self-propulsion and oscillatory flow advection—a coupling absent in passive or steady systems. Our results show that time-dependent flows can be used to tune the dispersion of active particles in confinement.

Transport and dispersion of active particles in structured environments such as corrugated channels and porous media are important for the understanding of both natural and engineered active systems. Owing to their continuous selfpropulsion, active particles exhibit rectified transport under spatially asymmetric confinement. While good progress has been made in experiments and particle-based simulations, a theoretical understanding of the effective long-time transport dynamics in spatially periodic geometries remains less developed. In this paper, we apply generalized Taylor dispersion theory (GTDT) to analyze the long-time effective transport dynamics of active Brownian particles (ABPs) in periodic channels and fields. We show that the long-time transport behavior is governed by an effective advection-diffusion equation. The derived macrotransport equations allow us to characterize the average drift and effective dispersion coefficient. For the case of ABPs in periodic channels without external fields, we show that regardless of activity, the average drift is given by the net diffusive flux along the channel. For ABPs, their activity is the driving mechanism that sustains a density gradient, which ultimately leads to rectified motion along the channel. Our continuum theory is validated against direct Brownian dynamics simulations of the Langevin equations governing the motion of each ABP.

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared to diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work, we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection-diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly-spherical particles, weak flow, and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

In microrheology, the local rheological properties, such as the viscoelasticity of a complex fluid, are inferred from the free or forced motion of embedded colloidal probe particles. Theoretical machinery developed for forced-probe microrheology of colloidal suspensions focused on either constant-force (CF) or constant-velocity (CV) probes, while in experiments, neither the force nor the kinematics of the probe is fixed. More importantly, the constraint of CF or CV introduces a difficulty in the meaningful quantification of the fluctuations of the probe due to a thermodynamic uncertainty relation. It is known that, for a Brownian particle trapped in a harmonic potential well, the product of the standard deviations of the trap force and the particle position is dkBT in d dimensions, with kBT being the thermal energy. As a result, if the force (position) is not allowed to fluctuate, the position (force) fluctuation becomes infinite. To allow the measurement of fluctuations in theoretical studies, in this work, we consider a microrheology model in which the embedded probe is dragged along by a moving harmonic potential so that both its position and the trap force are allowed to fluctuate. Starting from the full Smoluchowski equation governing the dynamics of N hard active Brownian particles, we derive a pair Smoluchowski equation describing the dynamics of the probe as it interacts with one bath particle by neglecting hydrodynamic interactions among particles in the dilute limit. From this, we determine the mean and the variance (i.e., fluctuation) of the probe position in terms of the pair probability distribution. We then characterize the behavior of the system in the limits of both weak and strong trap. By taking appropriate limits, we show that our generalized model can be reduced to the well-studied CF or CV microrheology models.

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.