Transient dispersion process of active particles

Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For the active dispersion process in confined flows, previous analytical studies focused on the long-time asymptotic values of dispersion characteristics. Only several numerical studies preliminarily investigated the temporal evolution. Extending recent studies of Jiang & Chen (J. Fluid Mech., vol. 877, 2019, pp. 1–34; vol. 899, 2020, A18), this work makes a semi-analytical attempt to investigate the transient process. The temporal evolution of the local distribution in the confined-section–orientation space, drift, dispersivity and skewness, is explored based on moments of distributions. We introduce the biorthogonal expansion method for solutions because the classic integral transform method for passive transport problems is not applicable due to the self-propulsion effect. Two types of boundary condition, the reflective condition and the Robin condition for wall accumulation, are imposed respectively. A detailed study on spherical and ellipsoidal swimmers dispersing in a plane Poiseuille flow demonstrates the influences of the swimming, shear flow, initial condition, wall accumulation and particle shape on the transient dispersion process. The swimming-induced diffusion makes the local distribution reach its equilibrium state faster than that of passive particles. Although the wall accumulation significantly affects the evolution of the local distribution and the drift, the time scale to reach the Taylor regime is not obviously changed. The shear-induced alignment of ellipsoidal particles can enlarge the dispersivity but impacts slightly on the drift and the skewness.

Transient dispersion regimes

Characterizing scalar dispersion is a key concern in a wide variety of applications, including both steady-state and time-dependent studies of wastewater outfalls, salinity distribution in estuaries, and the spreading of pollutants from industrial spills. As the size of a scalar plume grows with respect to the size of the containing water body, the effective dispersion varies, from the well-known ${ \sigma }_{x}^{2} \sim {t}^{3} $ behaviour for a plume enveloped in a region of linear shear, to the ${ \sigma }_{x}^{2} \sim t$ behaviour at the limit of a laterally well-mixed plume. We introduce an additional regime in which the plume extends across the full range of the available shear, but is not significantly affected by the lateral bounds of the water body. Through an analytic treatment we show that this regime exhibits a ${ \sigma }_{x}^{2} \sim {t}^{2} $ behaviour, independent of lateral mixing coefficient. Particle tracking results in an idealized, tidal channel–shoal basin demonstrate this regime as particle clouds straddle the channel–shoal interface. Quantitative analysis of spatial moments as plumes transition between regimes show good correlation between the observed parameters and parameters predicted by the analytic framework.