Migration of buoyancy-controlled active particles in a laminar open-channel flow

Abstract

The migration of active particles in a laminar open-channel flow is significant for many ecological and biological processes associated with ambient water, for example, the harmful algae blooms in the long-distance water conveyance project. This paper presents an analytical analysis of the migration of the buoyancy-controlled active particles in a laminar open-channel flow. The analytical solution of concentration distribution has also been derived for the active particle cloud in the laminar open-channel flow under the combined action of ambient flow and self-migration of active particles, by combing Chatwin’s long-time asymptotic expansion and central concentration moments. The analytical solution shows that the active particle cloud accumulates near the surface rather than uniform distribution in the cross-section even for the long-time evolution. The analytical solutions are also been rigorously derived to reflect the holistic characteristics of the active-particle cloud, including the total quantity over each streamline, moving velocity of mass center, longitudinal dispersivity, skewness, and kurtosis have been rigorously derived for an instantaneous point source in a laminar open-channel flow. It is found that the distribution of total active particles over each streamline, centroid motion, and longitudinal dispersion eventually reach a stable status with a necessary time dependent on the relative strength of the vertical migration velocity and the total vertical diffusion. However, the self-migration of active particles results in the deviation of the moving velocity of mass center from the depth-averaged flow velocity. The longitudinal dispersivity of the active particle cloud gradually increases to the peak and then decreases to a constant value, in contrast to the monotonous increase for the passive particle cloud. It is also found that the strong vertical self-migration can result in changes of skewness and kurtosis of concentration distribution: negative skewness and positive kurtosis during the initial stage. However, both of them gradually tend to zero for the long-time evolution of the active particle cloud.