Abstract
Presented in this paper is a theoretical analysis of migration of active particles in a surface flow constructed wetland, in which the longitudinal and vertical swimming behaviors are taken into account besides the hydrodynamic effects in the conventional analysis of passive particle dispersion. Theoretical solutions of mass distribution, moving velocity of the mass center, and the longitudinal dispersion coefficient are rigorously derived for a gyrotactic micro-organisms patch, based on Aris’ moment method and the method of separation of variables for linear partial differential equations. It is found that the mass distribution, motion of the mass center, and longitudinal dispersion of the gyrotactic micro-organisms patch can reach a stable status on the time scale 3/(π2+PeV2/4), where PeV is the Péclet number characterized by the total vertical diffusivity and vertical swimming velocity, and the necessary time corresponding to the dispersion of soluble contaminants can be embodied as a case of PeV=0. The total quantity of gyrotactic micro-organisms in each streamline satisfies exponential distribution for large PeV, while it approximately satisfies linear distribution for small PeV. The asymptotic velocity of mass center for gyrotactic micro-organisms in each streamline is, in general, not equal to the depth-averaged flow velocity, due to the longitudinal mean swimming caused by gyrotaxis in the shear flow. The increase of PeV and Ur (dimensionless flow velocity at free water surface relative to swimming speed) can enhance the motion of the mass center and longitudinal dispersion of the gyrotactic micro-organisms patch, while the increase of α (a dimensionless parameter to reflect shape of flow velocity profile) can weaken them.
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