Gyrotactic trapping of micro-swimmers in simple shear flows: a study directly from the fundamental Smoluchowski equation

Abstract

Thin phytoplankton layers with vertically compressed structures found in stratified lakes and coastal water bodies have been a hot-spot in marine science and fluid mechanics. Although extensive efforts have been made to explore gyrotactic trapping as a possible mechanism, obvious inconsistencies remain there between the generalised Taylor dispersion method and individual-based model. In this work, a study directly from the fundamental Smoluchowski equation is carried out on the gyrotactic trapping mechanism in representative simple shear flows. With no approximation made to the fundamental equation and applying a biorthogonal expansion to the local moments of the probability density function, the evolution and steady state of the transport process are in good agreement with results of individual-based model, thus removing the gap between the continuum approach and individual-based model. The influences of strongly varying shear rate and boundary effect are reasonably scrutinised. The average swimming orientation with fixed local shear is highlighted as time-dependent rather than steady as previously assumed. The variations of characteristic time of the transient thin layer as functions of the flow intensity, gyrotaxis intensity and swimming ability are characterised. Sedimentation of the micro-swimmers is considered in some cases to reflect that micro-organisms are possibly negatively buoyant, with corresponding boundary conditions devised. A steady thin layer can be realised in the solution by adding a small settling speed to counteract the gravitactic focusing at the upper surface.

Theoretical analysis for bedload particle deposition and hop statistics

“…2018; Guan et al. 2022; Wang, Jiang & Chen 2022 a ). In order not to disrupt the narrative flow, the solution procedure is summarized in Appendix A of § A.1.…”

“…The solutions of spatial moments have been systematically studied by Barton (1983Barton ( , 1984 using the Sturm-Liouville theory, with general expressions for the first four moments provided for direct use. Barton's results have been widely applied to studies on various dispersion phenomena (Zeng et al 2011;Li et al 2018;Guan et al 2022;Wang, Jiang & Chen 2022a). In order not to disrupt the narrative flow, the solution procedure is summarized in Appendix A of § A.1.…”

Environmental Transport of Gyrotactic Microorganisms in an Open‐Channel Flow

“…f n and K n can then be solved by the Galerkin method, which is similar to that used in previous papers (W. Wang et al, 2022b). Below is a summary of the solution procedure, and more details can be found in Section 3.2 of the work by W. Jiang and Chen (2021).…”

“…Based on the above assumptions and conditions, the governing equation for the transport can be adopted as (Wang et al., 2022b) $$$\frac{\partial P}{\partial t}=-\left[{Pe}_{s}\,\mathrm{cos}\,\theta +{Pe}_{f}U(z)\right]\frac{\partial P}{\partial x}-{Pe}_{s}\,\mathrm{sin}\,\theta \frac{\partial P}{\partial z}-\frac{\partial (\dot{\theta }P)}{\partial \theta }+{D}_{t}\frac{{\partial }^{2}P}{\partial {x}^{2}}+{D}_{t}\frac{{\partial }^{2}P}{\partial {z}^{2}}+\frac{{\partial }^{2}P}{\partial {\theta }^{2}},$$$ where t is time and P ( x , z , θ , t ) is the probability density function (pdf) of an individual of the microorganism found in the position and orientation space { x , z , θ , t }. In Equation we have already used the following dimensionless variables and parameters …”

Pre-asymptotic dispersion of active particles through a vertical pipe: the origin of hydrodynamic focusing

“…(2020) with inclusion of biased motility. From the fundamental Smoluchowski equation, an additional settling speed of gyrotactic swimmers is found efficient to maintain the presence of thin layers (Wang, Jiang & Chen 2022). However, much uncertainty still exists about the temporal evolution of hydrodynamic accumulation.…”