Biased swimming cells do not disperse in pipes as tracers: A population model based on microscale behaviour

Abstract

There is much current interest in modelling suspensions of algae and other microorganisms for biotechnological exploitation, and many bioreactors are of tubular design. Using generalized Taylor dispersion theory, we develop a population-level swimming-advection-diffusion model for suspensions of micro-organisms in a vertical pipe flow. In particular, a combination of gravitational and viscous torques acting on individual cells can affect their swimming behaviour, which is termed gyrotaxis. This typically leads to local cell drift and diffusion in a suspension of cells. In a flow in a pipe, small amounts of radial drift across streamlines can have a major impact on the effective axial drift and diffusion of the cells. We present a Galerkin method to calculate the local mean swimming velocity and diffusion tensor based on local shear for arbitrary flow rates. This method is validated with asymptotic results obtained in the limits of weak and strong shear. We solve the resultant swimming-advectiondiffusion equation using numerical methods for the case of imposed Poiseuille flow and investigate how the flow modifies the dispersion of active swimmers from that of passive scalars. We establish that generalized Taylor dispersion theory predicts an enhancement of gyrotactic focussing in pipe flow with increasing shear strength, in contrast to earlier models. We also show that biased swimming cells may behave very differently to passive tracers, drifting axially at up to twice the rate and diffusing much less.

Bioconvection under uniform shear: linear stability analysis

“…Finally, it should be pointed out that the translational diffusion model (2.6) with a 'constant' τ may not be a good approximation particularly when the shear rate is quite large. This issue has been addressed by several recent studies (Hill & Bees 2002;Malena & Frankel 2003;Bearon et al 2012;Croze et al 2013), which have proposed that the spatial dispersion of the cells in strong shear flows is better described by the so-called generalised Taylor dispersion theory. We note that, in practice, the difference between the present analysis and Taylor dispersion theory appears in calculating D * T .…”

“…We finally note that this instability mechanism may depend on the choice of the translational diffusivity model (2.6). However, the negative cross diffusivity induced by the shear has also been observed in other diffusivity models: for example, in the generalised Taylor dispersion theory (Bearon et al 2012). Therefore, the mechanism would probably be a robust instability mechanism for a real system.…”

“…It should be mentioned that the diffusivity tensor shown here (eq. (2.6)) may not be a good approximation except when S is quite small (Bearon et al 2012). The present results show a large discrepancy with those from generalised Taylor dispersion theory particularly when S 10 (Croze et al 2013, see also Appendix B for this estimation).…”

“…T 0 in this case (Hill & Bees 2002;Bearon et al 2012). Therefore, care needs to be taken in interpreting the results of the linear stability analysis for S > 10.…”

Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel

“…The assumption of a constant τ may not be a reasonable description particularly if the local vorticity (or the flow rate) is large (see also e.g. Bearon et al 2012;Croze et al 2013). It can be improved by the generalised Taylor dispersion theory as recently addressed (Hill & Bees 2002;Malena & Frankel 2003;Bearon et al 2011).…”