Transport and mixing in Stokes flow: the effect of chaotic dynamics on the blinking stokeslet

Abstract

Mixing and transport processes associated with slow viscous flows are studied in the context of a blinking stokeslet above a plane rigid boundary. Whilst the motivation for this study comes from feeding currents due to cilia or flagella in sessile micro- organisms, other applications in physiological fluid mechanics where eddying motions occur include the enhanced mixing which may arise in ‘bolus’ flow between red blood cells, peristaltic motion and airflow in alveoli. There will also be further applications to micro-engineering flows at micron lengthscales. This study is therefore of generic interest because it analyses the opportunities for enhanced transport and mixing in a Stokes flow environment in which one or more eddies are a central feature.

The central premise in this study is that the flow induced by the beating of microscopic flagella or cilia can be modelled by point forces. The resulting system is mimicked by using an implicit map, the introduction of which greatly aids the study of the system's dynamics. In an earlier study, Blake & Otto (1996), it was noticed that the blinking stokeslet system can have a chaotic structure. Poincaré sections and local Lyapunov exponents are used here to explore the structure of the system and to give quantitative descriptions of mixing; calculations of the barriers to diffusion are also presented. Comparisons are made between the results of these approaches. We consider the trajectories of tracer particles whose density may differ from the ambient fluid; this implies that the motion of the particles is influenced by inertia. The smoothing effect of molecular diffusion can be incorporated via the direct solution of an advection–diffusion equation or equivalently the inclusion of white noise in the map. The enhancement to mixing, and the consequent ramifications for filter feeding due to chaotic advection are demonstrated.