Two exact solutions in the theory of biogenic mixing by microorganisms
Suspensions of active particles, such as swimming microorganisms, turn out to be efficient stirrers of the surrounding fluid. This fact may be directly relevant to the feeding and evolutionary strategies of swimming cells. Microfluidic devices exploring swimmers-induced mixing have been proposed. The possibility of a significant biogenic contribution to the ocean circulation is currently under intense debate. However, understanding fluctuations and the effective tracer diffusion in these non-equilibrium systems remains a challenge.
In this talk we focus on the fundamentals of these processes. We discuss the impediments to stirring by force-free microswimmers and give a classification of the possible stirring mechanisms. We show that enhanced mixing may arise due to entrainment of the surrounding fluid by individual swimmers moving on infinite straight trajectories. Our first exact result shows that the total amount of fluid entrained by a swimmer, also know as its Darwin drift, is finite and can be decomposed into a universal and model-dependent parts that have a clear physical meaning.
A different stirring mechanism arises for swimmers having curved trajectories. We show that the previously suggested model of swimmers moving in straight finite runs interspersed with random reorientations can be solved exactly. In particular, we calculate the effective tracer diffusion coefficient for a suspension of dipolar swimmers and show that swimmers confined to a plane give rise to a Levy flight process.
Our results provide a quantitative description of the enhanced tracer mixing in dilute suspensions of microswimmers. They agree with the results of numerical simulations and recent experiments with suspension of E. coli.