Pairwise approximations in micro-swimming

Abstract

In this work we will examine several systems of swimmers with an increasing level of realism and sophistication. In several cases exact analytical solutions are found for the probability densities of such systems. For other models we derive approximations based on the diffusion or hydrodynamic interaction strength being small. Finally a model for hydrodynamical interacting swimmers in two dimensions is derived from first principles. This is analyzed using some of the techniques from previous chapters. We also develop a simulation technique for this system similar to Ewald summation in electrodynamics.

Suspensions of hydrodynamically interacting micro-swimmers have been observed to exhibit many interesting collective effects. Some phenomena that have been observed in experiments and simulations include enhanced diffusion of tracer particles (sometimes as much as two orders of magnitude above what would be expected based on thermal effects) as well as fluid point velocity correlations on length scales much larger than the size of individual swimmers. It has been demonstrated that a key in understanding the above effect is the orientational correlations of the swimmers themselves. However, theoretical computations up to now have not been successful in deriving estimates for such two-body statistics.