Stirring by swimmers in confined microenvironments

We consider the tracer diffusion $D_{rr}$ that arises from the run-and-tumble motion of low Reynolds number swimmers, such as bacteria. In unbounded dilute suspensions, where the dipole swimmers move in uncorrelated runs of length $\lambda$, an exact solution showed that $D_{rr}$ is independent of $\lambda$. Here we verify this result in numerical simulations for a particular model swimmer, the spherical squirmer. We also note that in confined microenvironments, such as microscopic droplets, microfluidic devices and bacterial microzones in marine ecosystems, the size of the system can be comparable to $\lambda$. We show that this effect alone reduces the value of $D_{rr}$ in comparison to its bulk value, and predict a scaling form for its relative decrease.