A kinetic approach to active rod dynamics in confined domains

Abstract: The study of active matter consisting of many self-propelled (active) swimmers in an imposed flow is important for many applications. Self-propelled swimmers may represent both living and artificial ones such as bacteria and chemically driven bi-metallic nano-particles. In this work we focus on a kinetic description of active matter represented by self-propelled rods swimming in a viscous fluid confined by a wall. It is well-known that walls may significantly affect the trajectories of active rods in contrast to unbounded or periodic containers. Among such effects are accumulation at walls and upstream motion (also known as negative rheotaxis). Our first main result is the rigorous derivation of boundary conditions for the active rods' probability distribution function in the limit of vanishing inertia. Finding such a limit is important due to (i) the fact that in many examples of active matter inertia is negligible, since swimming occurs in a low Reynolds number regime, and (ii) this limit allows us to reduce the dimension - and so computational complexity - of the kinetic description. For the resulting model, we derive the system in the limit of vanishing translational diffusion which is also typically negligible for active particles. This system allows for tracking separately active particles accumulated at walls and active particles swimming in the bulk of the fluid.