  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.
