  This work is devoted to the large-scale rheology of suspensions of
non-Brownian inertialess rigid particles, possibly self-propelling, suspended
in Stokes flow. Starting from a hydrodynamic model, we derive a semi-dilute
mean-field description in form of a Doi-type model, which is given by a
'macroscopic' effective Stokes equation coupled with a 'microscopic' Vlasov
equation for the statistical distribution of particle positions and
orientations. This accounts for some non-Newtonian effects since the viscosity
in the effective Stokes equation depends on the local distribution of particle
orientations via Einstein's formula. The main difficulty is the detailed
analysis of multibody hydrodynamic interactions between the particles, which we
perform by means of a cluster expansion combined with a multipole expansion in
a suitable dilute regime.
