  We study the multiscale viscoelastic Doi model for suspensions of Brownian
rigid rod-like particles, as well as its generalization by Saintillan and
Shelley for self-propelled particles. We consider the regime of a small
Weissenberg number, which corresponds to a fast rotational diffusion compared
to the fluid velocity gradient, and we analyze the resulting hydrodynamic
approximation. More precisely, we show the asymptotic validity of macroscopic
nonlinear viscoelastic models, in form of so-called ordered fluid models, as an
expansion in the Weissenberg number. The result holds for zero Reynolds number
in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish
several new well-posedness and regularity results for nonlinear fluid models,
which may be of independent interest.
