  We study the homogenization of the Dirichlet problem for the Stokes equations
in $\mathbb{R}^3$ perforated by $m$ spherical particles. We assume the
positions and velocities of the particles to be identically and independently
distributed random variables. In the critical regime, when the radii of the
particles are of order $m^{-1}$, the homogenization limit $u$ is given as the
solution to the Brinkman equations. We provide optimal rates for the
convergence $u_m \to u$ in $L^2$, namely $m^{-\beta}$ for all $\beta < 1/2$.
Moreover, we consider the fluctuations. In the central limit scaling, we show
that these converge to a Gaussian field, locally in $L^2(\mathbb{R}^3)$, with
an explicit covariance.
  Our analysis is based on explicit approximations for the solutions $u_m$ in
terms of $u$ as well as the particle positions and their velocities. These are
shown to be accurate in $\dot H^1(\mathbb{R}^3)$ to order $m^{-\beta}$ for all
$\beta < 1$. Our results also apply to the analogous problem regarding the
homogenization of the Poisson equations.
