  We provide a mathematical analysis of the effective viscosity of suspensions
of spherical particles in a Stokes flow, at low solid volume fraction $\phi$.
Our objective is to go beyond the Einstein's approximation
$\mu_{eff}=(1+\frac{5}{2}\phi)\mu$. Assuming a lower bound on the minimal
distance between the $N$ particles, we are able to identify the $O(\phi^2)$
correction to the effective viscosity, which involves pairwise particle
interactions. Applying the methodology developped over the last years on
Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the
$O(\phi^2)$-correction, and provide explicit formula for this limit when the
particles centers can be described by either periodic or stationary ergodic
point processes.
