  We prove a local variant of Einstein's formula for the effective viscosity of
dilute suspensions, that is $\mu^\prime=\mu (1+\frac 5 2\phi+o(\phi))$, where
$\phi$ is the volume fraction of the suspended particles. Up to now rigorous
justifications have only been obtained for dissipation functionals of the flow
field. We prove that the formula holds on the level of the Stokes equation
(with variable viscosity). We consider a regime where the number $N$ of
particles suspended in the fluid goes to infinity while their size $R$ and the
volume fraction $\phi=NR^3$ approach zero. We establish $L^\infty$ and $L^p$
estimates for the difference of the microscopic solution to the solution of the
homogenized equation. Here we assume that the particles are contained in a
bounded region and are well separated in the sense that the minimal distance is
comparable to the average one. The main tools for the proof are a dipole
approximation of the flow field of the suspension together with the so-called
method of reflections and a coarse graining of the volume density.
