  With rheology applications in mind, we present a fast solver for the
time-dependent effective viscosity of an infinite lattice containing one or
more neutrally buoyant smooth rigid particles per unit cell, in a
two-dimensional Stokes fluid with given shear rate. At each time, the mobility
problem is reformulated as a 2nd-kind boundary integral equation, then
discretized to spectral accuracy by the Nystrom method and solved iteratively,
giving typically 10 digits of accuracy. Its solution controls the evolution of
particle locations and angles in a first-order system of ordinary differential
equations. The formulation is placed on a rigorous footing by defining a
generalized periodic Green's function for the skew lattice. Numerically, the
periodized integral operator is split into a near image sum|applied in linear
time via the fast multipole method|plus a correction field solved cheaply via
proxy Stokeslets. We use barycentric quadratures to evaluate particle
interactions and velocity fields accurately, even at distances much closer than
the node spacing. Using first-order time-stepping we simulate, eg, 25 ellipses
per unit cell to 3-digit accuracy on a desktop in 1 hour per shear time. Our
examples show equilibration at long times, force chains, and two types of
blow-ups (jamming) whose power laws match lubrication theory asymptotics.
