  We apply lattice Boltzmann methods to study the segregation of binary fluid
mixtures under oscillatory shear flow in two dimensions. The algorithm allows
to simulate systems whose dynamics is described by the Navier-Stokes and the
convection-diffusion equations. The interplay between several time scales
produces a rich and complex phenomenology. We investigate the effects of
different oscillation frequencies and viscosities on the morphology of the
phase separating domains. We find that at high frequencies the evolution is
almost isotropic with growth exponents 2/3 and 1/3 in the inertial (low
viscosity) and diffusive (high viscosity) regimes, respectively. When the
period of the applied shear flow becomes of the same order of the relaxation
time $T_R$ of the shear velocity profile, anisotropic effects are clearly
observable. In correspondence with non-linear patterns for the velocity
profiles, we find configurations where lamellar order close to the walls
coexists with isotropic domains in the middle of the system. For particular
values of frequency and viscosity it can also happen that the convective
effects induced by the oscillations cause an interruption or a slowing of the
segregation process, as found in some experiments. Finally, at very low
frequencies, the morphology of domains is characterized by lamellar order
everywhere in the system resembling what happens in the case with steady shear.
