  Previously it was shown that there exists a class of viscous cosmological
models which violate the dominant energy condition for a limited amount of time
after which they are smoothly connected to the ordinary radiation era (which
preserves the dominant energy conditions). This violation of the dominant
energy condition at an early cosmological epoch may influence the slopes of
energy spectra of relic gravitons that might be of experimental relevance.
However, the bulk viscosity coefficient of these cosmologies became negative
during the ordinary radiation era, and then the entropy of the sources driving
the geometry decreases with time.
  We show that in the presence of viscous sources with a linear barotropic
equation of state $p=\gamma \rho$ we get viscous cosmological models with
positive bulk viscous stress during all their evolution, and hence the matter
entropy increases with the expansion time. In other words, in the framework of
viscous cosmologies, there exist isotropic models compatible with the standard
second law of thermodynamics which also may influence the slopes of energy
spectra of relic gravitons.
