  We apply the causal Israel-Stewart theory of irreversible thermodynamics to
model the matter content of the universe as a dissipative fluid with bulk and
shear viscosity. Along with the full transport equations we consider their
widely used truncated version. By implementing a dynamical systems approach to
Bianchi type IV and V cosmological models with and without cosmological
constant, we determine the future asymptotic states of such universes and show
that the truncated Israel-Stewart theory leads to solutions essentially
different from the full theory. The solutions of the truncated theory may also
manifest unphysical properties. Finally, we find that in the full theory shear
viscosity can give a substantial rise to dissipative fluxes, driving the fluid
extremely far from equilibrium, where the linear Israel-Stewart theory ceases
to be valid.
