  It is performed a detailed dynamical analysis for a bulk viscosity model in
the full Israel-Stewart formalism for a spatially flat
Friedmann-Robertson-Walker Universe. In our study we have considered the total
cosmic fluid constituted by radiation, dark matter and dark energy. The dark
matter fluid is treated as an imperfect fluid which has a bulk viscosity that
depends on its energy density in the usual form $\xi(\rho_{m})=\xi_{0}
\rho_{m}^{1/2}$, whereas the other components are assumed to behave as perfect
fluids with constant EoS parameter. We show that the thermal history of the
Universe is reproduced provided that the viscous coefficient satisfies the
condition $\xi_{0}\ll 1$, either for a zero or a suitable nonzero coupling
between dark energy and viscous dark matter. In this case, the final attractor
is a dark-energy-dominated, accelerating Universe, with effective EoS parameter
in the quintessence-like, cosmological constant-like or phantom-like regime, in
agreement with observations. As our main result, we show that in order to
obtain a viable cosmological evolution and at the same time alleviating the
cosmological coincidence problem via the mechanism of scaling solution, an
explicit interaction between dark energy and viscous dark matter seems
inevitable. This result is consistent with the well-known fact that models
where dark matter and dark energy interact with each other have been proposed
to solve the coincidence problem. Furthermore, by insisting in above, we show
that in the present context a phantom nature of this interacting dark energy
fluid is also favored.
