  We study the impact of the viscous effects of the primordial plasma on the
evolution of the primordial gravitational waves (pGW) spectrum from Inflation
until today, considering a self-consistent interaction that incorporates the
back-reaction of the GW into the plasma. We use a relativistic causal
hydrodynamic framework with a positive entropy production based on a
Second-Order Theory (SOT) in which the viscous properties of the fluid are
effectively described by a new set of independent variables. We study how the
spin-2 modes typical of SOTs capture the simplest GW-fluid viscous interaction
to first order. We consider that all non-ideal properties of the primordial
plasma are due to an extra effectively massless self-interacting scalar field
whose state becomes a many-particles one after Reheating and for which an
effective fluid description is suitable. We numerically solve the evolution
equations and explicitly compute the current GW spectrum obtaining two
contributions. On the one hand we have the viscous evolution of the pGW: For
the collision-dominated regime the GW source becomes negligible while in the
collisionless limit there exists an absorption of the pGW energy due to the
damping effect produced by the free-streaming spin-2 modes of the fluid and
driven by the expansion of the Universe. The latter effect is characterized by
a relative amplitude decrease of about 1 to 10 \% with respect to the GW free
evolution spectrum. On the other hand we get the GW production due to the decay
of the initial spin-2 fluctuations of the fluid that is negligible compared
with the above-mentioned contribution.
  This SOT framework captures the same qualitative effects on the evolution of
GW coupled to matter reported in previous works in which a kinetic theory
approach has been used.
