the index of the fluid.\
A simple algebra shows that expression (\[TmunuC\]) can be rearranged as $$\begin{aligned}
\label{Tmunu}
\begin{split}
T_{\mu\nu}&=(\rho+p')u_{\mu}u_{\nu}+p'g_{\mu\nu} \\
&-\eta\,F\,\left[\nabla_{\mu}u_{\nu}+\nabla_{\nu}u_{\mu}+u_{\mu}u^{\alpha}\nabla_{\alpha}u_{\nu}+u_{\nu}u^{\alpha}\nabla_{\alpha}u_{\mu}\right],
\end{split}\end{aligned}$$ where $p'$ is the total pressure containing the standard thermodynamical contribution and the negative component due to viscosity, i.e. $$\begin{aligned}
p'\equiv p-\lambda\nabla_{\alpha}C^{\alpha}, && \lambda\equiv\zeta-\frac{2}{3}\eta.\end{aligned}$$ The introduction of $F$ was first due by Lichnerowicz in order to describe viscous processes in relativistic dynamics, attempting to avoid superluminal signals. We can think of $F$ as a contribution which eliminates the non-causality features of the Eckart’s formulation by regularazing the velocity ($F$ will be therefore referred below as the *regulator* of the theory). The cosmological consequences of the Lichnerowicz description in isotropic cosmologies have been examined in [@Disc2]. In [@Disc1]-[@Disc4], the index of the fluid is parametrized as $$\label{F}
F=\frac{p+\rho}{\mu}\, ,$$ where $\mu$ is the rest mass-density which satifies the conservation law $$\label{restmass}
\nabla_{\alpha}(\mu u^{\alpha})=0.$$ The main advantages of the Lichnerowicz theory have been pointed out in [@Disc2]. A key-point consists in the fact that expression (\[Tmunu\]) reduces to the traditional description provided by Eckart upon setting $F=1$. Furthermore, one of the main assumptions of the well-posedness theorems requires that [@Disc3; @Disc4] $$\label{F1}
F>1.$$ Lichnerowicz was lead to introduce a 