by $$T_{i}^{k}=\left(  \rho +p+\Pi\right)  u_{i}u^{k}-\left(  p+\Pi\right)
\delta_{i}^{k},\label{1_a}$$ where $i,k$ takes $0,1,2,3$, $\rho$ is the mass density, $p$ the thermodynamic pressure, $\Pi $ the bulk viscous pressure and $u_{i}$ the four velocity satisfying the condition $u_{i}u^{i}=1$. The particle and entropy fluxes are defined according to $N^{i}=nu^{i}$ and $S^{i}=sN^{i}-\left(  \tau\Pi^{2}/2\xi T\right)
u^{i}$, where $n$ is the number density, $s$ the specific entropy, $T\geq0$ the temperature, $\xi$ the bulk viscosity coefficient, and $\tau\geq0$ the relaxation coefficient for transient bulk viscous effect (i.e. the relaxation time), respectively.

The evolution of the cosmological fluid is subject to the dynamical laws of particle number conservation $N_{\text{ };i}^{i}=0$ and Gibbs’ equation $Td\rho=d\left(  \rho /n\right)  +pd\left(  1/n\right)  $. In the following we shall also suppose that the energy-momentum tensor of the cosmological fluid is conserved, that is $T_{i;k}^{k}=0$.

The bulk viscous effects can be generally described by means of an effective pressure $\Pi $, formally included in the effective thermodynamic pressure $p_{eff}=p+\Pi $ [@Ma95]. Then in the comoving frame the energy momentum tensor has the components $T_{0}^{0}=\rho ,T_{1}^{1}=T_{2}^{2}=T_{3}^{3}=-p_{eff}$. For the line element given by Eq. (\[1\]), the Einstein field equations read $$\begin{aligned}
  \label{2}
\left( \frac{\dot{a}}{a}\right)^{2} &=& \frac{8\pi}{3}G \;\rho, \\
\frac{\ddot{a}}{a} &=& -\frac{4\pi}{3}G \; \left( 3p_{eff}+\rho \right),
\label{3}\end{aligned}$$ where 