Perturbation solutions of relativistic viscous hydrodynamics for longitudinally expanding fireballs

The solutions of the relativistic viscous hydrodynamics for longitudinally expanding fireballs are investigated with the Navier-Stokes theory and Israel-Stewart theory.The energy and the Euler conservation equations for the viscous fluid are derived in Rindler coordinates,by assuming that the longitudinal expansion effect is small.Under the perturbation assumption,an analytical perturbation solution for the Navier-Stokes approximation and numerical solutions for the Israel-Stewart approximation are presented.The temperature evolution with both shear viscous effect and longitudinal acceleration effect in the longitudinal expanding framework are presented.The specific temperature profile shows symmetric Gaussian shape in the Rindler coordinates.Further,we compare the results from the Israel-Stewart approximation with the results from the Bjorken and the Navier-Stokes approximations,in the presence of the longitudinal acceleration expansion effect.We found that the Israel-Stewart approximation gives a good description of the early stage evolutions than the Navier-Stokes theory.