Constraints on shear stress tensor in viscous relativistic hydrodynamics

We extend our hybrid model HydHSD by taking into account shear viscosity within the Israel-Stewart hydrodynamics. The influence of different forms of $\pi^{\mu\nu}$ constraints on observables is analyzed. We show that the form of the corresponding condition plays an important role for the sensitivity of viscous hydrodynamics to the ratio of shear viscosity to the entropy density, $\eta/s$. It is shown that the constraint used in the vHLLE model, results in most sensitivity of rapidity distributions and transverse momentum spectra to a change of the $\eta/s$ ratio; however, their applicability for large values of $\eta/s$ is doubtful. On the contrary, the strict constraints from \cite{MNR2010} are very strong but most established. We also found that $\eta/s$ as a function of the collision energy probably has an extremum at $E_{\rm lab}=10.7$ AGeV. However, we obtain that any considered condition does not allow to reproduce simultaneously pion and proton experimental data within our model.