  We present the hybrid hadron string dynamic (HydHSD) model connecting the
parton-hadron-string dynamic model (PHSD) and a hydrodynamic model taking into
account shear viscosity within the Israel-Stewart approach. The performance of
the code is tested on the pion and proton rapidity and transverse mass
distributions calculated for Au+Au and Pb+Pb collision at AGS--SPS energies.
The influence of the switch time from transport to hydro models, the viscous
parameter, and freeze-out time are discussed. Since the applicability of the
Israel-Stewart hydrodynamics assumes the perturbative character of the viscous
stress tensor, $\pi^{\mu\nu}$, which should not exceed the ideal
energy-momentum tensor, $T_{\rm id}^{\mu\nu}$, hydrodynamical codes usually
rescale the shear stress tensor if the inequality $\|\pi^{\mu\nu}\|\ll \|T_{\rm
id}^{\mu\nu}\|$ is not fulfilled in some sense. We show that the form of the
corresponding condition plays an important role in the sensitivity of
hydrodynamic calculations to the viscous parameter -- a ratio of the shear
viscosity to the entropy density, $\eta/s$. It is shown that the constraints
used in the vHLLE and MUSIC models give the same results for the observables.
With these constraints, the rapidity distributions and transverse momentum
spectra are most sensitive to a change of the $\eta/s$ ratio. As an
alternative, a strict condition is used. We performed global fits the rapidity
and transverse mass distribution of pion and protons. It was also found that
$\eta/s$ as a function of the collision energy monotonically increases from
$E_{\rm lab}=6A$GeV up to $E_{\rm lab}=40A$GeV and saturates for higher SPS
energies. We observe that it is difficult to reproduce simultaneously pion and
proton rapidity distribution within our model with the present choice of the
equation of state without a phase transition.
