Hybrid approaches based on relativistic hydrodynamics and transport theory have been successfully applied for many years for the dynamical description of heavy ion collisions at ultrarelativistic energies. In this work a new viscous hybrid model empl
oying the hadron transport approach UrQMD for the early and late non-equilibrium stages of the reaction, and 3+1 dimensional viscous hydrodynamics for the hot and dense quark-gluon plasma stage is introduced. This approach includes the equation of motion for finite baryon number, and employs an equation of state with finite net-baryon density to allow for calculations in a large range of beam energies. The parameter space of the model is explored, and constrained by comparison with the experimental data for bulk observables from SPS and the phase I beam energy scan at RHIC. The favored parameter values depend on energy, but allow to extract the effective value of the shear viscosity coefficient over entropy density ratio $eta/s$ in the fluid phase for the whole energy region under investigation. The estimated value of $eta/s$ increases with decreasing collision energy, which may indicate that $eta/s$ of the quark-gluon plasma depends on baryochemical potential $mu_B$.
Following the experimental program at BNL RHIC, we perform a similar energy scan using 3+1D viscous hydrodynamics coupled to the UrQMD hadron cascade, and study the collision energy dependence of pion and kaon rapidity distributions and $m_T$-spectra
, as well as charged hadron elliptic flow. To this aim the equation of state for finite baryon density from a Chiral model coupled to the Polyakov loop is employed for hydrodynamic stage. 3D initial conditions from UrQMD are used to study gradual deviation from boost-invariant scaling flow. We find that the inclusion of shear viscosity in the hydrodynamic stage of evolution consistently improves the description of the data for Pb-Pb collisions at CERN SPS, as well as of the elliptic flow measurements for Au-Au collisions in the Beam Energy Scan (BES) program at BNL RHIC. The suggested value of shear viscosity is $eta/sge0.2$ for $sqrt{s_{NN}}=6.3dots39$ GeV.
