Lattice-Boltzmann simulation of creeping generalized Newtonian flows: Theory and guidelines

•General guidelines to perform highly viscous lattice-Boltzmann simulations are discussed based on theoretical analysis.•The resulting numerical strategy is validated and analyzed for the flow past a square cylinder.•When the flow viscosity is constant (Newtonian case), simulations remain accurate even when major viscosities are employed.•When the viscosity is variable (non-Newtonian case), highly non-linear flows are accurately reproduced by the method.•Local control of the viscous incompressibility parameter allows considerable viscosity variations in the simulations.