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. The accuracy of the lattice-Boltzmann (LB) method is related to the relaxation time controlling the flow viscosity. In particular, it is often recommended to avoid large fluid viscosities in order to satisfy the low-Knudsen-number assumption that is essential to recover hydrodynamic behavior at the macroscopic scale, which may in principle limit the possibility of simulating creeping flows and non-Newtonian flows involving important viscosity variations. Here it is shown, based on the continuous Boltzmann equations, that a two-relaxation-time (TRT) model can however recover the steady Navier-Stokes equations without any restriction on the fluid viscosity, provided that the Knudsen number is redefined as a function of both relaxation times. This effective Knudsen number is closely related to the previously-described parameter controlling numerical errors of the TRT model, providing a consistent theory at both the discrete and continuous levels. To simulate incompressible flows, the viscous incompressibility condition Ma2/Re≪1 also needs to be satisfied, where Ma and Re are the Mach and Reynolds numbers. This concept is extended by defining a local incompressibility factor, allowing one to locally control the accuracy of the simulation for flows involving varying viscosities. These theoretical arguments are illustrated based on numerical simulations of the two-dimensional flow past a square cylinder. In the case of a Newtonian flow, the viscosity independence is confirmed for relaxation times up to 104, and the ratio Ma2/Re=0.1 is small enough to ensure reliable incompressible simulations. The Herschel-Bulkley model is employed to introduce shear-dependent viscosities in the flow. The proposed numerical strategy allows to achieve major viscosity variations, avoiding the implementation of artificial viscosity cut-off in high-viscosity regions. Highly non-linear flows are simulated over ranges