  In this article, we present a detailed asymptotic analysis of the lattice
Boltzmann method with two different collision mechanisms of BGK-type on the
D2Q9-lattice for generalized Newtonian fluids. Unlike that based on the
Chapman-Enskog expansion leading to the compressible Navier-Stokes equations,
our analysis gives the incompressible ones directly and exposes certain
important features of the lattice Boltzmann solutions. Moreover, our analysis
provides a theoretical basis for using the iteration to compute the
rate-of-strain tensor, which makes sense specially for generalized Newtonian
fluids. As a by-product, a seemingly new structural condition on the
generalized Newtonian fluids is singled out. This condition reads as "the
magnitude of the stress tensor increases with increasing the shear rate". We
verify this condition for all the existing constitutive relations which are
known to us. In addition, it it straightforward to extend our analysis to MRT
models or to three-dimensional lattices.
