  Non-Newtonian fluid flows, especially in three dimensions (3D), arise in
numerous settings of interest to physics. Prior studies using the lattice
Boltzmann method (LBM) of such flows have so far been limited to mainly to two
dimensions and used less robust collision models. In this paper, we develop a
new 3D cascaded LBM based on central moments and multiple relaxation times on a
three-dimensional, nineteen velocity (D3Q19) lattice for simulation of
generalized Newtonian (power law) fluid flows. The relaxation times of the
second order moments are varied locally based on the local shear rate and
parameterized by the consistency coefficient and the power law index of the
nonlinear constitutive relation of the power law fluid. Numerical validation
study of the 3D cascaded LBM for various benchmark problems, including the
complex 3D non-Newtonian flow in a cubic cavity at different Reynolds numbers
and power law index magnitudes encompassing shear thinning and shear thickening
fluids, are presented. Furthermore, numerical stability comparisons of the
proposed advanced LBM scheme against the LBM based on other collision models,
such as the SRT model and MRT model based on raw moments, are made. Numerical
results demonstrate the accuracy, second order grid convergence and significant
improvements in stability of the 3D cascaded LBM for simulation of 3D
non-Newtonian flows of power law fluids.
