A modified power-law (MPL) viscosity model of non-Newtonian fluid flow has been used for the multiple-relaxation-time (MRT) lattice Boltzmann methods (LBM) and then validated with the benchmark problems using the graphics process unit (GPU) parallel computing via Compute Unified Device Architecture (CUDA) C platform. The MPL model for characterizing the non-Newtonian behavior is an empirical correlation that considers the Newtonian behavior of a non-Newtonian fluid at a very low and high shear rate. A new time unit parameter ($\lambda$) governing the flow has been identified, and this parameter is the consequence of the induced length scale introduced by the power law. The MPL model is free from any singularities due to the very low or even zero shear-rate. The proposed MPL model was first validated for the benchmark study of the lid-driven cavity and channel flows. The model was then applied for shear-thinning and shear-thickening fluid flows through a backward-facing step with relatively low Reynolds numbers, $Re$ = 100–400. In the case of shear-thinning fluids ($n=0.5$), laminar to transitional flow arises while $Re\ge 300$, and the large vortex breaks into several small vortices. The numerical results are presented regarding the velocity distribution, streamlines, and the lengths of the reattachment points.

A two-dimensional (2D) laminar flow of nanofluids confined within a square cavity having localized heat source at the bottom wall has been investigated. The governing Navier–Stokes and energy equations have been non dimensionalized using the appropriate non dimensional variables and then numerically solved using finite volume method. The flow was controlled by a range of parameters such as Rayleigh number, length of heat source and nanoparticle volume fraction. The numerical results are represented in terms of isotherms, streamlines, velocity and temperature distribution as well as the local and average rate of heat transfer. A comparative study has been conducted for two different base fluids, ethylene glycol and water as well as for two different solids and . It is found that the ethylene glycol-based nanofluid is superior to the water-based nanofluid for heat transfer enhancement.