This study is concerned with the non-Newtonian fluid flow over stationary obstacles of different shapes and sedimentation of particles in non-Newtonian liquids. The direct forcing immersed boundary - Lattice Boltzmann method is used and the flows of non Newtonian fluids including the pseudo-plastic and dilatant fluids in the vicinity of circular, square and triangular disks are studied. The proposed direct numerical method employs the split-forcing algorithm for considering the presence of Lagrangian boundary points on a fixed Eulerian fluid domain. Unlike the previously used methods, the effects of added mass due to particle acceleration are included in the analysis. The validation tests for both stationary and moving cylinders with cross-sections of different shapes immersed in Newtonian fluids are presented. The effects of the number of forcing points on a generalized Reynolds number are investigated by utilizing two to six points in the diffuse interfaces. The results show that the increase of shear-thinning behavior and the number of sides of a cross-section's shape slightly decrease the accuracy of solutions. The number of forcing points, however, has no significant effect on the hydrodynamics parameters. The high accuracy and the simplicity of implementation of the presented method may make it attractive for solving problems involving non-Newtonian fluid flow near stationary and/or moving boundaries with different shapes with application to drug delivery in biological systems, fluidized bed reactors, and polymer processing operations. (C) 2015 Elsevier Ltd. All rights reserved.