Based on the Nernst-Planck model for ion transport, the electroosmotic flow of a non-Newtonian fluid near a surface potential heterogeneity is studied numerically. The objectives of this study are to highlight the limitations of the linear slip-model and the nonlinear Poisson-Boltzmann model at various flow conditions as well as to develop vortical flow to promote mixing of neutral solutes within the micro-channel. A power-law fluid, both shear-thinning and shear-thickening, for the pseudoplastic behavior of the non-Newtonian fluid or viscoplastic fluid with yield stress is adopted to describe the transport of electrolyte, which is coupled with the ion transport equations governed by the Nernst-Planck equations and the Poisson equation for electric field. The viscoplastic fluid is modeled as either Casson, Bingham or Hershel-Buckley fluid. A pressure-correction based control volume approach has been adopted for the numerical computations of the governing equations. The nonlinear effects are found to be pronounced for a shear thinning liquid, whereas, the electroosmotic flow is dominated by the diffusion mechanisms for the shear thickening liquid. A maximum difference of 39% between the existing analytic solution based on the Debye-Heckel approximation and the present numerical model is found for a shear thinning power-law fluid. A vortex, which resembles a Lamb vortex, develops over the potential patch when the patch potential is of opposite sign to that of the homogeneous surface potential. Enhanced mixing of a neutral solute is also analyzed in the present analysis. The yield stress reduces the electroosmotic flow however, promotes solute mixing.