Abstarct: Abstract Many fluids used in the food, medical, oil, and petrochemical industries, including solutions, molten polymers, and industrial oils, are non-Newtonian. Due to the significance and applications of these fluids in industry, various methods have been employed to analyze and simulate their behavior. In fluid dynamics, flow analysis requires solving the continuity and Cauchy momentum equations. Since pressure appears as an unknown variable in the Cauchy momentum equations, iterative methods and pressure correction equations are often used for its computation. To avoid calculating pressure in the Cauchy momentum equations, the curl operator is applied to these equations and integrated with the continuity equation for incompressible fluids, resulting in the vorticity equation, which does not contain a pressure term. One of the existing methods that has been used so far to solve the time-dependent Navier-Stokes equations over various ranges of Reynolds numbers for Newtonian fluids is the random vortex method, baxsed on solving the vorticity equation. Since solving unsteady non-Newtonian fluid flows, particularly in turbulent regimes, poses a major challenge in fluid dynamics, and given the success of the random vortex method in simulating incompressible Newtonian fluid flows, this thesis focuses on analyzing the behavior of non-Newtonian fluids under unsteady conditions in both laminar and turbulent regimes, aiming to extend the mentioned method for non-Newtonian fluids. In the random vortex method, the vorticity equation consists of two terms: convection and diffusion. The non-Newtonian nature of the fluid introduces a heterogeneity term in the diffusion component. This research thesis focuses on simulating flow in non-Newtonian fluids by solving the vorticity equation using the random vortex method. In this method for non-Newtonian fluid flow, the convection term is calculated by determining the induced velocity of vortices, and the heterogeneous diffusion term is solved using the Green’s function and random variables. By solving the convection and diffusion terms, the movement of vortices is distributed within the computational domain from a Lagrangian perspective, allowing the simulation of non-Newtonian fluid flow. In order to develop this method for non-Newtonian fluids, this study utilizes Fortran programming to simulate the unsteady flow of non-Newtonian fluid with a power-law model in a channel, around a cylinder, and over a backward-facing step as benchmark problems. The results obtained, including streamlines, velocity vectors, the length of the developed region within the channel, and the length of the recirculation zone around the cylinder and backward-facing step, are compared and validated against analytical, experimental, and other numerical methods. The accuracy of the obtained solutions demonstrates the potential for extending the random vortex method to non-Newtonian fluids with a power-law model in both laminar and turbulent regimes under unsteady conditions.