In industry, flow instabilities are important in journal bearings at high rotational speeds. For inviscid flows between two rotating coaxial cylinders, an instability criterion was introduced by Taylor-Couette. However, viscosity may play as stabilizer or destabilize in such flows. In this work, stability theories were reviewed for two coaxial rotating cylinders. Then, flow instability patterns, velocity, pressure, and vorticity fields were simulated using Fluent. In this method, the fluid flow was assumed laminar and by gradually increasing the relative rotational speeds between two coaxial cylinders, the critical Taylor number was calculated and compared with Taylor analytical solution. By increasing Taylor number, it is observed that vortices become stronger and instabilities appear in the form of cellular square patterns. The characteristics of these cellular square patterns were examined.