In this study, in a planar cavity geometry for some time-independent non-Newtonian fluids the stability of two-dimensional flow which is generated by different wall motions was investigated. The nonlinear equations defining the flow field were solved by Gauss-Seidel iteration method numerically by using the finite-difference technique. The velocity-pressure relationship was expressed via the vorticity-stream function relationship and special attention was paid to the computations for three different values of the aspect ratio and then the results were described. The calculations were carried out for different values of the Reynolds and Weissenberg numbers. The behaviour of the vortex flow in rectangular cavities was predicted. The results were then compared with these obtained in literature for Newtonian and non-Newtonian inelastic fluids, and non-Newtonian elastic fluids results were documented first time. The agreement between the other numerical results were reasonable.