Two-dimensional unsteady natural convection of a non linear fluid represented by Criminale-Erickson-Filbey (CEF) fluid model in a square cavity is studied in the fluid for Rayleigh Benard convection case. The governing vorticity and energy transport equations are solved numerically either simple explicit and A.D.I. methods respectively. The two-dimensional convective motion is generated by bouyancy forces on the fluid fluid in a square cavity, when the vertical walls are either perfectly insulated or conducted with Biot boundary conduction condition. The contributions of the elastic and shear dependent characteristics of the liquid to the non-Newtonian behaviour are investigated on the temperature distribution and heat transfer. The effect of the Weissenberg (which is a measure of the elasticity of the fluid), Rayleigh and Biot numbers on the temperature and streamline profiles are delineated and this has been documented first time for the viscoelastic fluid.