A numerical study is conducted to understand the effect of surface tension on buoyancy driven convection in a vertical cylindrical annular cavity filled with a low Prandtl number fluid. The inner and outer cylinders are maintained at different uniform temperatures and the horizontal top and bottom walls are thermally insulated. The upper free surface is assumed to remain flat and non-deformable. A finite difference scheme consisting of the Alternating Direction Implicit method and the Successive Line Over Relaxation method is used to solve the vorticity stream function formulation of the problem. Detailed numerical results of heat transfer rate, temperature and velocity fields have been presented for a wide range of physical parameters of the problem. The flow pattern and temperature distribution in the annular cavity are presented by means of contour plots of streamlines and isotherms. The rate of heat transfer is estimated by evaluating the average Nusselt number. Further, the present numerical results are compared with the existing results and are found to be in good agreement.