This article studies the pattern of heat lines in free convection non-Newtonian flow from a\ud semi-infinite vertical cylinder via Bejan’s heat function concept. The viscoelastic Jeffrey fluid\ud model is employed. The time-dependent, coupled, non-linear conservation equations for\ud momentum and energy (heat) are solved computationally with the unconditionally stable finite\ud difference Crank-Nicolson method. Extensive graphical results are presented for the influence\ud of Deborah number (viscoelastic parameter) and Prandtl number (with ranges 0 - 0.8 and 0.68\ud - 7.2, respectively) on thermal and flow characteristics including time histories of overall skin friction\ud and heat transfer rate. Lower values of Deborah number indicate that the material acts\ud in a more fluid-like manner whereas the higher values of Deborah number correspond to the\ud material showing characteristics more associated with a solid. The solutions indicate that the\ud time taken for the flow-field variables to achieve the steady-state is increased with higher\ud values of Deborah number. Boundary flow visualization is presented using heat lines,\ud isotherms and streamlines. It is observed that as Deborah number increases the intensity of heat\ud lines increases and they tend to deviate from the hot cylindrical wall. Furthermore, the flow field\ud variables for the Newtonian fluid case exhibit a significantly different pattern from that\ud of Jeffrey fluid.