Abstarct: In the present thesis, a numerical investigation of flow field past a circular cylinder and the heat transfer of viscoelastic fluid around it, have been studied by means of the open–source code OpenFOAM. The rheoFoam developed solver and the GiesekusLog structural model have been employed to solve the problem which is the novelty of the proposed approach herein. The polymeric viscosity and the stress relaxation time are defined as a function of temperature. The effects of several important dimensionless numbers such as Reynolds, Prandtl, Elastic, and Brinkman numbers, as well as the mobility factor and polymeric viscosity to total viscosity ratio on the characteristics of flow and heat transfer have been examined. The results indicated that an increase in the Reynolds number may increase the values of the Strouhal number, the amplitude of the fluctuations of lift and drag coefficients, and the average Nusselt. However with the increase of the mobility coefficient, the Strouhal number will not experience any significant changes, and the values of the fluctuation amplitudes of lift and drag coefficients and also the average Nusselt, may increase to negligible values. It is also shown that the increase in polymer viscosity causes an increase in the Strouhal numbers and the amplitude of the fluctuations of lift and drag coefficients, while the average Nusselt value will surprisingly decrease. Among other results, the reduction of the average Nusselt value with the increase of Prandtl number and the high impact of Brinkman number on the Nusselt value, could be pointed out.