Abstarct: In this study, the steady and unsteady flow of viscoelastic fluid around the sphere has been investigated in three-dimensional space. This numerical study has been carried out using the rheoFoam solver, with the openFoam open source code, and using the finite volume method. One of the assumptions considered in this research is that the viscoelastic fluid is an incompressible fluid at all stages. In order to avoid numerical divergence during the numerical solution of viscoelastic flow, especially in large elastic numbers, logarithmic mapping is used. Phan-Thien-Tanner (PTT) model with two parameters is used as one of the most accurate models to describe viscoelastic fluid behavior. The results of the present research include the numerical considerations of solving the viscoelastic fluid flow in two steady and unsteady states. It should be noted that viscosity changes and stress release time are more effective than other properties on fluid viscoelastic parameters. In this research, an increase in the drag coefficient, a decrease in the drag coefficient and a constant drag regime have been observed in different elasticity numbers. Also, the effect of dilution by increasing the elasticity number (El) and viscosity ratio (β) on changes in shear stress and related drag coefficients has been studied. In the study of changes in the drag coefficient with elasticity, it was concluded that, in general, elasticity has an effect on the drag coefficients: Increasing velocity and El, β and slip parameter (ξ) increase the number of drag coefficient. While the velocity decreases with the increase of the first parameter (ε) of the PTT model, viscous losses overcome this mechanism. The critical Reynolds number was obtained in the Newtonian state 〖Re〗_C=300 and it decreased in different elasticities, As an example, this number reaches 〖Re〗_C=300 in elasticity 100. An increase in the viscosity ratio causes a decrease in the critical Reynolds number, which has a strong impact on the viscosity ratio, while the parameters of the PTT model have a very small effect on the critical Reynolds number, and this number remains constant in different ranges of these parameters. Among other observations obtained in this research, we can mention the increase in the length and decrease in the separation angle of the vortexes due to the increase in the separation ratio. Finally, in the flow around the sphere, using the data obtained from the numerical solution, relations have been obtained for the drag coefficient changes in both stable and unstable regimes and the Strouhal number. The effect of different parameters such as Re, El, β, ε, ξ on the results of the research has also been investigated.