Abstarct: The study of fluid flow between the space of two cylinders (annulus space), considered as a basic and applied problems in fluid mechanics. Also, this type of flow has a particular importance in the oil and gas industry. In the process of drilling oil and gas wells, drilling fluid exiting the drilling bit and encompasses the cuttings, then going through the annulus and moving out of the well head. Studies and experiments which were carried out on most of the oil-baxsed and water-baxsed drilling fluids determine the viscoelastic behavior of drilling fluids in the extensive ranges of shear rates. Adding polymeric materials to drilling fluids cause them improving in their quality and results in improving the drilling efficiency. In this study, the numerical investigations of helical fluid flow between two vertical concentric cylinders with inner cylinder rotation will be done. Here for the first time the study of viscoelastic fluid behavior in this geometry is done by using the Giesekus constitutive equation model. The numerical and analytical solutions former usually ignored the rotation of inner cylinder and the viscoelastic fluid was skipped. First, with regard to the relationship between modulus of absorption and dissipation of energy and angular frequency of oscillation, the experimental curve fitting is discussed. The minimum of the difference between the experimental data and the results can range from relationships constants and viscosity of the fluid when it is achieved. To optimize this error function and find the coefficients or the constant problem of evolutionary genetic algorithm is used. To find the coefficients of the mobility factor spectrum used in the Giesekus equation, the Giesekus equation which the stress components are tensor should be opened first. The amount of each component of the tension, the sum of the individual components of the stress comes in all modes. Ode45 method is used for solving differential equations in MATLAB to obtain optimal dynamic coefficients of PSO evolutionary algorithm. In order to model the flow of open source software, Open Foam, a toolbox of Computational Fluid Dynamics (CFD) is used. The application of the finite volume method (FVM) uses for solving partial differential equations. baxsed on numerical simulation, numerical solution of a number of independent elements (network) is investigated. In this study the effect of Reynolds number, Taylor number, Wisenberg number and mobility factor on the drilling fluid in annulus space is studied. Dimensionless numbers to see the effect on the axial pressure drop of the fluid; can be a way to reduce the energy required for pumping drilling fluid in the annulus space to be achieved.