# Different Formulations to Solve the Giesekus Model for Flow between Two Parallel Plates
## 5 Conclusions  
This paper presents three different formulations to obtain the solution of a two-dimensional viscoelastic fluid flow between two parallel plates, modeled by the Giesekus constitutive equation.  
The first formulation presented is based on the work by [PERSON] and [PERSON] [10]. The second formulation presented is based on the idea of the independent variable change. This change allows the flow components to be obtained analytically, except for the velocity profile, obtained using a high-order numerical integration method. The third formulation presented was called high-order simulation (HOS), i.e., the numerical simulation of the flow modeled by the Navier-Stokes equations and the constitutive equation, using high-order methods to obtain the solution.  
The [PERSON] and [PERSON] [10] formulation is efficient to obtain the flow components accurately and quickly. However, using many numerical methods for the solution makes convergence difficult for specific parameter values. Each step is verified if the value inside the square root is negative, making the solution complex and no longer real. However, this formulation proved efficient in obtaining the flow components for purely polymeric fluids (\(\beta=0\)), converging to the solution for almost all values proposed. It did not work for high values of Reynolds \(Re\) and Weissenberg \(Wi\) numbers.  
The formulation based on the independent variable change proved to be very efficient, as it solves all of the flow components analytically, but the velocity profile. The flow components are obtained quickly and accurately, for any values of dimensionless numbers of Reynolds \(Re\), Weissenberg \(Wi\), \(\alpha_{G}\), and \(\beta>0\). The only limitation of this formulation is when the fluid is composed of purely polymeric fluid flows, or near it \(\beta<0.01\).  
The HOS formulation is based on the complete solution of the Navier-Stokes equations and the considered constitutive equation. This formulation has a high computational cost since the simulations take a long time to be solved. However, the numerical methods used proved to obtain good results for the simulations carried out for all of the proposed dimensionless parameters, with a high Weissenberg number, \(\beta=0\), \(\alpha_{G}\), and Reynolds number \(Re\).  
It could be observed that the formulations presented proved to be efficient at obtaining the components of the desired flow. The results of the formulations were explored and analyzed, and their respective limitations and efficiencies were commented on. These results could be used to clarify and help researchers with which formulation is most suitable, depending on the fluid and flow parameters adopted.  
Methodology, L.J.d.S.F., M.T.d.A., A.C.B., D.O.d.A.C. and L.F.d.S.; investigation, L.J.d.S.F., M.T.d.A., A.C.B., D.O.d.A.C. and L.F.d.S.; writing--original draft preparation, L.J.d.S.F., M.T.d.A., A.C.B., D.O.d.A.C. and L.F.d.S.; writing--review and editing, L.J.d.S.F., M.T.d.A., A.C.B., D.O.d.A.C. and L.F.d.S. All authors have read and agreed to the published version of the manuscript.  
Research was carried out using the computational resources of the Center for Mathematical Sciences Applied to Industry (CeMEAI), funded by FAPESP grant 2013/07375-0.  
Not applicable.  
L.J.d.S.F., M.T.d.A. and L.F.d.S. acknowledge the Department of Applied Mathematics and Statistics--University of Sao Paulo, Sao Carlos. A.C.B. acknowledges the Department of Mathematics and Computer Science--Sao Paulo State University, Presidente Prudente and D.O.d.A.C. acknowledges the Department of Mechanical Engineering--Federal University of Rio de Janeiro, Rio de Janeiro.  
The authors declare no conflict of interest.