# Different Formulations to Solve the Giesekus Model for Flow between Two Parallel Plates
## 1 Introduction  
The solution for the velocity and extra-stress tensor distribution in a viscoelastic fluid flow using a specific model can be obtained numerically and, sometimes, analytically. Each specific model has its own complexity and limitations, compared to the real viscoelastic fluids. The numerical solution of a laminar viscoelastic fluid flow is necessary for many flow analyses, for instance, in laminar-turbulent transition flow studies. The fluid flow components of this laminar flow can be obtained easily with some viscoelastic models, for instance, the velocity and extra-stress tensor field solutions for the Oldroyd-B model for the flow between two parallel plates. For some other models, the solution can require more profound analysis and mathematical and numerical tools to be obtained, even for simplified geometries. In general, practical problems do not allow for analytical solutions due to their complexity. For this reason, numerical methods for simulating non-Newtonian fluid flows have been part of a very active area of research. Techniques for simulating viscoelastic flows have been used to solve different constitutive models such as Oldroyd-B [1, 2], FENE-P [3, 4], FENE-CR [5, 6], PTT [4, 7, 8], and Giesekus [9, 10]. The fluids that are elastic and have a constant viscosity are known as Boger fluids and the Maxwell, Oldroyd-B, and Giesekus models are suitable to simulate these fluids type [11]. For the Giesekus constitutive model [12], [PERSON] and [PERSON] [9] studied the analytical solution for Couette and Poiseuille flows. For the Poiseuille flow, they obtained a semi-analytical solution for the mean velocity. The values were obtained by the integration (numerical, due to the complexity of the resulting equation) of the expression obtained for the derivative of the mean velocity. [PERSON] and [PERSON] [10] continued [PERSON] and [PERSON]'s [9] studies, presenting a weak and classical solution for Poiseuille flow, with and without solvent viscosity contribution. However, just like [PERSON] and [PERSON] [9], the presented solution for the average velocity is obtained implicitly. More recently, [PERSON] [13] presented an approximation for the solution of the Couette-Poiseuille flow for the Giesekus model. However, its results depend on the numerical solution of the shear stress at the stationary plate by the Newton-Raphson method. [PERSON] [8] also carried out studies for the solution of the Poiseuille flow using the Giesekus model, and showed an agreement between the results obtained by the mentioned works and the guarantees of the solution existence for the branch \(1/2<\alpha_{G}\leq 1\). [PERSON] et al. [14] presented a solution method for the Giesekus viscoelastic fluid flow based on work by [PERSON] and [PERSON] [10], where an analytical solution for the flow between two parallel plates problem was proposed. In their work, the authors considered a purely polymeric fluid flow. Among the differential constitutive models, the laminar flow solution with the Giesekus model can not be obtained directly, because the extra-stress tensor appears non-linearly through the quadratic term. On the other hand, the model is considered to better approximate the rheology of polymers [15; 16] and has the advantage of simplicity where only three parameters are involved: the temporal relaxation \(\lambda\), the mobility parameter \(\alpha_{G}\), and the viscosity of the polymer \(\eta_{p}\). Furthermore, the Giesekus model is able to predict the first and second normal-stress differences. The current study presents different formulations to obtain the solution for the Giesekus constitutive model considering the flow between two parallel plates (Poiseuille flow for Newtonian fluid flow). The first one is the formulation proposed by [PERSON] and [PERSON] [10]. The second formulation is the independent variable change, a new method proposed here, and the last method is the numerical simulation of the governing equations using high-order approximations.