# Different Formulations to Solve the Giesekus Model for Flow between Two Parallel Plates
## 3 Different Formulations to Obtain the Fully Developed Laminar Flow with the Giesekus Model
### [PERSON] and [PERSON] Formulation  
Here, we present the steps to solve analytically the non-linear system of equations that represent the steady-state of the isothermal, incompressible flows of a Giesekus fluid with a Newtonian solvent between two parallel plates. The formulation is based on [9; 10], who presented them using different dimensionless forms among each other. [PERSON] and [PERSON] (SW) [10] solved and discussed the solutions mathematically, considering the Giesekus fluid with and without Newtonian solvent, and they commented about the Giesekus fluid without the Newtonian solvent for the axisymmetric case. In their paper, the solution is not obtained explicitly, i.e., the derivative of the solution may not be checked out directly by the reader. Hence, this section will provide a detailed explanation to achieve the analytical solution and a numerical algorithm for solving the implicit equation, which is named herein the \"semi-analytical solution\". The system of equations--Equations (4)-(8) is analogous to the system (2.1)-(2.5), solved by [PERSON] and [PERSON] [10], with some differences in the dimensional system. Rewriting Equation (8) in an equivalent form, we get:  
\[\left(T_{yy}+\frac{(1-\beta)}{2\alpha_{G}ReWi}\right)^{2}+T_{xy}^{2}=\frac{(1 -\beta)^{2}}{4\alpha_{G}^{2}Re^{2}Wi^{2}}, \tag{9}\]  
which provides two expressions to \(T_{yy}\) as a function of \(T_{xy}\),  
\[T_{yy}=\frac{-(1-\beta)\pm\sqrt{(1-\beta)^{2}-4\alpha_{G}^{2}Re^{2}Wi^{2}T_{ xy}^{2}}}{2\alpha_{G}ReWi}. \tag{10}\]  
Since the extra-stress tensor should be equal to zero along the centerline of the channel, the best choice in Equation (10) is the plus sign:  
\[T_{yy}=\frac{-(1-\beta)+\sqrt{(1-\beta)^{2}-4\alpha_{G}^{2}Re^{2}Wi^{2}T_{xy}^ {2}}}{2\alpha_{G}ReWi}. \tag{11}\]  
Adding Equations (6) and (8), one can obtain:  
\[(T_{xx}+T_{yy})+\frac{\alpha_{G}ReWi}{(1-\beta)}[(T_{xx}+T_{yy})^{2}-2T_{xx}T_ {yy}+2T_{xy}^{2}]-2 WiT_{xy}u^{{}^{\prime}}=0, \tag{12}\]  
or equivalently,  
\[ \begin{split}\frac{\alpha_{G}ReWi}{(1-\beta)}(T_{xx}+T_{yy})^{2} +(T_{xx}+T_{yy})-\frac{2\alpha_{G}ReWi}{(1-\beta)}T_{xx}T_{yy}+\\ +\frac{2\alpha_{G}ReWi}{(1-\beta)}T_{xy}^{2}=2 WiT_{xy}u^{{}^{ \prime}}.\end{split}  \tag{13}\]  
The value of \((T_{xx}+T_{yy})\) can be obtained from Equation (7):\[(T_{xx}+T_{yy})=\frac{\frac{(1-\beta)}{Re}u^{{}^{\prime}}+WiT_{yy}u^{{}^{\prime}}- T_{xy}}{\frac{\alpha_{G}ReWi}{(1-\beta)}T_{xy}}. \tag{14}\]  
Moreover, Equation (14) implies that:  
\[T_{xx}=\frac{(1-\beta)[(1-\beta)+ReWiT_{yy}]u^{{}^{\prime}}}{\alpha_{G}Re^{2} WiT_{xy}}-\frac{(1-\beta)+\alpha_{G}ReWiT_{yy}}{\alpha_{G}ReWi}. \tag{15}\]  
Equations (14) and (15) are valid for all \(y\
eq 0\). The values of the extra-stress tensor components \(T_{xx}\), \(T_{yy}\), \(T_{xy}\) and the streamwise velocity component \(u^{{}^{\prime}}\) are known at centerline \(y=0\). Substituting Equations (14) and (15) into Equation (13) and carrying on some algebraic manipulations, and with the use of Equation (8), one can obtain:  
\[u^{{}^{\prime}}=\frac{[\frac{(1-\beta)}{Re}+bWiT_{yy}]T_{xy}}{[\frac{(1-\beta) }{Re}+WiT_{yy}]^{2}}, \tag{16}\]  
where \(b=2\alpha_{G}-1\). Equation (16) shows that \(u^{{}^{\prime}}\) is a function of the extra-stress tensor component \(T_{yy}\) and \(T_{xy}\). As Equation (11) shows that the extra-stress tensor component \(T_{yy}\) is a function of the extra-stress component \(T_{xy}\), it is possible to use Equation (11) in Equation (16); after some algebraic manipulations, again, a equation for \(u^{{}^{\prime}}\) can be obtained:  
\[u^{{}^{\prime}}=\frac{2\alpha_{G}ReT_{xy}[(1-\beta)+b\sqrt{(1-\beta)^{2}-4 \alpha_{G}^{2}Re^{2}Wi^{2}T_{xy}^{2}]}}{[b(1-\beta)+\sqrt{(1-\beta)^{2}-4 \alpha_{G}^{2}Re^{2}Wi^{2}T_{xy}^{2}]^{2}}}. \tag{17}\]  
We should comment about the sign of the square root term presented in Equation (17). The solution of the [PERSON] model needs to satisfy all equations, in particular Equation (8), which was rewritten as Equation (9).