# 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
### Independent Variable Change  
The formulation proposed here is based on the change of the independent variable in the equation system. The equation system is rewritten in terms of the component tensor \(T_{xy}\). This change in the equation system allows us to find a solution for \(y\) as a function of \(T_{xy}\) analytically. From the Equations (6) and (8), two solutions for each equation can be obtained:  
\[T_{xx}=\frac{(-1+\beta)\Bigg{(}1\pm\sqrt{1-\frac{4 ReT_{xy}W\dot{\mu}_{G}(ReT_{ xy}\alpha_{G}+2(-1+\beta)u^{{}^{\prime}})}{(-1+\beta)^{2}}}\Bigg{)}}{2 ReW \dot{\alpha}_{G}}, \tag{22}\]  
and  
\[T_{yy}=\frac{(-1+\beta)\Bigg{(}1\pm\sqrt{1-\frac{4 Re^{2}T_{xy}^{2}W\dot{\mu}_{ G}^{2}}{(-1+\beta)^{2}}}\Bigg{)}}{2 ReW\dot{\alpha}_{G}}. \tag{23}\]  
In the channel center, the extra-stress tensor should be zero. Therefore, the solution adopted is the one with the signal of the minus \((-)\) before the square root. From the Equation (4), by integrating in the \(y\) direction, one can obtain an equation for \(u^{{}^{\prime}}(y)\):  
\[u^{{}^{\prime}}=\frac{Re}{\beta}(p_{x}y-T_{xy}).\]  
Substituting the last equation and the Equations (22) and (23) into Equation (7), the resulting equation is a function of the variables \(p_{x},y\) and the extra-stress tensor component \(T_{xy}\):  
\[ \begin{split}\big{(}T_{xy}-p_{x}y\big{)}\Bigg{[}-2+\frac{2}{ \beta}+\frac{1}{\alpha_{G}}\Bigg{(}1-\sqrt{1-\frac{4 Re^{2}T_{xy}^{2}W\dot{\mu} _{G}^{2}}{(-1+\beta)^{2}}}\Bigg{)}+\\ +\frac{1}{\alpha_{G}\beta}\Bigg{(}\sqrt{1-\frac{4 Re^{2}T_{xy}^{2 }W\dot{\mu}_{G}^{2}}{(-1+\beta)^{2}}}-1\Bigg{)}\Bigg{]}+T_{xy}\sqrt{1-\frac{4 Re^{2}T_{xy}^{2}W\dot{\mu}_{G}^{2}}{(-1+\beta)^{2}}}+\\ +T_{xy}\sqrt{1-\frac{4 Re^{2}T_{xy}^{2}W\dot{\mu}_{G}^{2}}{(-1+ \beta)^{2}}}+\frac{8 Re^{2}T_{xy}W\dot{\mu}^{2}\big{(}T_{xy}-p_{x}y\big{)} \alpha_{G}}{(-1+\beta)\beta}=0.\end{split}  \tag{24}\]The aim of this formulation is to solve the Equation (24) for \(y\) and then obtain a solution \(y\) as a function of \(T_{xy}\). After some algebraic manipulations and simplifications, two equations can be obtained, one is given by:  
\[y=\frac{T_{xy}}{p_{x}}.\]  
This equation satisfies the hypothesis in the channel center, but from the equation of the model, the relation between \(y\) and \(T_{xy}\) is non-linear, so this equation should not be adopted. The interesting equation is the second one, which expresses the non-linear relation between the tensorial forces and the width of the channel, as one can see in the equation:  
\[y=\Bigg{[}T_{xy}\Bigg{(}-2\alpha_{\rm C}^{2}\Bigg{(}(\beta-1) \Bigg{(}\beta\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i} ^{2}}{(\beta-1)^{2}}}-1\Bigg{)}+1\Bigg{)}+ \tag{25}\] \[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\] \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\] \[\Bigg{[}p_{x}\Bigg{(}2\alpha_{\rm C}(\beta-1)^{2}\Bigg{(}\sqrt{1 -\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}-1 \Bigg{)}+\] \[(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}-\] \[-2\alpha_{\rm C}^{2}(-\beta+ReT_{xy}Wi+1)(\beta+ReT_{xy}Wi-1) \Bigg{)}\Bigg{]}.\]  
The Equation (25) allows to obtain a distribution of \(y\) from a given \(T_{xy}\). Starting from the channel center, where \(T_{xy}\) is zero, the values for \(T_{xy}\) are increased to obtain, respectively, \(y\), until the channel boundaries, where \(y=\pm 1\). For this procedure, it is necessary to find a step size that is able to capture the extra-stress tensor component \(T_{xy}\) distribution until \(y=\pm 1\). For that, an assumption that gives us a \(6^{th}\) degree function is accomplished, in which coefficients are all the variables involved in the flow.