# 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
In order to find the pressure gradient, we adopted the existing condition of the solution in the real plane given by [PERSON] and [PERSON] [10]:  
\[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\]4  
Using this condition in the Equation (25), at the wall \(y=1\), and solving the inequation for \(p_{x}\), we obtained an existence condition for the flow in terms of the pressure gradient:  
\[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\]5  
Note that if \(Re\), \(Wi\) or \(\alpha_{G}\) is zero (UCM and Oldroyd-B), the pressure gradient does not have a limiting value, the same happens if \(\alpha_{G}=\frac{1}{2}\). From the inequation (28), it is possible to see that the hypothesis of the pressure gradient component \(p_{x}\) lower than zero is satisfied. After these calculations, it is possible to obtain all of the flow components using the equations written in terms of the component tensor \(T_{xy}\). The last step is the calculation of the velocity profile \(u(y)\) as a function of \(T_{xy}\). Equation (29) shows the relation between the component tensor \(T_{xy}\) and the derivative of the velocity profile \(u(y)\),  
\[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\]6  
For the calculation of the velocity profile, it is necessary to rewrite these equations in terms of the tensor component \(T_{xy}\). For that, an expression for \(\frac{du}{dT_{yy}}\) is needed. Using the chain rule, the Equation (29) can be rewritten as:\[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\]7  
Integrating Equation (30) the following expression can be obtained:  
\[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\]8  
Solving \(\int ydT_{xy}\), it can be obtained:  
\[+Re^{2}T_{xy}^{2}W\dot{i}^{2}\Bigg{)}+\alpha_{\rm C}(\beta-1)(3 \beta-2)\Bigg{(}\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_{xy}^{2}W\dot{i}^{2} }{(\beta-1)^{2}}}-1\Bigg{)}+\]9 \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\]0 \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\]1 \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\]2 \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\]3 \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\]4 \[+(\beta-1)^{2}\Bigg{(}1-\sqrt{1-\frac{4\alpha_{\rm C}^{2}Re^{2}T_ {xy}^{2}W\dot{i}^{2}}{(\beta-1)^{2}}}\Bigg{)}\Bigg{]}\Bigg{/}\]5  
where the logarithm term does not have a solution in the real plane for any value of \(\alpha_{G},Re,Wi,T_{xy}\), and \(\beta\). This information, about the equation for the velocity \(u\), shows that it is not possible, using this formulation, to obtain an analytical solution for velocity. Therefore, the calculation sequence to obtain the main flow components is:  
1. From Equation (28), it is possible to obtain the \(p_{x}\) max, used to start the simulation and the recursive process;
2. Equation (26) allows one to obtain the step size required to find the point distribution to increase \(T_{xy}\), to obtain the coordinate \(y\), respectively;
3. Find \(u^{{}^{\prime}}\) by the Equation (29);
4. Integrate numerically \(u^{{}^{\prime}}\) to obtain the velocity \(u\).