# Different Formulations to Solve the Giesekus Model for Flow between Two Parallel Plates
## 2 Mathematical Formulation  
In this paper, we consider a non-Newtonian, two-dimensional, and incompressible fluid flow, which is assumed to be unsteady and without body forces. The dimensionless governing equations are given by the continuity equation:  
\[\
abla\cdot\mathbf{u}=0, \tag{1}\]  
and the momentum equation:\[\frac{\partial\mathbf{u}}{\partial t}+\
abla\cdot(\mathbf{u}\mathbf{u})=-\
abla p+ \frac{\beta}{Re}\
abla^{2}\mathbf{u}+\
abla\cdot\mathbf{T}, \tag{2}\]  
where \(\mathbf{u}\) is the velocity field, \(t\) is the time, \(p\) is the pressure, \(\beta=\frac{\eta_{s}}{\eta_{0}}\) is the coefficient that controls the solvent viscosity contribution (where \(\eta_{0}=\eta_{s}+\eta_{p}\), with \(\eta_{0}\) being the total viscosity, \(\eta_{s}\) and \(\eta_{p}\) being the solvent and polymer viscosity, respectively), \(Re=\frac{\rho UL}{\eta_{0}}\) is the Reynolds number (\(\rho\) is the fluid density) and \(\mathbf{T}\) is the non-Newtonian extra-stress tensor that must obey an appropriate constitutive equation.  
The Giesekus constitutive model [12; 13; 14; 15] is given by the following equation,  
\[\mathbf{T}+W\mathbf{i}\mathbf{T}+\frac{\alpha_{G}WiRe}{(1-\beta)}(\mathbf{T} \cdot\mathbf{T})=\frac{2(1-\beta)}{Re}\mathbf{D}, \tag{3}\]  
where \(\mathbf{D}\) is the rate of deformation tensor, \(Wi=\frac{\lambda U}{L}\) is the Weissenberg number, \(\lambda\) is the relaxation-time of the fluid, \(L\) is the channel half-width, \(U\) is the velocity scale, \(\alpha_{G}\) is the so-called mobility parameter (\(0\leq\alpha_{G}\leq 1\)) and \(\frac{\
abla}{\mathbf{T}}\) is the upper-convected derivative of \(\mathbf{T}\). This model is based on molecular concepts and it reproduces the characteristics of polymeric fluids [8].  
The system of Equations (1) and (2) with the Giesekus constitutive Equation (3) in two-dimensional Cartesian coordinates \((x,y)\) are adopted. For the formulation where the solution is analytic or semi-analytic, some simplifications are carried out in the governing equations. The assumptions of such flow for these formulations are: steady-state flow \(\left(\frac{\partial(\cdot)}{\partial t}=0\right)\), no variation of the velocity and tensor in the streamwise direction \(\left(\frac{\partial(\cdot)}{\partial x}=0\right)\), \(\mathbf{u}=\mathbf{u}(y)\), \(\mathbf{T}=\mathbf{T}(y)\right)\), normal velocity equal zero, and a constant streamwise pressure gradient \(\left(\frac{\partial p(x,y)}{\partial x}=p_{x}<0\right)\). The value of the streamwise pressure gradient is achieved considering the integral \(\int_{-1}^{1}udy=4/3\). This value is obtained for the Newtonian velocity profile with a maximum value equal to 1 in the channel center. According to these assumptions, it is considered a horizontal channel where the fluid flows in the streamwise direction \(x\); hence, the following system is taken into account:  
\[p_{x}=\frac{\beta}{Re}u^{\prime\prime}+T^{\prime}_{xy}\quad, \tag{4}\]  
\[\frac{\partial p(x,y)}{\partial y}=T^{\prime}_{yy}\quad, \tag{5}\]  
\[T_{xx}-2 WiT_{xy}u^{\prime}+\frac{\alpha_{G}ReWi}{(1-\beta)}\Big{(}T^{2}_{xx} +T^{2}_{xy}\Big{)}=0\quad, \tag{6}\]  
\[T_{xy}-WiT_{yy}u^{\prime}+\frac{\alpha_{G}ReWi}{(1-\beta)}T_{xy}\big{(}T_{xx }+T_{yy}\big{)}=\frac{(1-\beta)}{Re}u^{\prime}\quad, \tag{7}\]  
\[T_{yy}+\frac{\alpha_{G}ReWi}{(1-\beta)}\Big{(}T^{2}_{xy}+T^{2}_{yy}\Big{)}=0 \quad. \tag{8}\]  
In addition, it will be considered \(-1\leq y\leq 1\) and, thus, \(T_{xx}(0)=T_{xy}(0)=T_{yy}(0)=u^{\prime}(0)=0\) at the center (\(y=0\)).