# Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity
## 2 Model Equations
### The Oldroyd 8-Constant Model  
The most general linear viscoelastic model is the Oldroyd 8-constant model [16] described by  
\[ \begin{split}\mathbf{T}+\lambda_{1}\overset{\circ}{\mathbf{T}} +\mu_{0}&(tr\mathbf{T})\mathbf{D}-\mu_{1}(\mathbf{T}\mathbf{D}+ \mathbf{D}\mathbf{T})+\
u_{1}[tr(\mathbf{T}\mathbf{D})]\mathbf{I}\\ &=2\eta_{0}[\mathbf{D}+\lambda_{2}\overset{\circ}{\mathbf{D}}-2 \mu_{2}\mathbf{D}^{2}+\
u_{2}tr(\mathbf{D}^{2})\mathbf{I}],\end{split}  \tag{5}\]  
where \(\lambda_{1},\lambda_{2},\mu_{0},\mu_{1},\mu_{2},\
u_{1},\
u_{2}\) are material constants with the dimension of time. \(\eta_{0}\) is the total viscosity split by \(\eta_{0}=\eta_{p}+\eta_{s}\) and \(\eta_{p}\) is the polymer contribution to the viscosity. \(\overset{\circ}{\mathbf{T}}\) is the corotational objective time derivative of \(\mathbf{T}\) defined as  
\[\overset{\circ}{\mathbf{T}}:=\frac{d\mathbf{T}}{dt}-\mathbf{WT}+\mathbf{TW}, \tag{6}\]  
where \(\mathbf{W}=\frac{1}{2}(\
abla\mathbf{u}^{T}-\
abla\mathbf{u})\) is the skew-symmetric vorticity tensor. [PERSON] [1] has extended the Oldroyd 8-constant model by adding the term \(\
u_{0}(tr\mathbf{T})\mathbf{I}\) to the left-hand side of (5), but this extension will not be considered in the present investigation. Most viscoelastic models can be derived from the Oldroyd 8-constant model [15]. For example, setting \(\mu_{1}=\lambda_{1}\), \(\mu_{2}=\lambda_{2}=\frac{\eta_{c}}{\eta_{0}}\lambda_{1}\), and \(\mu_{0}=\
u_{1}=\
u_{2}=0\) in (5) gives the Oldroyd-B model:  
\[\mathbf{T}+\lambda_{1}\overset{\triangledown}{\mathbf{T}}=2\eta_{0}\left( \mathbf{D}+\lambda_{2}\overset{\triangledown}{\mathbf{D}}\right) \tag{7}\]  
where the symbol \(\triangledown\) is the upper-convected time derivative, defined by  
\[\overset{\triangledown}{\mathbf{T}}:=\frac{d\mathbf{T}}{dt}-\
abla\mathbf{u} ^{T}\cdot\mathbf{T}-\mathbf{T}\cdot\
abla\mathbf{u}. \tag{8}\]  
Usually, a retardation parameter is defined by \(\beta=\frac{\eta_{p}}{\eta_{0}}\in[0,1]\). \(\beta=0\) corresponds to a Newtonian fluid, \(\beta=1\) to a Maxwell fluid, and in between a Oldroyd fluid. If \(\frac{\eta_{s}}{\eta_{0}}=1\), hence \(\lambda_{2}=\lambda_{1}\), the Oldroyd-B fluid reduces to the Newtonian fluid with the constitutive equation \(\mathbf{T}=2\eta_{s}\mathbf{D}\). If \(\eta_{s}=0\), hence \(\lambda_{2}=0\), \(\eta_{0}=\eta_{p}\), \(\mathbf{T}=\mathbf{T}_{p}\), and the Maxwell-B model is obtained:  
\[\mathbf{T}_{p}+\lambda_{1}\overset{\triangledown}{\mathbf{T}}=2\eta_{p} \mathbf{D}. \tag{9}\]  
Combining the constitutive Equation (7) and the relation \(\mathbf{T}=2\eta_{s}\mathbf{D}+\mathbf{T}_{p}\) for an Oldroyd fluid results in a constitutive relation in terms of \(\mathbf{T}_{p}\), which is completely consistent with the expression of (9). However, in contrast to the Maxwell fluid, the Newtonian viscosity \(\eta_{s}\) in the momentum equation of an Oldroyd-type fluid is not zero. More examples of viscoelastic models derived from the Oldroyd 8-constant model can be found in the Table II of [15]. We employed the dimensionless variables, which are scaled by the characteristic length \(L_{0}\), strain rate \(a\) and viscosity \(\eta_{0}\) as following:  
\[ \begin{split}\tilde{\mathbf{x}}=\frac{\mathbf{x}}{L_{0}},\quad \tilde{\mathbf{u}}=\frac{\mathbf{u}}{aL_{0}},\quad\tilde{p}=\frac{p}{a\eta_{0}},\quad\tilde{\mathbf{T}}=\frac{\mathbf{T}}{a\eta_{0}},\quad Re=\frac{\rho aL_{ 0}^{2}}{\eta_{0}},\quad W_{1}=a\lambda_{1},\\ W_{2}=a\mu_{0},\ W_{3}=a\mu_{1},\ W_{4}=av_{1},\ W_{5}=a\lambda_{2 },\ W_{6}=a\mu_{2},\ W_{7}=av_{2},\end{split}  \tag{10}\]  
where \(W_{1},...,W_{7}\) are Weissenberg numbers and \(Re\) the Reynolds number. Omitting the tilde symbol, the constitutive Equation (5) can be rewritten in the following dimensionless form:  
\[