# Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity
## 4 Analytical Solutions of a Near-Wall Stagnation-Point Flow  
Analytic investigations of near-wall stagnation-point flows of viscoelastic fluids have rarely been dealt with. These investigations only treat the constitutive stress equations, however, the momentum equation is not satisfied, as can be seen, e.g., in [12; 13]. The difficulty is the accessibility of analytic solutions. In contrast to a wall-free stagnation-point flow, the similarity solution of the velocity field (12) in a near-wall stagnation-point flow must satisfy the no-slip conditions at the wall \(y=0\). This requires \(f^{\prime}(0)=0\) in addition to \(f(0)=0\). To satisfy this condition, the simplest choice is \(f(y)=y^{2}\) as have been employed in [12,13]. Here, we investigate a more general form \(f(y)=y^{\alpha}\) with \(\alpha>1\) and \(y>0\). The corresponding dimensionless velocity field takes the form:  
\[\mathbf{u}=n(axy^{\alpha-1},-y^{\alpha}), \tag{72}\]  
where \(n=\pm 1\) denotes the direction of the flow. As displayed in Figure 4, \(n=1\) corresponds to the inflow toward the stagnation point and \(n=-1\) indicates the outflow away from the stagnation point.  
For such a velocity field, an analytical solution is only attainable for less complex viscoelastic models. In this paper, we consider the most commonly employed Oldroyd-B model. As described in Section 2, the constitutive model equation for an Oldroyd-B fluid can be expressed for \(\mathbf{T}_{p}\), which is the polymeric contribution to the stress tensor. Omitting the index \(p\), the dimensionless model equation takes the form:  
\[\mathbf{T}+W\overset{\
abla}{\mathbf{T}}=2\beta\mathbf{D}, \tag{73}\]  
where \(W\) is the only Weissenberg number in this model, and \(\beta=\frac{\eta_{p}}{\eta_{p}+\eta_{s}}\in[0,1]\) is the retardation parameter.  
As has been shown, the Reynolds number does not appear in the constitutive model equations, so its value does not affect the singularities of the models. Therefore, it is convenient and common to assume the Reynolds number \(Re=1\). In this case, the corresponding momentum equation is given as  
\[\mathbf{u}\cdot\
abla\mathbf{u}=-\
abla p+(1-\beta)\
abla\cdot\mathbf{D}+ \
abla\cdot\mathbf{T}. \tag{74}\]  
Inserting the velocity field (72) into the constitutive Equation (73), we obtain a PDE system, consisting of three equations as follows:  
\[Waxy^{\alpha-1}\frac{\partial T_{11}}{\partial x}-Wy^{\alpha} \frac{\partial T_{11}}{\partial y} =(2 Way^{\alpha-1}-n)T_{11}+2W\alpha(\alpha-1)xy^{\alpha-2}T_{12}+ 2\beta ay^{\alpha-1}, \tag{75}\] \[Waxy^{\alpha-1}\frac{\partial T_{22}}{\partial x}-Wy^{\alpha} \frac{\partial T_{22}}{\partial y} =-(2 Way^{\alpha-1}+n)T_{22}-2\beta ay^{\alpha-1},\] (76) \[Waxy^{\alpha-1}\frac{\partial T_{12}}{\partial x}-Wy^{\alpha} \frac{\partial T_{12}}{\partial y} =-nT_{12}+W\alpha(\alpha-1)xy^{\alpha-2}T_{22}+\beta\alpha(\alpha -1)xy^{\alpha-2}. \tag{77}\]  
Figure 4: Sketch of velocity field in a near-wall stagnation-point flow.  
The solutions of this PDE system must also satisfy the momentum Equation (74). Applying the curl operator to (74) in order to eliminate the pressure \(p\) results in the compatibility condition:  
\[\begin{split}&\frac{\partial^{2}T_{11}}{\partial x\partial y}-\frac{ \partial^{2}T_{22}}{\partial x\partial y}-\frac{\partial^{2}T_{12}}{\partial x ^{2}}+\frac{\partial^{2}T_{12}}{\partial y^{2}}\\ =& 2\alpha(\alpha-1)xy^{2\alpha-3}-n(1-\beta)\alpha( \alpha-1)(\alpha-2)(\alpha-3)xy^{\alpha-4}.\end{split} \tag{78}\]  
Then, we will give the analytical solution of the PDE system (75)-(77) and discuss its compatibility with Equation (78).