# Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity
## 4 Analytical Solutions of a Near-Wall Stagnation-Point Flow
### Compatibility Condition  
Substituting the solutions of the stress components (80)-(82) into the compatibility condition (78) yields the following restriction on the functions \(F_{1}\), \(F_{2}\), and \(F_{3}\):  
\[-\alpha F_{1}^{\prime}\cdot y^{\alpha-1}-\big{(}F_{2}^{\prime\prime} +\frac{1}{nW}F_{1}^{\prime}\big{)}\cdot y^{2\alpha}+\alpha\big{(}\alpha F_{1}^{ \prime}\psi+2(\alpha-1)F_{1}\big{)}\psi y^{\alpha-3} \tag{83}\] \[+\alpha\big{(}-\alpha F_{2}^{\prime\prime}\psi^{2}-(3\alpha-1)F_ {2}^{\prime}\psi+2F_{2}-\frac{\alpha}{nW}F_{1}^{\prime}\psi^{2}-\frac{\alpha-2 }{nW}F_{1}\psi\big{)}\cdot y^{-2}\] \[+\alpha\big{(}F_{3}^{\prime\prime}\psi-F_{3}^{\prime}-\frac{3}{nW }F_{2}-\frac{1}{W^{2}}F_{1}\psi\big{)}\cdot y^{-\alpha-1}+\frac{1}{nW}\big{(} \frac{1}{nW}F_{2}+F_{3}^{\prime}\big{)}\cdot y^{-2\alpha}\] \[+\psi\cdot G(y)=0,\]  
where \(\psi=xy^{\alpha}\) and:  
\[G(y)= 2\beta\big{[}\frac{1}{nW^{2}}\alpha(\alpha-1)y^{-2\alpha}+\frac{ 1}{W}\alpha^{2}(\alpha+1)y^{-\alpha-1}+n\alpha^{2}(\alpha+1)^{2}y^{-2} \tag{84}\] \[+4W\alpha^{2}(\alpha+1)(\alpha-1)y^{\alpha-3}\big{]}\cdot\int y^ {-3}e^{\frac{y^{1-\alpha}}{nW(\alpha-1)}}dy\] \[+\big{[}n\alpha\big{(}\alpha(\alpha^{2}-2\alpha+5)-2(1-\beta)(2 \alpha^{2}-3\alpha+3)\big{)}y^{-4}-2\alpha(\alpha-1)y^{\alpha-3}\] \[+4\beta W\alpha^{2}(\alpha+1)(\alpha-1)y^{\alpha-5}+\frac{2}{W} (1-\beta)\alpha(\alpha-1)y^{-\alpha-3}\big{]}\cdot e^{\frac{y^{1-\alpha}}{nW( \alpha-1)}}.\]  
The condition (83) could only be satisfied for any value of \(y\) along the characteristic line \(\psi=0\), which denotes the symmetric line \(x=0\). Along this line, the compatibility Equation (83) will be greatly simplified to:  
\[-\alpha F_{1}^{\prime}\cdot y^{\alpha-1}-\big{(}F_{2}^{\prime \prime}+\frac{1}{nW}F_{1}^{\prime}\big{)}\cdot y^{2\alpha}+2\alpha F_{2}\cdot y ^{-2} \tag{85}\] \[-\alpha\big{(}F_{3}^{\prime}+\frac{3}{nW}F_{2}\big{)}\cdot y^{- \alpha-1}+\frac{1}{nW}\big{(}\frac{1}{nW}F_{2}+F_{3}^{\prime}\big{)}\cdot y^{ -2\alpha}=0,\]  
which leads to:  
\[F_{1}(\psi)=F_{1}(\psi=0)=C_{1},\qquad F_{2}=0,\qquad F_{3}(\psi)=F(\psi=0)=C_ {3}, \tag{86}\]  
where \(C_{1}\) and \(C_{3}\) are arbitrary constants. The corresponding stress components along the \(y\)-axis are given by  
\[T_{11}= C_{3}y^{-2\alpha}e^{-\frac{y^{1-\alpha}}{nW(\alpha-1)}}-\frac{2 \beta}{W}xy^{-2\alpha}e^{-\frac{y^{1-\alpha}}{nW(\alpha-1)}}\int y^{2\alpha-1 }e^{\frac{y^{1-\alpha}}{nW(\alpha-1)}}dy, \tag{87}\] \[T_{22}= C_{1}y^{2\alpha}e^{-\frac{y^{1-\alpha}}{nW(\alpha-1)}}+\frac{2 \beta}{W}xy^{2\alpha}e^{-\frac{y^{1-\alpha}}{nW(\alpha-1)}}\int y^{-2\alpha-1 }e^{\frac{y^{1-\alpha}}{nW(\alpha-1)}}dy,\] \[T_{12}= 0.\]  
Furthermore, applying the [PERSON] rule, we can analyze the behavior of the stress components near the stagnation point, respectively. Since \(\alpha>1\) and \(y>0\), we obtain:  
\[\lim_{y\to 0}y^{\pm 2\alpha}e^{-\frac{y^{1-\alpha}}{nW(\alpha-1)}}= \left\{\begin{aligned} 0&\qquad\text{for $n=1$,}\\ \infty&\qquad\text{for $n=-1$,}\end{aligned}\right. \tag{88}\]  
\[\lim_{y\to 0}y^{\pm 2\alpha}e^{-\frac{y^{1-\alpha}}{nW(\alpha-1)}}\int y^{\mp 2 \alpha-1}e^{\frac{y^{1-\alpha}}{nW(\alpha-1)}}dy=0. \tag{89}\]  
To avoid the singularity for the case of outflow, we can choose \(C_{1}=C_{3}=0\). Hence, for both the cases of inflow and outflow, regular solutions exist near the stagnation point.  
As mentioned above, the solutions under the velocity assumption (72) can satisfy the conservation of the momentum equation only on the symmetric axis. In order to obtain a solution that satisfies the compatibility equation, a more suitable function \(f(y)\) to describe the velocity field has to be proposed. This is an interesting topic for future study. Actually, in many previous investigations for both wall-free and near-wall stagnation-point flows, the singularity of the stress field was analyzed by forcing the solution to be independent of \(x\), see, e.g., [8; 12; 13].