# Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity
## 3 Analytical Solutions of a Wall-Free Stagnation-Point Flow and Their Singularities
### Case \(k_{2}=0\)
#### 3.1.1 Case \((k_{1}-1)(k_{1}+1)\
eq 0\)  
In this case, \(k_{1}\
eq\pm 1\), we can obtain the following solutions by solving the Equations (19)-(21):  
\[T_{11} =F_{1}(\psi)\cdot y^{\frac{1-k_{1}}{W_{1}}}+\frac{k_{3}-2}{k_{1}- 1}, \tag{22}\] \[T_{22} =F_{2}(\psi)\cdot y^{\frac{1+k_{1}}{W_{1}}}-\frac{k_{3}+2}{k_{1}+ 1},\] (23) \[T_{12} =F_{3}(\psi)\cdot y^{\frac{1}{W_{1}}}. \tag{24}\]  
\(F_{1}(\psi)\), \(F_{2}(\psi)\) and \(F_{3}(\psi)\) are arbitrary functions of \(\psi=xy\), and \(\psi=const.\) represents the characteristic lines of the solutions. This result is consistent with that obtained by [PERSON] [14] for the stagnation-point flow of an upper convected Maxwell fluid, in which \(k_{1}=2W_{1}\) and \(k_{3}=0\). Further satisfying the compatibility condition (18) yields the following restriction on functions \(F_{1}\), \(F_{2}\), and \(F_{3}\):  
\[ \begin{split}&\left[\psi F_{1}^{\prime\prime}+\left(\frac{1-k_{1}}{W _{1}}+1\right)F_{1}^{\prime}\right]y^{-\frac{k_{1}}{W_{1}}}-\left[\psi F_{2}^{ \prime\prime}+\left(\frac{1+k_{1}}{W_{1}}+1\right)F_{2}^{\prime}\right]y^{k_{ 1}}\\ &+\left[\psi^{2}F_{3}^{\prime\prime}+\frac{2}{W_{1}}\psi F_{3}^{ \prime}+\frac{1}{W_{1}}\left(\frac{1}{W_{1}}-1\right)F_{3}\right]y^{-2}-F_{3}^ {\prime\prime}y^{2}=0.\end{split}  \tag{25}\]  
the only way to satisfy this condition for any value of \(y\) is that all coefficients of \(y^{k}\) vanish. This induces three different cases depending on the value of \(\frac{k_{1}}{W_{1}}\). The case of \(\frac{k_{1}}{W_{1}}=\pm 2\) corresponds to the Oldroyd and Maxwell fluids. Their solutions for Weissenberg number \(W_{1}\
eq\frac{1}{2}\) are covered in this investigation. 1. If \(\frac{k_{1}}{W_{1}}=2\), only two restriction conditions for \(F_{1}\), \(F_{2}\), and \(F_{3}\) can be obtained from (25), meaning that: \[\psi F_{1}^{\prime\prime}+\left(\frac{1}{W_{1}}-1\right)F_{1}^{ \prime}+\psi^{2}F_{3}^{\prime\prime}+  
Substituting (30) into (28), (29) and then into (22)-(24), results in the final solutions of the stress tensor:  
\[T_{11}= -x^{2-\frac{1}{W_{1}}}F^{\prime\prime}+C_{1}x^{2-\frac{1}{W_{1}}}+C_{2} y^{\frac{1}{W_{1}}-2}+\frac{k_{3}-2}{k_{1}-1}, \tag{31}\] \[T_{22}= -x^{-\frac{1}{W_{1}}}y^{2}F^{\prime\prime}+\frac{2}{W_{1}}x^{- \frac{1}{W_{1}}-1}yF^{\prime}-\frac{1}{W_{1}}\left(\frac{1}{W_{1}}+1\right)x^{ -\frac{1}{W_{1}}-2}F\] \[+C_{3}x^{-\frac{1}{W_{1}}-2}+C_{4}y^{\frac{1}{W_{1}}+2}-\frac{k_{ 3}+2}{k_{1}+1},\] (32) \[T_{12}= x^{1-\frac{1}{W_{1}}}yF^{\prime\prime}+\left(1-\frac{1}{W_{1}} \right)x^{-\frac{1}{W_{1}}}F^{\prime}. \tag{33}\]  
In a recent investigation by [17], the solution of a wall-free stagnation-point flow for the Maxwell fluid with \(W_{1}\
eq\frac{1}{2}\) satisfying the momentum equation was discovered. This corresponds to our solutions (31)-(33) with \(k_{1}=2W_{1}\) and \(k_{3}=0\). Physically, the components of the stress tensor should be limited everywhere, including at the stagnation point \((x,y)=(0,0)\) and at infinity. Hence, further restrictions on the arbitrary constants \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\) and the arbitrary function \(F(xy)\) in the solutions are needed. To avoid the singularity of the stress tensor, Ref. [17] suggested the choice:  
\[F(xy)=(xy)^{2+\frac{1}{W_{1}}+\delta}e^{-b(xy)^{2}},\quad C_{1}=C_{2}=C_{3}=C_ {4}=0, \tag{34}\]  
where \(\delta>0\) and \(b>0\). However, this choice (34) cannot prevent the singularity as was claimed. To demonstrate this, as an example, we substitute (34) into (33) resulting in:  
\[T_{12}=\left[a_{1}+a_{2}(xy)^{2}+a_{3}(xy)^{4}\right]e^{-b(xy)^{2}}\cdot(xy)^ {1+\delta}\cdot y^{\frac{1}{W_{1}}}=G(xy)\cdot y^{\frac{1}{W_{1}}}, \tag{35}\]  
where \(a_{1}=(2+\delta)(2+\delta+\frac{1}{W_{1}})\), \(a_{2}=-2b(6+2\delta+\frac{1}{W_{1}})\) and \(a_{3}=4b^{2}\) are constant coefficients. For \(W_{1}>0\), the particular solution (33) has the following properties:  
* Along the characteristic curves \(xy=0\) and \(xy\to\infty\), \(T_{12}\to 0\), no singularity occurs.