# 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)\
* Along the characteristic curve \(xy=c_{0}\), where \(c_{0}\) is a non-zero finite constant, the value of \(G(xy)\) is a bounded constant \(G(c_{0})\), while \(y^{\frac{1}{W_{1}}}\) is singular at \(y\to\infty\). An infinite shear stress \(T_{12}\) arises in the region far away from the stagnation point \((y\to\infty)\) and near the \(y\)-axis \((x\to 0)\). With the choice (34), similar singularities also appear in the stress tensor components \(T_{11}\) and \(T_{22}\) far away from the stagnation point, for \(y\to\infty\) (but \(x\to 0\)) or \(x\to\infty\) (but \(y\to 0\)). As an example, the corresponding stress components \(T_{11}\) and \(T_{12}\) excluding the constant part \(\frac{k_{3}-2}{k_{1}-1}\) are presented in Figures 2 and 3 for the case \(W_{1}=0.25\), \(\delta=2\) and \(b=12\) with a much finer spatial resolution than [17] used. The present figures show that a singularity in the stress field may arise in the region far away from the stagnation point \((y\to\infty)\) and near the \(y\) axis \((x\to 0)\), as previously analytically recognized, while in [17], this tendency was invisible due to the rather coarse resolution employed. Actually, the appearance of the singularity at \((x,y)\rightarrow(0,\infty)\) or \((x,y)\rightarrow(\infty,0)\) is independent from the choice of the arbitrary function \(F(xy)\). This can be easily recognized by observing the distribution of the stresses (31)-(33) along an arbitrary characteristic curve \(xy=c_{0}\), where \(c_{0}\) is non-zero finite constant. As an example, the term \(x^{-\frac{1}{W_{1}}}F^{\prime}(xy)\) tends to be infinite at \((x,y)\rightarrow(0,\infty)\) for \(W_{1}>0\) or at \((x,y)\rightarrow(\infty,0)\) for \(W_{1}<0\). These singularities are independent from the choice of \(F(xy)\) and thus non-removable. However, possible singularity near the stagnation point \((x,y)\rightarrow(0,0)\) can be effectively prevented by choosing a reasonable function \(F(xy)\), e.g., (34). This ensures that no singularity occurs in the stress field near the stagnation point. For a stagnation-point flow with the velocity field given by (13), the velocity becomes unbounded at \(x\rightarrow\infty\) or \(y\rightarrow\infty\) and is thus physically no longer meaningful. The singularity arising in the far field at \(x\rightarrow\infty\) or \(y\rightarrow\infty\) may not be relevant when a bounded stagnation-point flow is investigated. We should then focus on the analysis of singularity in a bounded area near the stagnation point. 2. If \(\frac{k_{1}}{W_{1}}=-2\), performing the similar steps as for the above case gives the solutions: \[T_{11}= x^{-\frac{1}{W_{1}}}y^{2}F^{\prime\prime}-\frac{2}{W_{1}}x^{-\frac{1} {W_{1}}-1}yF^{\prime}+\frac{1}{W_{1}}\left(\frac{1}{W_{1}}+1\right)x^{-\frac{ 1}{W_{1}}-2}F\] \[+C_{1}x^{-\frac{1}{W_{1}}-2}+C_{2}y^{\frac{1}{W_{1}}+2}+\frac{k_{ 3}-2}{k_{1}-1},\] (36) \[T_{22}= x^{2-\frac{1}{W_{1}}}F^{\prime\prime}+C_{3}x^{2-\frac{1}{W_{1}}}+C _{4}y^{\frac{1}{W_{1}}-2}-\frac{k_{3}+2}{k_{1}+1},\] (37) \[T_{12}= x^{1-\frac{1}{W_{1}}}yF^{\prime\prime}+\left(1-\frac{1}{W_{1}} \right)x^{-\frac{1}{W_{1}}}F^{\prime}.\] (38) Similarly, by reasonably choosing of \(C_{i}\) (\(i=1,2,3,4\)) and \(F(xy)\), singularity near the stagnation point can be avoided. \[T_{22}= -x^{-\frac{1}{W_{1}}}y^{2}F^{\prime\prime}+\frac{2}{W_{1}}x^{- \frac{1}{W_{1}}-1}yF^{\prime}-\frac{1}{W_{1}}\left(\frac{1}{W_{1}}+1\right)x^{ -\frac{1}{W_{1}}-2}F\]0 Stress component \(T_{12}\) with a spatial resolution of \(x\in[0:0.001:0.2]\), \(y\in[0:0.001:10]\). 3. If \(\frac{k_{1}}{W_{1}}\
eq\pm 2\), we obtain four equations for \(F_{1}\), \(F_{2}\) and \(F_{3}\) from (25).