# 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 \(k_{2}\
eq 0\)  
For some of the viscoelastic models derived from the Oldroyd 8-constant model \(k_{2}\) is not zero, such as the [PERSON] 3-constant Oldroyd model, Oldroyd-4-constant model, etc., details see [15]. For this case, the solution of \(T_{12}\) remains the same as in (24). \(T_{11}\) and \(T_{22}\) need to be solved from the coupled PDE system consisting of (15) and (16) with non-constant coefficients. Applying the method of characteristics, the PDE system reduces to the following ODE system:  
\[W_{1}y\frac{dT_{11}}{dy} =(1-k_{1})T_{11}+k_{2}T_{22}+k_{3}-2, \tag{51}\] \[W_{1}y\frac{dT_{22}}{dy} =-k_{2}T_{11}+(1+k_{1})T_{22}+k_{3}+2, \tag{52}\]  
along the characteristic curves \(xy=const.\) Furthermore, using the transformation:  
\[y=e^{W_{1}y} \tag{53}\]  
Equations (51) and (52) can be transformed into an ODE system with constant coefficients:  
\[\frac{dT_{11}}{d\hat{y}} =(1-k_{1})T_{11}+k_{2}T_{22}+k_{3}-2, \tag{54}\] \[\frac{dT_{22}}{d\hat{y}} =-k_{2}T_{11}+(1+k_{1})T_{22}+k_{3}+2, \tag{55}\]  
or in matrix form:  
\[\frac{d\mathbf{t}}{d\hat{y}}=\mathbf{At}+\mathbf{b} \tag{56}\]  
with:  
\[\mathbf{t}= \begin{pmatrix}T_{11}\\ T_{22}\end{pmatrix} \text{,}\quad\mathbf{A}= \begin{pmatrix}1-k_{1}&k_{2}\\ -k_{2}&1+k_{1}\end{pmatrix} \text{,}\quad\mathbf{b}= \begin{pmatrix}k_{3}-2\\ k_{3}+2\end{pmatrix} \text{.} \tag{57}\]Eliminating \(T_{22}\) by combining (54) and (55), gives an second-order ODE with constant coefficients for \(T_{11}\):  
\[\frac{d^{2}T_{11}}{d\bar{y}^{2}}-2\frac{dT_{11}}{d\bar{y}}+(1-k_{1}^{2}+k_{2}^{2 })T_{11}=k_{2}(k_{3}-2)-(k_{1}+1)(k_{3}+2). \tag{58}\]  
The homogeneous solution of this ODE depends on the type of solutions of the corresponding characteristic equation:  
\[\lambda^{2}-2\lambda+(1-k_{1}^{2}+k_{2}^{2})=0. \tag{59}\]  
Its eigenvalues are:  
\[\lambda_{1,2}=1\pm\sqrt{k_{1}^{2}-k_{2}^{2}}. \tag{60}\]  
Depending on the value of \(k_{1}^{2}-k_{2}^{2}\), the homogeneous solution of (58) has the following three different cases:  
(i) If \(k_{1}^{2}-k_{2}^{2}>0\), \(\lambda_{1}\) and \(\lambda_{2}\) are real numbers with \(\lambda_{1}\
eq\lambda_{2}\), we obtain:  
\[T_{11,h}=C_{1}\cdot e^{\lambda_{1}\bar{y}}+C_{2}\cdot e^{\lambda_{2}\bar{y}}; \tag{61}\]  
(ii) If \(k_{1}^{2}-k_{2}^{2}=0\), i.e., \(\lambda_{1}=\lambda_{2}=1\), we obtain:  
\[T_{11,h}=(C_{1}+C_{2}\bar{y})\cdot e^{\bar{y}}; \tag{62}\]  
(iii) If \(k_{1}^{2}-k_{2}^{2}<0\), \(\lambda_{1/2}=1\pm i\omega\) with \(\omega=\sqrt{k_{2}^{2}-k_{1}^{2}}\) are conjugates complex numbers, the homogeneous solution of \(T_{11,h}\) is given by  
\[T_{11,h}=e^{\bar{y}}[C_{1}\sin{(\omega\bar{y})}+C_{2}\cos{(\omega\bar{y})}]. \tag{63}\]  
The particular solution of the ODE (58), depending on the value of the coefficient \(1-k_{1}^{2}+k_{2}^{2}\), includes two cases as follows:  
(i) If \(1-k_{1}^{2}+k_{2}^{2}\
eq 0\), we obtain:  
\[T_{11,p}=\frac{k_{2}(k_{3}+2)-(k_{1}+1)(k_{3}-2)}{1-k_{1}^{2}+k_{2}^{2}}; \tag{64}\]  
(ii) If \(1-k_{1}^{2}+k_{2}^{2}=0\), the particular solution takes the form:  
\[T_{11,p}=-\frac{1}{2}[k_{2}(k_{3}+2)-(k_{1}+1)(k_{3}-2)]\bar{y}. \tag{65}\]  
The general solution of the ODE (58) is expressed as  
\[T_{11}=T_{11,h}+T_{11,p}. \tag{66}\]  
Obviously, the second case of the particular solution, (65), can only be combined with the first case of the homogeneous solution, (61). The solution of \(T_{22}\) can be directly determined by inserting the solution for \(T_{11}\), (66), into (54):  
\[T_{22}=\frac{1}{k_{2}}\bigg{(}\frac{dT_{11}}{d\bar{y}}+(k_{1}-1)T_{11}-k_{3}+2 \bigg{)}. \tag{67}\]  
To obtain the final solutions of the original PDE system consisting of (15) and (16), we only need to replace the variable \(\bar{y}\) with \(y\) by the transformation \(\bar{y}=\ln{y}^{\frac{1}{\overline{w}_{1}}}\), and all arbitrary constants \(C_{i}\) with arbitrary functions \(F_{i}(xy)\). Finally, together with the solution of \(T_{12}\) in (24), we obtain the complete solutions of the PDE system (15)-(17) as follows. \[