# Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity
## 1 Introduction  
The working fluids encountered in practical applications and industry are often non-Newtonian, and research on this type of fluids has been conducted for decades. In theoretical research, a variety of non-Newtonian fluid models has developed [1]. One group of these models is of the rate type which involves differential transport equations for the stress tensor. As these models are highly complex and mostly non-linear, exact analytical solutions can only be obtained for very special flow cases. Many theoretical works limit themselves to investigating the distribution of stress tensor in simple canonical viscoelastic flows by means of relatively simple models, such as the Oldroyd-B and Maxwell-B model, see, e.g., [2, 3, 4]. A classical model problem in this context, and also to be considered presently, is the similarity solution for the velocity field in a stagnation-point flow described, e.g., in [5]. Therein, the velocity distribution is usually assumed in the form \((u,v)=(xf^{\prime}(y),-f(y))\), though for a wall-free stagnation-point flow or a creeping stagnation-point flow far away from the wall, the velocity profile reduces to \(f(y)=ay\), i.e., \((u,v)=(ax,-ay)\), where \(a\) is constant rate of the strain. Under this assumption, [PERSON] [6, 7] obtained exact solutions to the plane and axisymmetric stagnation-point flows for both Maxwellian and Oldroyd-B fluids, respectively, where the governing equations were reduced to a system of ordinary differential equations. It was shown that in a stagnation-point flow with a certain Weissenberg number, there is a singularity in the stress field. [PERSON] [8] analyzed a steady creeping flow of the upper convected Maxwell (UCM)fluid, in which the shear stress was assumed to be zero, and the normal stress depended only on the transverse coordinate of the outflow. His solution demonstrated that a singularity exists not only at the stagnation point, but also along the entire streamline passing the stagnation point downstream. [PERSON] et al. [9] used the same ansatz for the velocity field and solved the stress field of an Oldroyd-B fluid by using the method of characteristics. Their results showed that the behavior of the solutions is very sensitive to the Weissenberg number. However, the exact analytical solutions were only constructed for the model equations of the stress field without considering the compatibility with the momentum conservation equation. [PERSON] et al. [10] obtained a general analytical solution for a steady planar extensional wall-free stagnation-point flow of a viscoelastic fluid described by the UCM model. This solution depends on both space coordinates, and represents an extension of the previous solutions considering only the dependence of the transverse coordinate. Recently, [PERSON] et al. [11] extended the analysis of the stress distribution in a wall-free stagnation-point flow from the UCM model to the Johnson-Segalman model and also took the momentum conservation equation into account. Their solution demonstrates that the Johnson-Segalman model has a non-removable logarithmic singularity. In a near-wall stagnation-point flow, the velocity must satisfy the no-slip condition at the wall, so the velocity profile for the wall-free stagnation-point flow, i.e., \((u,v)=(xf^{\prime}(y),-f(y))\) with \(f(y)=ay\), is here no longer suitable. In this case, the function \(f(y)\) could take the form of \(f(y)=ay^{n}\) with \(n>1\) for an impermeable wall located at \(y=0\). Under the simplest assumption \(f(y)=ay^{2}\), [PERSON] et al. [12] and [PERSON] et al. [13], respectively, considered an UCM fluid and presented the exact solutions to the coupled PDEs of the viscoelastic stress. However, they did not analyze the compatibility of their solutions with the momentum conservation equation. [PERSON] [14] analyzed the same problem and showed that the solution for the stress components fails to satisfy the momentum conservation equation except in the linear case \(f(y)=-ay\), corresponding to a wall-free stagnation-point flow.