We characterise the stress singularity of the Oldroyd-B, Phan-Thien–Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces. By integrating along streamlines, we verify the polymer stress behavior of r−4/11 for PTT and r−5/16 for Giesekus, where r is the radial distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity....

中文翻译：

---

粘滑奇点附近牛顿速度场中 PTT、Giesekus 和 Oldroyd-B 流体的应力

我们在稳定的平面粘滑流中表征了 Oldroyd-B、Phan-Thien-Tanner (PTT) 和 Giesekus 粘弹性模型的应力奇异性。对于存在溶剂粘度的 PTT 和 Giesekus 模型，渐近线表明速度场在粘滞面和滑移面连接处的奇点附近以牛顿为主。聚合物应力边界层存在于粘着表面和滑动表面上。通过沿流线积分，我们验证了 PTT 的 r-4/11 和 Giesekus 的 r-5/16 的聚合物应力行为，其中 r 是离奇点的径向距离。PTT 和 Giesekus 的这些渐近结果在 Oldroyd-B 的消失二次应力项的极限内不成立。然而，我们可以在规定的牛顿速度场的固定运动学中考虑 Oldroyd-B 模型。