This paper explores two-dimensional flows near a free critical point in an incompressible viscoelastic Maxwell medium, governed by a rheological constitutive law. While stagnation point flow problems have been widely studied, general exact analytical solutions for stresses in cylindrical coordinates - more practical and suitable for certain experiments—remain undiscovered. In this study, we derive the general solution for the Maxwell model with the Johnson-Segalman convected derivative in cylindrical coordinates for an arbitrary model parameter α. The analysis reveals the necessity of separately considering the upper-convected, lower-convected, and Jaumann derivatives when solving the stagnation point flow problem.

中文翻译：

---

Johnson-Segalman 导数中参数的所有值具有圆柱对称性的停滞点问题的 Maxwell 方程的一般解


本文探讨了不可压缩粘弹性 Maxwell 介质中自由临界点附近的二维流动，该流动受流变学本构定律支配。虽然停滞点流动问题已被广泛研究，但圆柱坐标中应力的一般精确解析解 - 更实用且适用于某些实验 - 仍未被发现。在本研究中，我们推导出了 Maxwell 模型的通解，其中 Johnson-Segalman 对流导数在圆柱坐标中为任意模型参数α。分析揭示了在解决停滞点流问题时，有必要分别考虑上对流、下对流和 Jaumann 导数。