We review progress and open challenges for a number of problems related to the Oldroyd B model which are of current interest to the mathematical community. We discuss existence results for initial value problems and steady flows, flow stability, and the problem of understanding the asymptotics of the high Weissenberg number limit. We also discuss efforts and challenges in numerical simulation, exemplified by three problems: extensional flow in a four roll mill geometry, flow past a confined cylinder, and locomotion of microorganisms in a viscoelastic fluid.

中文翻译：

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数学家对Oldroyd B模型的看法：进展和未来挑战

我们回顾了与Oldroyd B模型相关的许多问题的进展和面临的挑战，这些问题目前在数学界引起了人们的兴趣。我们讨论了初值问题和稳定流量，流量稳定性以及了解高Weissenberg数极限的渐近性的问题的存在性结果。我们还讨论了数值模拟中的工作和挑战，以三个问题为例：四辊轧机几何结构中的拉伸流，通过密闭圆柱体的流以及粘弹性流体中微生物的运动。