A mathematician’s perspective on the Oldroyd B model: Progress and future challenges

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.