High Weissenberg number boundary layer structures for UCM fluids

We describe three distinct stress boundary layer structures that can arise in the high Weissenberg number limit for the upper convected Maxwell (UCM) model. One is a single layer structure previously noted by M. Renardy, High Weissenberg number boundary layers for the upper convected Maxwell fluid J. Non-Newtonian Fluid Mech. 68 (1997), 125–132. The other two are double layer structures. These latter two structures extend the core flows that can be accommodated by the UCM model in the high Weissenberg regime. The three structures taken together, represent the main dominant balances that occur for the UCM equations near solid boundaries. For each structure, the leading order equations are derived in each region together with particular exact solutions when available. Importantly, the matching conditions between respective regions for each structure are also derived and explained. These stress boundary layers can arise in order one Reynolds number flows and are independent of the velocity boundary layers that can arise in high Reynolds number flows.