The Asymptotic Behaviour at a Re-Entrant Corner for a PTT Fluid in the Limit of Small kappa

We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien-Tanner (PTT) equations for steady planar flow around re-entrant comers. The PTT equations give the UCM equations in the limit of vanishing model parameter kappa, this dimensionless parameter being associated with the quadratic stress terms in the PTT model. The critical length scale local to the comer is r = O(kappa (1/2(1-alpha))) as kappa -> 0, where pi/alpha a is the re-entrant comer angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PTT kappa = 1 problem, whilst on distances greater (but still small) we obtain the UCM problem kappa = 0. This critical length scale is that on which intermediate behaviour of the PTT model is obtained where both linear and quadratic stress terms are present in the wall boundary layer equations. The double limit kappa -> 0, r -> 0 thus yields a nine region local asymptotic structure.