The UCM limit of the PTT equations at a re-entrant corner

We consider the Upper Convected Maxwell (UCM) limit of the Phan-Thien–Tanner (PTT) equations for steady planar flow around re-entrant corners. The PTT equations give the UCM equations in the limit of vanishing model parameter κ  , this dimensionless parameter being associated with the quadratic stress terms in the PTT model. We show that the critical length scale local to the corner is r=Oκ12(1−α) as κ → 0, where π/α is the re-entrant corner angle with α ∈ [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PTT κ = 1 problem, whilst on distances greater (but still small) we obtain the UCM problem κ = 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 κ → 0, r → 0 thus yields a nine region local asymptotic structure.