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 kappa, 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 = O (kappa(1/2(1-alpha))) as kappa -> 0, where pi/alpha is the re-entrant corner angle with alpha is an element of [1/2, 1) and r the radial distance. On distances far smaller than this we obtain the PIT K = 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.
- re-entrant corner
- stress singularity
- upper convected Maxwell
- elastic boundary layers
Evans, J. D., & Sibley, D. N. (2010). The UCM limit of the PIT equations at a re-entrant corner. Journal of Non-Newtonian Fluid Mechanics, 165(21-22), 1543-1549. https://doi.org/10.1016/j.jnnfm.2010.06.018