Re-entrant corner behaviour of the PTT fluid with a solvent viscosity

Abstract

The local asymptotic behaviour is given for planar re-entrant corner flows of Phan-Thien–Tanner fluids with a solvent viscosity. The solvent stress and Newtonian velocity field dominate in all regions, with the polymer stress being uniformly subdominant. At small radial distances r to the corner, the velocity field vanishes as O ( r λ 0 ) whilst the solvent stress behaviour is O ( r − ( 1 − λ 0 ) ) . The polymer stress has the less singular behaviour O ( r − 4 ( 1 − λ 0 ) / ( 5 + λ 0 ) ) , where λ 0 ∈ [ 1 / 2 , 1 ) is the Newtonian flow-field eigenvalue. Stress boundary layers are needed at the walls for the polymer stress solution, which are of thickness O ( r ( 4 − λ 0 ) / 3 ) . These results confirm the order of magnitude estimates previously obtained by Renardy [14], the alternative derivation given here using the method of matched asymptotic expansions. Further, we complete previous analysis by providing solutions (particularly for the polymer stresses) in the asymptotic regions that arise. These results breakdown in the limit of vanishing solvent viscosity as well as the Oldroyd-B model limit.