Re-entrant corner flows of oldroyd-B fluids in the natural stress basis

We consider the steady planar flow of Oldroyd-B type fluids for re-entrant corners of angle π/α where 1/2≤α<1. Additional mathematical results to Rallison and Hinch [9] are presented, where we place their physical arguments more systematically within the method of matched asymptotic expansions. The asymptotic regions are known to consist of an outer region in which the upper convected stress derivative dominates, together with inner (boundary) layers at the walls. An artificial small parameter is introduced (based on the assumption that we are considering small length scales) and used to systematically compare the sizes of terms and facilitate the matching procedure between the regions. Additional results relate to the balance for the natural stress variable v in the outer region (so that it is no longer a constant along streamlines), as well as the asymptotic behaviors of the boundary layer equations at the wall and in the far-field towards the outer region. Further results on the parameter dependence of the solution are given in terms of similarity parameters.